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NEGOTIATING MEANING: A CASE OF TEACHERS DISCUSSING
MATHEMATICAL ABSTRACTION IN THE BLOGOSPHERE
Judy Larsen
Simon Fraser University
ajlarsen@sfu.ca
Many mathematics teachers engage in the practice of blogging. Although they are separated
geographically, they are able to discuss teaching-related issues. In an effort to better understand
the nature of these discussions, this paper presents an analysis of one particular episode of such
a discussion. Wenger’s theoretical framework of communities of practice informs the analysis by
providing a tool to explain the negotiation of meaning in the episode. Results indicate that the
blogging medium supports continuity of discussions and can allow for the negotiation of
meaning, but that a more nuanced treatment of the construct is necessary.
Keywords: Teacher Education-Inservice/Professional Development, Technology, Learning
Theory, Informal Education
Introduction
I’ve been throwing around the term “pyramid of abstraction” recently, and there was some
great pushback on Twitter this evening. This post is my attempt to clarify what I mean, and
why I think it is a useful perspective to building students’ knowledge. (Kane July 31, 2015)
Mathematics teacher Dylan Kane wrote the passage quoted above in a blog post after a series
of Twitter interactions with mathematics teacher bloggers Dan Meyer and Michael Pershan. The
series of interactions were prompted by Dylan’s original blog post in which he explained ideas
about a metaphor for mathematical abstraction as pertaining to his teaching practice. The
discussion that resulted from this formulation presents an instance of a negotiation of meaning
around a metaphor related to the teaching and learning of mathematics. Most importantly, it
provides an example of how mathematics teachers are engaging in the practice of blogging,
which is the focus of this paper. It should be noted, however, that this paper is not concerned
with teacher knowledge (Ball, Thames, & Phelps, 2008; Shulman, 1986), but rather with the
social interaction among mathematics teachers that the blogging practice affords.
Mathematics Teacher Blogging
Mathematics teachers around the world are choosing out of their own will to create, organize,
and manage their own personal blog pages. On these publicly visible virtual pages, they make
relatively regular posts, which are most often related to their work as mathematics teachers.
These posts can include written and/or media enhanced recounts from their teaching experiences,
links to interesting resources, and responses to posts that other bloggers have made.
Since blog pages are individually managed and no universal blogging platform exists, finding
like-minded bloggers may at times be difficult. To mitigate this issue, many bloggers extend
their practice through Twitter, a universal micro-blogging platform that allows users to create
searchable profiles. There are currently 431 profile listings on the Math Twitter Blogosphere
directory, most of whom have both a Twitter handle and a blog page listed on their profile.
Although micro-blogging on Twitter is slightly different in nature than blogging due to the 140
character limit on each post, which forces users to communicate ideas more concisely, it is still
considered a form of blogging (Ebner, 2013). Blog pages allow users to explore ideas deeply, but
Twitter makes it easier for like-minded bloggers to connect. Mathematics teacher bloggers often
use both mediums to express ideas, linking between the two when necessary. For instance, some
bloggers include snips of Twitter conversations in their blog posts, and other bloggers link to
their blog posts within their Twitter posts. In this way, teachers are communicating with peers
across the world.
This sort of collegial interaction is unusual for teachers because teaching is generally an
isolated profession (Flinders, 1988; Lortie, 1975). Mathematics teachers’ typical contact with
other mathematics teachers is limited to that of department meetings and teacher lunchrooms,
where teachers usually plan lessons and grade papers individually (Arbaugh, 2003). Sparse one-
time professional development initiatives are not generally found to be effective in stimulating
teacher collaboration (Ball, 2002), and they don’t often extend into more regular professional
development opportunities because they require time, funding, and facilitation.
However, there are hundreds of mathematics teacher bloggers who seem to be overcoming
these constraints. It is evident that they spend hours writing publicly about their daily practice,
posting resources, and sharing their dilemmas with no compensation and no mandate. This
unprompted, unfunded, and unevaluated teacher activity is a rich phenomenon of interest that
deserves attention. Ironically, this phenomenon is largely unstudied. Empirical investigations
related to blogs in mathematics education are limited to studies on the utility of blogging as a
pedagogical tool within either a mathematics course (Nehme, 2011) or a mathematics education
course (Silverman & Clay, 2010; Stein, 2009). These studies do not account for the autonomous
and self-driven nature of the blogosphere and there is no clear work in mathematics education
exploring the activities of these teachers.
Research Question
As such, this study is guided by the overall research question of what the mathematics
teacher blogosphere affords for teachers who engage in blogging in relation to their practice. To
this end, in this paper I pursue an investigation of one episode that exemplifies the type of
interaction that is possible within the mathematics teacher blogosphere.
Theoretical Framework
Since blogging is an individual practice that is made public (Efimova, 2009), a mid-level
theory that accounts for situated participation is desirable. Communities of practice (Wenger,
1998) is one such theory: it is a social theory of learning where learning is considered as
increasing participation in the pursuit of valued enterprises that are meaningful in a particular
social context. Practice is at the heart of Wenger’s (1998) communities of practice, and a key
aspect of practice is the ability to motivate the social production of meaning. The continuous
production of meaning is termed as the negotiation of meaning, and is further defined by the
duality between participation and reification. For Wenger (1998), participation is “a process of
taking part [as well as] the relations with others that reflect this process” (p. 55), and reification
is “the process of giving form to our experience by producing objects that congeal this
experience into ‘thingness’” (p. 58). Both participation and reification shape the participant and
the community in which they participate in an ongoing manner.
According to Wenger (1998), participation and reification imply each other, require and
enable each other, and interact with each other. Wenger (1998) notes that the benefit to viewing
the negotiation of meaning as a dual process is that it allows one to question how the production
of meaning is distributed. He notes that there is “a unity in their duality, [because] to understand
one, it is necessary to understand the other, [and] to enable one, it is necessary to enable the
other” (Wenger, 1998, p. 62). Various combinations of the two will produce different
experiences of meaning, and together they can create dynamism and richness in meaning if a
particular balance is struck.
Wenger (1998) notes that “when too much reliance is placed on one at the expense of the
other, the continuity of meaning is likely to become problematic in practice” (p. 65). If
participation dominates, and “most of what matters is left unreified, then there may not be
enough material to anchor the specificities of coordination and to uncover diverging
assumptions” (p. 65). However, if reification dominates, and “everything is reified, but with little
opportunity for shared experience and interactive negotiation, then there may not be enough
overlap in participation to recover a coordinated, relevant, or generative meaning” (p. 65). In
essence, participation allows for re-negotiation of meaning, and reification creates the conditions
for new meanings. In this interwoven relationship, the two aspects work together to drive the
process of negotiation of meaning.
The interplay between participation and reification is different in each unique social
situation, contributing to a different experience of meaning for participants of that practice. What
is of interest in this paper is how mathematics teacher bloggers experience the social production
of meaning within the blogging practice.
Method
In order to be able to view mathematics teacher blogger conversations on Twitter, I have
spent over a year collecting subscriptions. Every user sees different posts based on who they
subscribe to, and it is important to note that my pervasively subjective position influences what I
can notice in this ultra-personalized and dense virtual environment.
Initially, my position was predominantly that of a ‘lurker’ in that I had not made significant
contributions to the blogosphere. This position changed slightly after my attendance to the
MTBoS conference ‘Twitter Math Camp 2015’ (TMC15). Physically meeting many of these
bloggers connected me to them more than before. As my subscription list grew, I was also able
to view more of their conversations. It was during this time after my return from TMC15 that I
encountered a particular conversation between Dylan, Michael, and Dan that I flagged as
interesting based on my theoretical framework and for its power to illustrate a possible mode of
interaction in the blogosphere.
After the conversation took place, I used storify.com
1
to identify all tweets related to the
conversation from the feeds of each of the participants (Dylan, Michael, and Dan), and
rearranged them chronologically. I copied the written content of each post, the name of the
participant, and the time stamp, and pasted it into an offline spreadsheet document. I also
included the written content of any blog post that was linked to in the Twitter posts within this
document. This reconstructed conversation made of Twitter and blog posts comprises the data set
for this paper.
This data set was then coded according to Wenger’s (1998) negotiation of meaning so that
conclusions could be drawn about participant experiences of the social production of meaning, as
embedded within the practice of teachers in the blogosphere. As part of the analysis, moments of
participation and reification were coded and labelled at each instance. In general, participation
1
Storify.com is an online service that allows users to curate content from various social media sources, and arrange
it in any order, with narration if desired. It also makes it possible to automatically arrange content chronologically.
was considered to be any action that a blogger took as part of the blogging practice, and
reification was considered to be any trace that was left from a participation. The interplay
between these aspects was then considered in relation to the data, and conclusions were drawn
about the nature of the negotiation of meaning as exemplified in this case of mathematics teacher
blogging.
In what follows, a reduced version of the reconstructed conversation is presented in the
results section, and is then reviewed in terms of Wenger’s (1998) negotiation of meaning
construct in the analysis and discussion section.
Results
On June 25, 2015, Dylan writes a blog post about his ideas regarding how teachers can help
students deal with solving difficult mathematical problems by helping them become better
equipped to transfer prior knowledge to new contexts in mathematics. He refers to the popular
‘ladder of abstraction’ metaphor, and claims that it is incomplete.
I often hear references to the “ladder of abstraction” — the idea that students’ understanding
begins with the concrete, and climbs a metaphorical ladder as it becomes more and more
abstract. I think this is a useful metaphor, but is also incomplete. (Kane June 25, 2015)
Dylan then suggests an alternative to the metaphor that would make it more useful in classroom
practice and more reflective of how he has experienced students learning mathematics.
I think the metaphor of a ladder of abstraction would be better replaced by a pyramid of
abstraction . . . I worry that the ladder of abstraction metaphor leads me to believe that, once
a student understands one concrete example of a function, they are ready for a more abstract
example. While some students may be, I want to focus on building a broad base first, and
then moving up the pyramid after we have spent time analyzing the connections between the
examples and the underlying structure. (Kane June 25, 2015)
A month later, on July 17th, Dylan writes a reflection about his experiences at PCMI, a three-
week summer mathematics institute. In this post, he discusses the importance of letting students
engage in productive struggle within mathematical problem solving, without oversimplifying.
Just after making this point, he uses a hyperlink (italicized below) to refer back to the June 25th
blog post in which he had introduced the idea of a ‘pyramid of abstraction.’
This gets at something I wrote about recently that I called the pyramid of abstraction – that
students build abstract ideas from looking at connections between a wide variety of
examples, rather than simply jumping from concrete to abstract. (Kane July 17, 2015)
Just as for his June 25th post, Dylan publishes a link to his July 17th post on Twitter (Figure 1).
Figure 1: Dylan’s Twitter post linking to his blog post
Shortly after publishing the link to his July 17th blog post on Twitter, Michael Pershan responds
in agreement with Dylan’s reference to the ‘pyramid of abstraction’ metaphor (Figure 2).
Figure 2: Michael’s response to Dylan’s post
Fifteen days later on July 31st, Dan Meyer challenges Dylan and Michael on Twitter by asking
about the meaning of the apex of the ‘pyramid of abstraction’ (Figure 3).
Figure 3: Dan’s question to Dylan and Michael
This is followed by a series of interactions on Twitter between Dan, Michael, and Dylan that
occurs over the course of a few hours. This series of interactions is presented in transcript form
below. Some comments have been removed for brevity.
3:55 Dan Q: What does the apex represent?
4:13 Michael Like, the uber-apex? Or the apex for a skill family?
4:23 Dylan an abstract principle that can be transferred among multiple contexts-think of
all math as nested pyramids of abstraction
4:24 Michael And at the very tippy tippy top something like "do math," I guess.
4:28 Dan I'm asking about the significance of the pyramid's tippy-top in this newfangled
metaphor.
4:29 Dan I don't think Dylan's explanation works. I can always abstract the "abstract
principle" he puts at the top.
4:30 Dan Abstraction has no end. Seems to me you guys are in a jam.
4:31 Dylan maybe abstract isn't the best word. Heart of this for me is knowledge that will
transfer to a new context
4:35 Dan Pyramid of something-other-than-abstraction then? Curious what it is you're
trying to describe.
4:38 Dylan I'm defining abstract as knowledge that transfers. I want to stick with that, but
it's worth defining more carefully
4:39 Michael I'm staying away from such a tough problem as trying to define "abstract"!
4:40 Michael When I am interested in pyramids of abstraction, it's an attempt to describe
mathematical thinking.
4:40 Michael There are strategies that we use that often represent bundles of strategies, and
so on.
4:55 Dylan my premise is that students need many representations to abstract from, hence
a pyramid
5:07 Dan But abstraction requires multiple instances /by definition/.
5:08 Dan Just saying this pyramid thing complicates an already complicated concept.
A few hours later, at 9:28PM on July 31st, Dylan publishes a blog post in response to this
conversation, starting with the post initially quoted in the introduction of this paper, and
proceeding to explain how he has defined ‘abstraction.’
I’m defining abstraction very specifically. A student abstracts a concept, or builds abstract
knowledge, if they can apply that knowledge in multiple contexts. (Kane July 31, 2015)
He then discusses why he thinks a ‘pyramid of abstraction’ is useful in teaching mathematics and
why this metaphor matters to him as a teacher.
There are a set of teacher actions we can take to facilitate transfer — that moment when a
student applies a concept they understand to a context they haven’t seen before. That’s what
I’m chasing. (Kane July 31, 2015)
In the last section of his blog post, he lists issues he still has with the metaphor, acknowledging
that it is possible the metaphor obfuscates the concept he wants it to represent, and asking the
reader if it makes sense to them. He also concedes that there may be no end to abstraction, but
that he personally doesn’t believe this is true.
If points in the coordinate plane are an abstract concept, and linear relationships are another
abstract concept, and functions are another abstract concept, do we end up in an infinite
pyramid of abstraction? I don’t think so. I think each of those ideas can then be a building
block for broader mathematical concepts. (Kane July 31, 2015)
Finally, he admits he is integrating knowledge transfer and abstraction into one idea, and
implicates that this may be a result of his own understanding of ‘abstraction.’
Do I Actually Understand Abstraction? I’m redefining abstraction a bit, and I’m also lumping
knowledge transfer as one giant idea, which it might not be. (Kane July 31, 2015)
Analysis and Discussion
Initially, Dylan participates in thinking about his past formulations about problem solving
and writes these ideas in the blog post that he publishes on June 25th. Within this process of
participation, Dylan engages in a process of reification when formulating his ‘pyramid of
abstraction’ metaphor. He also produces a public trace of this process, which can be seen as the
product of his reification. This trace is made even more public than a blog post because Dylan
also links to it on Twitter.
On July 27th, Dylan participates again in this same manner by posting a blog post and linking
to it on Twitter, but this time with a focus on recapturing his experiences from a professional
development encounter. In this process of participation, he uses a hyperlink to reference a
reification he made on June 25th regarding his formulation of the ‘pyramid of abstraction’
metaphor. In this way, the June 25th reification has prompted further participation around this
topic on July 27th, resulting in a new reification, and a re-negotiation of meaning. The concept
now has a rich history, and is traceable through the use of hyperlinks.
When Michael publicly agrees on Twitter with Dylan’s metaphor, a participation and
reification in itself, it makes Dylan’s ‘pyramid of abstraction’ reification even more public, and
catches the attention of Dan fifteen days later, who then challenges the metaphor. The power of
Dan’s reification (3:55) is that it is pervasively public. Dan currently has 46,324 followers,
Michael has 4,761, and Dylan has 1190. The nature of Twitter is that if one member tags another
in a post, only those who follow both members see the post in their feed. Since Dan, Michael,
and Dylan share followers as mathematics teacher bloggers, the number of people who see such
a post is large.
This now very visible reification prompts a series of participations and reifications from all
three of these members as well as from any ‘lurkers’ who may be reading. In particular, Dan’s
question about the apex of the pyramid (3:55) is a reification that prompts participation from
both Dylan and Michael, who reify their interpretations of the ‘pyramid of abstraction’ metaphor.
Dylan reifies his focus on knowledge transfer as the implication of the pyramid metaphor (4:23),
and Michael reifies his vision of the pyramid metaphor as a skill family (4:13) that ultimately
comprises mathematics (4:24). In this way, the meaning of the metaphor is being negotiated.
Dan subsequently participates by posting that he does not think Dylan’s explanation works
(4:29) and that there can be no ‘top’ to abstraction (4:30). This reification stimulates a re-
negotiation of the term ‘abstraction.’ Dylan participates further by reifying his focus on
knowledge transfer (4:38) while Michael participates by reifying that the focus should not be on
defining abstraction, but rather on how students bundle strategies in mathematics (4:39-4:40).
This shifts the negotiation of meaning to a focus on the applicability of this metaphor to
mathematical learning, and Dylan reifies that his intent with the metaphor is to explain that
students need multiple representations of a concept in order to be able to abstract mathematical
meaning (4:55). Dylan then restates his impressions of the conversation and refines his concept
of the ‘pyramid of abstraction’ metaphor in the blog post he publicizes a few hours later. This
post may be seen as a reflective form of participation in which Dylan reifies a heightened level
of awareness regarding the ‘pyramid of abstraction’ metaphor. Perhaps the most powerful
reification Dylan makes in this blog post is that ‘abstraction’ is not ubiquitously defined. This
ultimately reflects Dylan’s experience of meaning in this context.
Wenger (1998) states that “having a tool to perform an activity changes the nature of that
activity” (p. 60). It is clear that the blogging tool has done just that for these mathematics
teachers because unlike in a face to face discussion, the practice of blogging results in
participations that directly produce permanent, public, and traceable reifications, making them
prime contenders for prompting participation and holding participants accountable for their
statements. As such, the continuity of meaning within this medium clearly does not suffer as
Wenger (1998) warns can happen if too much emphasis is placed on either participation or
reification.
Further, the asynchronicity of the medium allows participants to take time between
responses, which may imply various degrees of participation in the practice, and in turn, various
degrees of reification. Unfortunately, Wenger’s (1998) construct does not provide a mechanism
for such differentiation. He defines each component broadly, including a wide variety of
interactions, most of which would occur in a face to face setting. However, in the case of
blogging, participations include not only writing posts, but also restating, questioning, and
devising examples, while reifications include not only publicized posts, but also ideas that have
now attained a state of ‘thingness’ in the community, which here is most prominently the
‘pyramid of abstraction’ metaphor.
Conclusions
I have illustrated that the blogging medium changes the nature of mathematics teacher
discussions because its design allows for participation and reification to be closely intertwined
in a way that their co-evolving and co-implicated relationship drives the negotiation of meaning
among participants, ensuring continuity of negotiation. Such continuity is an important
affordance of the mathematics teacher blogosphere experience, and it can be attributed to three
important features: asynchronicity, permanency, and publicity. Asynchronicity allows users to
participate and reify to different degrees depending on how long they take to respond,
permanency allows reifications to prompt further participation even days or months later, and
publicity allows reifications of participations to be visible by many ‘lurkers,’ whose presence
makes participants accountable for their reifications.
I have also revealed that a more nuanced treatment of participation and reification is needed
for mathematics teacher blogging in particular. Some participations are quick and spontaneous
tweets, while others are prolonged and accompanied by reflective activity. Some reifications are
mere traces, while others are concepts such as a ‘pyramid of abstraction.’ Heightened levels of
awareness resulting from a prolonged negotiation of meaning can also be considered reifications.
There is currently no terminology within Wenger’s (1998) construct to refer to such higher order
participations and reifications.
Finally, the construct of negotiation of meaning also has the power to expose what is being
negotiated. In this case, teachers found it valuable to invest time into engaging in negotiating the
meaning of ‘abstraction’ and the metaphor of a ‘pyramid of abstraction’ as opposed to a ‘ladder
of abstraction.’ Looking for more of these instances may help identify teaching-related issues of
interest to mathematics teachers. Most evidently, the development of ideas can be traced within
the blogging medium. Each idea has a history and a future. The history is traceable, and the
future is quickly unfolding.
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