My Perspective on Constructs

[Keith]

What did you think of this idea of constructs? In what ways does it
help (or not help) you think about the issues related to learning and
teaching mathematics with technology that we have discussed in this
class?

[CJ]

I would like to say "Bravo" to the lovely people who took the time to
review the scattered and random research concerning technology in
mathematics education and not only synthesize it into main themes/
constructs, but also take some time to better define those constructs
and make a case for why they may be useful. If researchers would take
time to read this article and then try to use the names of the
constructs to describe their future research, communication would
probably improve and research might make quicker progress toward the
development of guiding principles for teaching with technology that
teachers can use to think about their teaching, curriculum, and
learning. I especially enjoyed the comment on page 1201, "Realizing
the plethora of constructs relating to inappropriate uses of
technology, for example, researchers may be motivated to study and
develop characterizations of judicious use of technology." Research
that reports misuse seems to paralyze me as a teacher. Research that
reports and describes judicious uses of technology empowers me with
ideas and options.
As I read this report, I kept wondering if it would have been helpful
if we had read it at the beginning of this course. That way, we could
have spent the whole semester looking for examples of these
constructs. However, I think that it also would have been very
intimidating trying to read it back then. Half of the constructs would
have been meaningless because I wouldn't have been able to relate to
them at the time. Now, after reading articles, attending class, and
using technology to learn mathematics on my own, I feel that each new
construct that is introduced in the paper is a label for something
that I have thought about lately, or closely related to ideas that
have been important in this course. Here's an example:
The constructs of externalized representations, mathematical fidelity,
and cognitive fidelity take me back to my virtual manipulatives lesson
plan that I used with my 306 students. As I wrote the lesson plan, I
kind of followed this intuitive feeling that seeing good pictures
(visualizing?) of fraction addition would help my students to develop
conceptual understanding of the idea of equivalent fractions (re-
naming before you add) and convince them that good pictures are worth
taking the time to draw. However, as we discussed my lesson plan in
608, it was mentioned that the manipulative was set up for a limited
number of solution processes, and therefore didn't promote a lot of
agency or discovery or even algorithm development because the
algorithm was basically built in. Looking back, I would now say that a
strength of the manipulative was mathematical fidelity; the applet
made nice, accurate pictures that demonstrated the relationships and
differences between fifteenths and sixteenths, fifteenths and thirds
or any other denominator pair (up to fortieths). The conventions of
fraction addition were built in as the students were guided to name
their sum in terms of one whole (i.e., two fifths plus one fifth is
three fifths, it's not three tenths, even though you may have drawn
ten equal sized pieces in your representation.) However, the cognitive
fidelity of the manipulative was quite low, because the pre-determined
method for renaming fractions (increasing the denominator one unit at
a time until the two fractions had common, compatible denominator) was
not reflective of my student's personal methods and thought processes.
I am more convinced of this as I reflect back on the semester and
realize that nobody complained about drawing pictures after we used
the virtual manipulatives, and pictures for equivalent fractions were
much more appropriate than they were before this point. At the same
time, I can't remember a time where anybody chose to use the
computers' method for finding a common denominator on written tasks.
So the mathematical fidelity of the technology made a lasting
impression on my students while the cognitive infidelity(?) didn't
seem to stick with them.
Although these ideas have kind of been floating around in my head,
this report has given me the language to better conceptualize and
describe them. Similar things have happened with the other constructs,
but I will let others have the opportunity to expound on them first.
If someone wants to give an example that highlights the differences
between exploratory and expressive activity that would be great,
because those two seem to keep switching places in my mind, possibly
because of my current ideas on the meaning of "exploration."

[Janellie]

I have mixed feelings on the idea of constructs. It does seem
wonderful that there is a synthesis of a lot of the research on
technology in mathematics education. I like constructs because they
can help researchers and students be able to talk about issues
relevant to the current research in technology. They can use the
constructs to categorize the types of research on technology that
exists. I like how they had the constructs under various headings to
organize them. Something that I didn't like about constructs is that
there seems to be too many of them. I'm sure the authors did not
capture all of the constructs that exists in all of technology
research. I am wondering if it is possible to do this.

A quick question, what would be some other ways of saying
"construct"? In the definition the authors said that their
"constructs are concepts that have explanatory power regarding
technology and the teaching and learning of mathematics" (p. 1172).
So, can theories be constructs?

One of the constructs that I liked was "mathematical fidelity." In
this construct, technology generated representations must be "faithful
to the underlying mathematical properties of that object" (p. 1174).
This makes me think of Geometer's sketchpad and some of the pitfalls
we talked about. Because of this construct of mathematical fidelity,
it is not wise to let students construct objects in Sketchpad before
they know of the properties. As teachers, we also have to be careful
when we construct representations with technology.

Another construct that I thought helped me understand more about
learning mathematics with technology was that of "instrumental
genesis." One cannot just know of the technology, they have to be
able to develop ways to use the technology effectively. One of my
favorite quotes in the whole paper is: "technology does not have the
same automatic power for all users" (p. 1179). We can't just expect
that we can put technology in front of a student and expect that they
would use it in a useful and powerful manner.

I briefly mention one last construct: Match between technology and
practice. A teacher's practice must align with the nature of the
technology. We cannot just use technology and hope it will magical
transform our students' mathematical capabilities without changing our
teaching practices to match the technology.

On Apr 2, 4:15 pm, Keith <LethaLeat...@gmail.com> wrote:

[erin...@byu.edu]

Of course I can't help but notice that Chris wrote first and is
thus far the only one to respond. Did anybody else have a difficult
time reading this article? I felt like I was climbing through mud...
I agree with Chris' "bravo" to the people who took the time to compile
this. I would also like to give each and every one of you a huge
"bravo" for reading the resulting article as I know it was a lot to
wade through. Before I get into Keith's instructed response, I wanted
to say just a few things on my own. First, I really enjoyed the quote
on page 1169 where Kaput (1992) predicted that the "major limitations
of computer use in the coming decades are likely to be... a result of
limited human imagination and the constraints of old habits and social
structures." I wished the authors had discussed this further - will
human imagination become limited because of computer use? Have we
become creatively dependent on computers and thus have limited our own
future potential? Or (and I think this was their intent), once the
computer and its programs can actually do what we originally imagined/
intended for them to do, are we going to be out of ideas? Is this
even possible? I bet that at the beginning of time, people never even
dreamed of cars or microwaves as we know them today; surely innovation
will continue to inspire/spawn new ideas. Human imagination will
continue unlimited and constraints of old habits/social structures
will give way to new habits and social structures conducive of
expansions of computer use. Second, I felt that the description of
technical versus conceptual mathematical activity had many parallels
with Skemp's ideas of relational versus procedural understanding.
Anyone else?
What do I think of this idea of constructs? From my
understanding, a construct is a concept, model, schematic idea OR
concrete image or idea, that has explanatory power regarding
technology and the teaching and learning of mathematics. So
basically, there's a lot of things that qualify as constructs. As
such, I had a difficult time because I felt like a construct should be
more of a teaching scheme or a tangible object than a concept or
idea. Regardless, I appreciated the authors' work to combine previous
research in such a way.
In particular, I appreciated how under externalized representation,
the authors state that "meaning may go unnoticed or be misinterpreted
by the student... The student must work to extract feedback from
interactions with a physical tool." Too often we (as teachers and
students) expect significance/meaning to be apparent through use of
cognitive tools. The construct of mathematical fidelity is obviously
related and something that I think we've touched on a lot during the
semester. It is definitely necessary to be aware of these constructs
in the classroom or teachers and students will certainly misuse
technology and walk away believing incorrect things (like when we
tried to estimate big exponents on our calculators or in excel) or
that if no meaning appears it is not there (which reminds me of the
article on what students learn besides the mathematics).
Two others that I particularly felt drawn to were pedagogical
fidelity and role of the teacher. If a teacher believes that
cognitive tools do not allow students to act mathematically or that
they do not do so "in ways that correspond to the nature of
mathematical learning that underlies a teacher's practice" (p 1187),
then that teacher will certainly not implement technology in the
classroom. The teacher's role(s) and experience with technology
provides the teacher's framework and belief system regarding
technology use, and so it is vital to consider this construct when
deciding to/not to use technology in the mathematics classroom.
I don't really have questions to ask, but am eager to respond to
everybody else's take on these constructs. Hopefully you choose to
discuss constructs that i didn't discuss and then I can respond to
your comments!

On Apr 2, 4:15 pm, Keith <LethaLeat...@gmail.com> wrote:

[Tenille]

I thought that this chapter was a good capstone reading. Kudos to
amazing thinkers who are able to not only synthesize the vast amount
of research but also articulate constructs. I think that the only
thing more difficult than reading a chapter from the Handbook would be
writing a chapter for the Handbook.  Although I was somewhat bogged
down by the number of different constructs, overall I found the idea
of constructs to be helpful. I like how the authors put it: “In
mathematics education, identifying a construct brings the phenomenon
or particular distinctions related to the phenomenon into awareness.
Constructs give researchers a finer grained descriptive language for
variables in the teaching and learning process. . .” (p. 1201).
Keeping in mind the first few articles we read in class, I hope that
the development of these constructs will increase the quality of
technology research in the field. I think that if researchers are
aware of issues and have a way to talk about the issues then they will
be able to design studies that really get at the issues.

For me, reading about the different constructs increased my awareness
of certain issues. For example, the discussion on mathematical
fidelity reminded me of Paige, Seshaiyer and Toda (2007). From this
article, the limitations of the technology was the only way that I
conceived of the emergence of a lack of mathematical fidelity.
However, Zbiek et al.’s (2007) discussion of this issue has made me
more aware of mathematical and cognitive fidelity and the implication
of these constructs in my classroom.

The constructs also provided me with a way to talk about the different
issues that I have been thinking about this semester as I have
observed student teachers. Lately, I have been particularly concerned
with the construct of representational fluency (although I previously
did not have a word to describe my concern). In a class I observed,
the students were solving systems of equations by graphing without the
use of technology. The students were unable to focus on the
mathematics of systems of linear equations because they couldn’t
transfer from one representation of a linear equation to another.
They were not fluent in their ability to transfer from on
representation to another. But I have only identified the problem
here; I do not have a solution for increasing the representational
fluency of the students. Do you think that by incorporating
technology into the classroom would have increased the students’
representational fluency? And is speed a consideration in the
construct of representational fluency?

On Apr 2, 4:15 pm, Keith <LethaLeat...@gmail.com> wrote:

[Rachelle]

I thought this paper was incredibly thick and hard to read. I'm not
exactly sure if I understood much, but I'll try my best. I'll admit
that at times I felt like the authors' only goals were to provide a
lot of vocabulary. But after a closer look, I've decided that the
words are important because they are being used to identify critical
aspects of the field of mathematics education and technology.

First, I like the idea of identifying the critical aspects of teaching
and learning mathematics with technology. That is an important
contribution to the field and could help to synthesize previous
studies, and would certainly aid in the design and analysis of future
studies. If the field can be focused on the same critical aspects (as
long as there are still active thinkers on the lookout for other
possible important aspects), then things could be made much more
efficient and useful for developing an overall understanding of how
learners and teachers deal with mathematics in technology
environments.

Next, although I really like the idea of identifying critical aspects,
I'm not sure I understand what a construct is. Is it just a word for
something that I already understand? Or does it mean something more?
The authors write that "our constructs are concepts that have

explanatory power regarding technology and the teaching and learning

of mathematics" (p. 1172). I'm not sure I understand exactly what that
means. Could anyone explain further? I may understand but just be lost
in the language, but if constructs are anything more than the critical
aspects of what/how students and teachers deal with mathematics and
technology, then I think I missed it.

Last, there are some specific constructs mentioned in the article that
helped me synthesize much of what we talked about throughout the
semester. Mathematical fidelity is a critical issue that we touched
upon, but I feel this article really brought the idea to light and
helped me understand how important the alignment is between technology
and the underlying mathematics. Mathematical concordance hit me as an
important construct, especially between the mathematics the student
engages in and the mathematics the teacher intends him to engage in.
It reminded me of our discussions of the enacted curriculum versus the
intended curriculum. Concordance of all mathematics is important for
an effective classroom. The constructs of amplifiers and reorganizers
gave a wonderful distinction to the ways in which new technology may
require change in the classroom. It extended and improved upon de
Villiers' "no change" pitfall.

I guess the idea of constructs helped me think critically about many
of the issues we've discussed this semester, but I still don't know
exactly what a construct is. Can anyone help to illuminate the meaning
of the word?

On Apr 2, 4:15 pm, Keith <LethaLeat...@gmail.com> wrote:

[Rachelle]

Janelle,
It looks like you and I have a similar question! I think my feelings
are mixed, too, but mostly because I'm unsure what constructs are.

Thanks for mentioning the match between technology and practice
construct. I sometimes with technology could magically transform
students' capabilities, but I think you and the authors are right on
to say that it just doesn't work that way. I think that's what makes
introducing technology in my classroom so scary.. . I don't know how I
need to change my practice. But I think the amplifiers and
reorganizers constructs at least give a starting place to think
critically about the changes that the technology should bring.

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[Tenille]

Janelle, Rachelle, and everyone,

I agree; this paper was very difficult to read. I'm still trying to
make sense of everything that we read. I also had a hard time trying
to decide what a construct is and how the authors were using it. I
didn't find the definitions of constructs given in the article helpful
in the least. After wrestling with the concept for a time, I think
that I understand it better. The test will be trying to articulate it
here. First, I found considering the verb "to construct", meaning to
build or frame, helpful. In essense, the authors were trying to build
a theory that will frame the technology research so far. I also
looked up "construct" on dictionary.com. Here are a couple of
definitions I found helpful: "an image, idea, or theory, esp. a
complex one formed from a number of simpler elements" and "a concept,
model, or schematic idea". So in a broad sense, these constructs are
ideas formed from collections of smaller ideas (research). In this
article, the authors created different concepts that unified the
research on teaching and learning mathematics with technology. I
think that they provide a lens for looking at research. I found these
definitions very helpful and I wanted to share them with all of you.
I hope you find them helpful too. Unfortunately, eventhough I have a
broad understanding of constructs now, I am still trying to understand
the individual constructs presented in the article. All of your
responses have helped me understand the constructs themselves better.
Thank you to everyone.

[Rachelle]

Thanks for sharing, Tenille! So very helpful. I like using both the
definition of "construct" and "to construct" to help get at the
meaning.

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[Janellie]

I also thought that this was a paper that I would have to spend weeks
on to really understand it. It was good though to get a feel on some
of the issues in research on technology in mathematics education.
Tenille, thanks for helping us better understand the meaning of
"construct". Dictionaries are wonderful, and I should have thought to
look there myself.

Also I would like attempt to address Tenille's question about
representational fluency and technology. Just having technology will
not necessarily enhance students' representational fluency.
Especially if the teacher dictates exactly how to use the technology,
students are not gaining the ability to move among representations
with ease. However, I do think that having technology available
increases the amount of representations that students have in their
repertoire of tools. Also the representations allowed by technology
are sometimes dynamic, which I would think would increase
representational fluency. You also asked about whether speed is a
consideration in fluency. I don't think I have an answer for that
one. I would think that speed is sort of important just for the sake
of being able to continue learning and building upon current
knowledge.

P.S.--I hate technology when it shuts down to "update" itself right
when you are trying to respond to a blog...

On Apr 10, 9:13 am, Tenille <tenille.can...@gmail.com> wrote:

[erin...@byu.edu]

To my most wonderful math ed graduate student friends,
I just wanted a cool salutation to kick it off. Then I wanted to
thank chris for her really great comment on the difference between
research that reports the misuse of technology (and thus has a
paralyzing effect) versus research that reports/describes judicious
uses of technology (providing an empowering effect). I feel like
there are so many things in life that either provide paralyzing or
empowering effects and at this moment in my life felt like you said it
exactly how it was resounding in my soul when I couldn't find the
proper words. Janelle - I definitely felt like the authors would say
that yes, theories can be constructs. I think everybody else would
probably agree, especially after Tenille's clarification. Tenille
also mentioned (although since I wrote it down on paper I can't find
where it was in your response) that researches need an awareness and a
way to talk about these constructs; isn't that the necessity of any
kind of communication? I think it's pretty cool how we can generalize
things to the broader perspective and really appreciate the things all
of you say that resonate with my own beliefs/theories...

On Apr 10, 9:13 am, Tenille <tenille.can...@gmail.com> wrote:

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[CJ]

Hi Erin,
Yes, I did think of Skemp's relational and instrumental understanding
when I read about technical and conceptual activity. It's kind of
funny because I think a lot of people view mathematics and
mathematical activity and mathematical understanding in some variation
of these two categories and it is interesting how many different pairs
of words have been applied to this distinction that to me is basically
the same. Is this making any sense? Basically, what I am trying to say
is that it seems like Skemp identified these constructs long ago, and
we still have no consensus on how to talk about and call these things.
I had hope that this article on technology/mathematics/teaching
constructs would help to standardize language for these important
ideas, but if Skemp is any example, that is not necessarily going to
happen. Sometimes I wonder why we can't just all use the same words to
describe related things. To partially answer my wondering, I think
researchers make up new words to focus on subtle differences in their
way of seeing the world, or they don't want to be accused of mis-using
the words of another person. I have actually wondered about this in
the past . . . are we doomed to just keep thinking of new ways to
describe closely related ideas and never come to language consensus? I
think that we often talk about ideas that psychologists and educators
and linguists and people in other fields think a lot about, but we
don't collaborate because we don't realize that we are talking about
related ideas. Language is crazy. It's a miracle that communication
works at all.