Some responses to Shandy's and Tim's recent posts:
Shandy,
You write that "included in 'pedagogical content knowledge,' as we
are conceiving it for this project, is substantive subject matter or
content knowledge about topics, procedures, and concepts along with a
comprehension of the relationships among them."
Why are choosing to bundle substantive knowledge of a subject into
pedagogical content knowledge (PCK)?
Shulman categories both substantive and syntactic knowledge as
subject matter content knowledge (SMCK). I recognize that other
researchers categorize the subject-specific knowledge used in
teaching somewhat differently than Shulman (e.g., Ball, Lubienski, &
Mewborn (2001) refer to knowledge of mathematics, knowledge about
mathematics, and PCK; Kennedy (1990) refers to knowledge of the
subject's content, of the organization of that content, and of the
methods of inquiry used in the subject; Grossman, Wilson, & Shulman
(1989) refer to content knowledge, substantive knowledge, syntactic
knowledge, and beliefs about the subject matter), so one isn't
required to use Shulman's 1987 categorization scheme. I also
recognize that categorization schemes can be problematic in the sense
that as ideas get elaborated, the boundaries between categories that
at first seemed pretty distinct can start to become fuzzy, sometimes
even leading people to abandon the initial categorization scheme in
favor of a new one. I think this boundary issue is at play a bit now
in the area that's referred to as "mathematical knowledge for
teaching" (Ferrini-Mundy and colleagues) and "pedagogically useful
mathematical knowledge" (Ball and Bass) -- the mathematical knowledge
that teachers need and that other users of mathematics, although they
might have it, typically don't need (e.g., moving beyond the ability
to work fluently across different uses of equals signs and literals
to having much more detailed and explicit knoweldge of these
different uses/interpretations, as they create difficulty for
students and some are central to understanding different curricular
approaches to school algebra). (In my earlier email, I also included
Usiskin's term, "teachers' mathematics." I'm leaving that out of
my
list here because my understanding of it is that it's a larger
category than what Ferrini-Mundy and colleagues and Ball and Bass are
talking about.) Those developing this construct have chosen to treat
this kind of knowledge as a subset of SMCK, and although I understand
that decision, I also think an argument could be made for saying it's
a subset of PCK instead, since the specific mathematical
understandings involved are so closely linked with elements of PCK
(e.g., with understanding "what makes the learning of specific topics
easy or difficult," Shulman, 1986, p. 9). Consistent with this fuzzy
boundary, you see some people referring to Ma's profound
understanding of fundamental mathematics construct as SMCK and some
people referring to it as PCK (and my own interpretation is that her
"knowledge packages" incorporate both kinds of content knowledge, as
well as elements of curricular content knowledge). Still, I don't
understand why it's useful to merge PCK with SMCK.
As an aside, it's also not clear to me that PCK develops out of
general pedagogical knowledge and subject matter knowledge. The
relationship among the development of these different kinds of
knowledge is an interesting question.
Tim,
Preservice K-12 teachers arrive in their teacher education courses w/
a large set of beliefs about what constitutes good teaching, what
they need to learn in order to teach well, ..., so by itself, their
desire to become teachers doesn't guarantee that they want to learn
or see the value of learning all that teacher educators think is
important.
Also, while I think that it's valuable for all teachers to reflect on
their teaching with others and is perhaps especially important for
novice teachers, I think other kinds of experiences can also be
valuable in helping convince novice teachers of the need to better
understand their students as learners of the subject. One assignment
we've used with the preservice secondary math teachers at Michigan
State is a "listening project" -- something of a clinical interview
of a single student, trying to get at some of his/her understanding
of a mathematical topic (in our case, an understanding of fractions).
Our preservice teachers are often anxious to start "teaching," and it
can be challenging for them to work both on asking probing questions
and listening not for a correct/incorrect answer but in order to
learn how the student is actually reasoning, as they often don't yet
think of these things as part of teaching. But, I think that becoming
curious about one's students as learners is important in developing
PCK.
Dara
At 2:48 AM -0700 2/9/05, S. Hauk wrote:
Hi all,
I posted to the list a note on a grant project I am developing
which would produce a collection of video and textual case materials
for the development of college mathematics instructors (Hauk, 2005).
In that message, as Dara Sandow has noted, I mentioned a difference
between the focus of IMAP materials for prospective K-12 teachers
and the materials the proposal would develop: pre-service school
teachers, particularly for grades K-8, struggle with mastering the
mathematical content they will teach, graduate teaching assistants
(GTAs) tend not to do so. Prospective school teachers take several
classes about the nature of thinking and learning, GTAs tend not to
do so. What both prospective teacher populations have in common is
the struggle to develop robust pedagogical content knowledge from an
uneven base of content and pedagogical understandings. Here's where
I get to the short version of what "pedagogical content knowledge"
means in the proposed work, at the moment....
Researchers have been focusing increasing attention on
"knowledge for teaching," a constellation of ideas sometimes
associated with the phrase "pedagogical content knowledge" (Shulman,
1986, 1987). Included in "pedagogical content knowledge," as we are
conceiving it for this project, is substantive subject matter or
content knowledge about topics, procedures, and concepts along with
a comprehension of the relationships among them. In its most robust
form, this part of mathematics pedagogical content knowledge is what
Ma (1999) called "profound understanding of fundamental
mathematics." Additionally, pedagogical content knowledge
incorporates syntactic knowledge of the culturally embedded nature
and forms of discourse in a subject (both in and out of educational
settings) as well as anticipatory knowledge, an awareness of the
diverse ways in which learners may engage with content, processes,
and concepts. Finally, the ability to adapt teaching according to
content and socio-cultural context and use content, syntactic, and
anticipatory understandings in the classroom can be called knowledge
for action (Ball & Bass, 2000).
A teacher with well-developed pedagogical content knowledge has
the ability to foster deep understanding among students while also
sidestepping misunderstanding. For example, a third grade teacher
helping students learn about fractions can engage students with the
ideas of quarter of an hour and quarter of a dollar in such a way
that learners are unlikely to confuse the 15 minute-units of a
quarter-hour with the 25 cent-units of a quarter-dollar. Similarly,
a college teacher with rich pedagogical content knowledge is aware
that though her students are adults, they may not share her
mathematically enculturated views and may never have experienced
mathematics as interesting or clear. Moreover, this awareness may be
enacted in many ways in her teaching of college mathematics: in
proscriptions about learning established (Davis & Simmt, 2003), in
the classroom milieu fostered (Yackel, Rasmussen, & King, 2000), or
in the ways questioning is used (Hufferd-Ackles, Fuson, & Sherin,
2004) (just to name a few).
One step in examining the growth of pedagogical content
knowledge among college mathematics teachers is to understand the
perceptions and conceptions that a graduate student learner who is
also an in-service college teacher might construct in developing
content, syntactic, anticipatory, and action knowledge for
mathematics teaching. A multi-pronged approach to such investigation
includes (at least) examination of individual teacher views of
content, syntactic, anticipatory, and action knowledge; awareness of
the challenges and potential changes to those views engendered by a
professional development program; subsequent review of the teachers'
views and in-class actions; and analysis of learning outcomes for
the students of those teachers. These research and evaluation
strands are woven into the proposed video-case development work.
...
At 8:44 AM -0500 2/9/05, Tim Gutmann wrote:
I want to add a thought to Shandy and Dara's
comments about pedagogical
content knowledge (PCK) as it referrs to TAs. The primary issue that
Shandy and Dara have discussed is that of the importance of PCK and how
the related K-12 teacher research might be adapted to tell us something
about what kinds of PCK TAs need. A second and related question is how
TAs perceive their need for PCK and what motivations they have for
picking it up.
Unlike K-12 teachers and preservice teachers, many TAs are not working
as TAs only because want to teach. (For many this may be an extremely
important motivation, but it generally is not the only one.) As such
they can be expected to have a number of goals competing for their
attention. In this setting, we have to think carefully about how to
make PCK available and desirable to TAs (the desirable component being
self-provided by K-12 professionals).
In writing this I feel as though I am suggesting TAs don't want to be
good at teaching and therefore we need to find ways to provide this want
for them. I don't mean to be saying this. Rather, I think we need to
recognize their specialized motivations for being in the field and find
ways to integrate an understanding of the need for PCK into the
existing environment. Perhaps this is as simple as structuring
experiences into the time TAs do devote to teaching to help them reflect
on the process. (At many institutions TAs work with minimal guidance
during the semester, or receive instructions about how or what to
present. What is most frequently lacking is a time when TAs discuss
what they have done, what results they saw, what they struggled with,
(all past tense) and then think about the mechanics or the PCK issues
that might address concerns they have.) If nothing else, this approach
asks TAs to consider PCK at a time when they are devoting energies to
teaching and may help those whose primary reason for being in a graduate
program is not teaching to recognize their needs while performing in
this aspect of their work.
Tim.
...
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Tim Gutmann; tgutmann(a)une.edu
faculty.une.edu/cas/tgutmann
Decary 302, 207-283-0170 x 2764
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The more we complain, the longer God makes us live.