UC Berkeley

# Abstract

How can designers of instructional materials for mathematics
learning best support students' progress from intuition to
inscription? This paper explains the *embodied-design* methodology
for creating *cognitively ergonomic learning tools*. Two case
studies are contrasted in which individual participants in a
design-based research study of mathematical cognition engaged in
problem-solving situations pertaining to the content of probability. I
analyze the microgenesis of the reflexive, mediated interplay between
students' multimodal, intuitive, presymbolic notions and the available
multimedia tools (cf. Radford, 2003). The first case demonstrates how
the interviewer-student dyad coped with the *ontological
imperialism* (Bamberger & diSessa, 2003) inherent to some forms of
notation. Following a brief overview of how embodied mathematical
reasoning, and particularly gesture studies, can contribute to a deeper
and more nuanced understanding of students' reasoning processes as
they problem solve, I recount the second case, in which close listening
to a student's truncated expressivity directly informed the design of
a computer-based module that enables electronic gesture through which
students can elaborate their reasoning and interlink spatial and
symbolical referents. I advocate a measured balance between streamlined
and frictive learning tools, because it is in breakdown that creative
reasoning and creative habits of mind are fostered.

## Keywords

# 1. Introduction

In inscribing—writing, drawing, notating—we project presymbolic notions into and through a representational system hosted in an available medium, thus rendering the phenomenological semiotic (Radford, 2003; Vygotsky, 1978/1930). Doing so, we are tacitly relying both on the availability of suitable media and on our basic fluency with representational systems. Yet what happens in the case of explorative problem solving, where requisite media are not necessarily at hand and representational systems are yet to be created let alone mastered? This paper tracks the evolution of a design rationale for supporting mathematical problem-solving processes in which media and representational systems emerge as negotiated products in a structured collaboration between a designer and research participants. In a sense, it is a design-theory paper on methodology for creative design enabling creativity.

Mathematical thinking is vacuous without the machinery of the mathematical semiotic system, be it embodied in the human body, oral tradition, or scripted notations, and this machinery is a cultural legacy into which individual cognizers are acculturated through participation in organized discourse-based practices (Cobb & Bauersfeld, 1995; Ernest, 1988; Gee, 1996; Lemke, 1998; Radford, 2003; Saxe, 1981; Wittgenstein, 1956). Yet, as Olson (1994) has argued, the inscribed is no neutral receptacle; the medium, or more precisely a person’s intentional representational interaction with the medium, participates in the construction, elaboration, and, at times, modification of the presymbolic experience (see also Rotman, 2000), such that “content“ and “form“ of mathematical thought become ontologically and epistemologically inextricable. In the hands of fluent agents in a semiotic system, this dynamic reciprocity of intuitive content and representational form may be advantageous to the development of source notions (for example, the word processor I am using to edit this text is facilitating my reasoning). However, this content–form reciprocity may challenge novices to a representational system, making for learning impediments and communication breakdowns. Whereas it would be absurd to critique the very idea of introducing learners into pre-existing representational systems - such a line of argument would imply a critique of teaching children to adopt natural language - this paper is intended to illuminate cases where careful design and facilitation may support students in experiencing a safer and more empowering passage from intuition to inscription, i.e., from “raw“ phenomenology to participation in cultural semiotic practice. Moreover, because the context of the studies discussed below is that of technological design in the service of learning, students participating in explorative usability testing contribute to the ongoing development of new mathematical artifacts. As such, designers and students are co-dependent partners in pedagogical creativity' the designer applies professional acumen to enable students to voice their incipient notions, but the designer cannot, in principle, anticipate all these emergent notions.

The contribution of this paper is in the suggestion, supported by
empirical data, that in order to engender students' creative
reasoning, we should attend very closely to what they are attempting to
express, and where we detect the unavailability of optimal *semiotic
means of objectification* (Radford, 2003), we should respond with
design that enables the students to coalesce their burgeoning thoughts.
Specifically, I will demonstrate how scrutinizing students' hand
gestures from an embodied-cognition perspective informed the design of
mixed-media pedagogical materials that I call *cognitively ergonomic
learning tools* (Abrahamson, in 2008a). Thus, in submitting this
paper to the CreativeIT workshop, I am implicitly arguing for a breadth
of theoretical perspective in the study of creativity: the creative
act, proverbially an individual's spark of de-situated genius, is
inherently situated within a socio-cultural context and can be
coconstructed by multiple agents (e.g., Sawyer, 2007). One such agent is
a design-based researcher.

The data discussed in this paper
were collected in the context of the *Seeing Chance* project
(Abrahamson, 2006a, 2007b, in 2008b; Abrahamson & Cendak, 2006;
Abrahamson & Wilensky, 2007), an investigation of students' problem
solving in situations pertaining to the mathematical study of
probability. These studies were conducted using the meta-methodology of
*design-based research*, a paradigm, first explored in the late
1980's (Brown, 1992; Collins, 1992). Initially inspired by engineering
practices and attempting to address the complexity of real learning
ecologies, design-based research is now firmly established as a powerful
approach to researching diverse cognitive and social phenomena occurring
in learning environments through studying the iterative implementation
of experimental materials and/or pedagogical practices (Barab et al.,
2007; Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Confrey, 2005).
The learning materials used in my studies are part of *ProbLab*
(Probability Laboratory, Abrahamson & Wilensky, 2002), an
under-development experimental middle-school unit that continues the
*Connected Probability* project (Wilensky, 1997) and utilizes the
*NetLogo* multi-agent modeling-and-simulation environment
(Wilensky, 1999). The theoretical and pedagogical frameworks underlying
ProbLab's unique design rationale as well as empirical findings from
implementing the design are detailed elsewhere (e.g., Abrahamson, in
2008b); selected interactive materials are on http://edrl.berkeley.edu/
design.shtml.

I begin, below, by introducing the ProbLab
learning materials used in these studies. Next, I present and analyze
the case study of LF 'a middle-school student' that has been revealing
to me deep dilemmas underlying constructivist design (for a critique of
constructivist design that fails to deliver, see Kirschner, Sweller, &
Clark, 2006). I will discuss the case of LF as demonstrating the
*ontological imperialism* (Bamberger & diSessa, 2003) we are liable
to impose on students, and how we must *listen closely* to
students' voices (Confrey, 1991) in order to improve our design and
teaching. Next, following a brief overview of the embodied-reasoning
perspective on mathematical reasoning, I build on insight from the LF
case to interpret the case study of MK'a college senior' whose close
encounter with the *obduracy of the world*'the shortcoming of
immediately available media to enable expression of an elaborate
mathematical thought was detected through *microgenetic* video
analysis (Siegler & Crowley, 1991) so that we could create new media
tailored to her, and presumably other students', needs.

# 2. Design: Learning Materials for an Interview-Based Investigative Mini-Lesson in Basic Probability

Leading education researchers of probabilistic cognition generally agree as to the pedagogical value of enabling students to explore the complementarity of two inquiry activities, 'theoretical' and 'empirical,' targeted at random generators: (a) combinatorial analysis as an instrument for determining expected outcome distributions in actual experiments with the generators; and (b) experimentation, typically computer simulated (Jones, Langrall, & Mooney, 2007). ProbLab offers one plausible methodology for facilitating this theoretical-empirical complementarity.

ProbLab mathematical objects are embodied in
mixed media - both traditional and computer-based. A researcher, working
with a flexible interview protocol (Ginsburg, 1997), engages students in
activities using ProbLab materials. The interview begins with students
analyzing an experimental procedure in which four marbles are drawn out
randomly from a boxful of mixed green and blue marbles of equal numbers.
To draw out these samples, we use a special utensil, the *marble
scooper*, consisting of four concavities arranged in a 2-by-2 array
(a *4-Block*; see Figure 1a). Thus, unlike classical urns from
which we draw order-less samples, e.g., a combination of 1 green and 3
blue marbles that roll around in no particular order, the scooper has
inherent structural order, such that any draw perforce is a particular
permutation on the order-less combination, e.g., 'blue, green, blue,
blue.' Students are asked to guess what the most common draw might be.
According to probability theory, the most likely draw is the event, or
combination, with exactly 2 green marbles and 2 blue marbles in any
order. Specifically, the distribution of expected events by number of
green marbles is 1:4:6:4:1, corresponding to 0 green, 1 green, 2 green,
3, green, and 4 green, respectively (these 16 outcomes are arranged, in
Figure 1b, in a histogram-like distribution). Typically, middle-school
students correctly predict the 2-green-blue sample as the most
common draw and warrant this guess by referencing the 'half-half'
distribution of green and blue marbles in the box (Abrahamson, 2007c;
Abrahamson & Cendak, 2006). In stating their prediction, these
students' who have not studied probability 'make no allusion to the
order of the marbles in the scooper. Prompted, the students keep
referring to their visual impressions and offer no analytic warrant.

a | b | c |

*Figure 1*. ProbLab materials used in the study's theoretical
and empirical embodiments of the 4-Block mathematical object: (a) The
marble scooper; (b) the combinations tower; and (c) an actual
experimental outcome distribution produced by a computer-based
simulation of this probability experiment.

Next, students are guided to use a set of stock-paper cards, each
depicting an empty 4-block (a blank 2-by-2 grid) as well as green and
blue crayons so as to create all the different 4-block patterns one
could possibly draw out of the box with the scooper. This general
procedure is called in probability theory *combinatorial analysis*,
albeit the students may or may not be aware of the mathematical
implications of the activity in which they are engaging. There are 16
unique green/blue configurations (2^{4}), yet if one does not
attend to the order of the green and blue marbles in the samples, there
are only five possible objects - 4 blue; 1 green and 3 blue, 2 green and
2 blue, 3 green and 1 blue, and 4 green (hence 0g-4b, 1g-3b, 2g-2b,
3g-1b, 4g-0b)- and these five events (appear to) exhaust "what you
can get" when you operate the device

By way of a negotiation
with the interviewers, all students eventually construct all 16 unique
outcomes and are then guided to arrange these cards in columns by number
of green cells in the 4-blocks. The result is the *combinations
tower* (see Figure 1b), the sample space of the 4-block stochastic
device that is distributed such that it anticipates *the shape of
things to come* (Abrahamson, 2006b) in actual or simulated
experiments with the random generator (see Figure 1c; by default, the
computer simulations are set at .5 *p* value, reflecting the equal
numbers of green and blue marbles in the box). This hybrid nature of the
combinations tower - that it shares figurative properties with two
constructs that students purportedly need to coordinate toward deep
conceptual understanding, here a sample space and an experimental
outcome distribution (compare Figures 1b & 1c - is a hallmark of the
*learning axes and bridging tools* design framework for mathematics
education (Abrahamson, 2004; Abrahamson & Wilensky, 2007).

Students who have worked with the combinations tower are then asked to describe outcome distributions they expect to receive in a computer-based probability experiment that simulates the operation of the 4-block stochastic device and accumulates the experimental outcomes according to the number of green in each scoop (the ProbLab 'Sample Stalagmite' model, Abrahamson, 2006b). We ask, "What will the histogram look like?"; "Now, what if we change the probability of getting green from .5 to a greater number?" As we will see in the data excerpts, students spontaneously elect to use the combinations tower, on the desk in front of them, so as to describe the outcome distribution they anticipate in a computer-based simulation. Thus the activity design creates opportunities for students to juxtapose the theoretical and empirical aspects of random process and, through supported insight, to coordinate them (Abrahamson, in 2008b; Abrahamson & Cendak, 2006).

We now turn to the case studies. In the first case study, LF
negotiates with the interviewer modes of engagement with the
combinations tower - the interviewer *disciplines* LF's
*perception* (Stevens & Hall, 1998) toward the combinations tower
such that LF comes to see the tower as representing anticipated results
from running the marbles-box experiment. I will interpret LF's
difficulty as a *rupture* caused by the unavailability of suitable
*semiotic means of objectification* (Radford, 2003) that would
enable him to reify his presymbolic psychological objects. In the second
case study, MK attempts to use the combinations tower to express a
prediction for an experiment with unequal numbers of green and blue
marbles, but the medium falls short of conveying the message. Still
within that round of research studies, though, we created a new
computer-based module, *Histo-Blocks*, that appears to enable
students to express that message, i.e., to concretize their envisioned
manipulation of the combinations tower, if virtually.

# 3. A Case of Ontological Imperialism Causing Inadvertent Metonymy, or, "Try to See Things My Way, Do I have to Keep on Talking Till I Can't Go on? While You See It Your Way(We Can Work It Out!)"

Gazing
at the box (see Movie 1), LF concluded that '2g-2b' would be the
most common draw. He then works with the empty 4-block cards. He builds
the five possible events (see Figure 2, on the right, the bottom row)
and is content that they exhaust the sample space, e.g., LF sees the
card that has a single light-colored square in the top-right corner as
objectifying the 1g *event* (LF inadvertently sees the outcome as
metonymical of an entire event class). The interviewer, however,
perceives this same card as objectifying one particular
*permutation* out of all four possibilities (see, in Figure 2,
three additional grayed out permutations above that card). That is, the
card is an ambiguous or polysemous semiotic object, and the dyad tacitly
disagrees over its meaning - whether it is an event or an outcome. LF
does not initially realize the implications of his representation' that
he has unwittingly been subjected to *ontological imperialism*
(Bamberger & diSessa, 2003) through interacting with the media put at
his disposal as expressive means. That is, LF is not aware that by
virtue of using the available media, he has represented his intuitive
judgment such that a skilled user of the medium reads into the
representations meanings, new layers of signification, that LF had not
intended or even harbored. Namely, the emergent cues are 'visible'
only to practitioners in this disciplinary domain (see Goodwin, 1994, on
'professional vision'), for whom order counts in combinatorial analysis.
Thus, the available semiotic means of objectification were coerced upon
LF. His presymbolic categories were order-less combinations, and yet the
cards' structural properties add specificity, rendering order-less
combinations ordered permutations. LF was seeing combinations as the
'things' in the experiment, yet the media imposed upon him
permutations as the privileged elemental unit.

*Figure 2*. "These don't really matter" (+ gesture): the
sample space, as the student initially sees it.

The cards
appear to the interviewer (hence, Int.) as outcomes, because he is
attending to the particular configuration of little squares (the four
independent outcomes), whereas LF is not. Yet attending to order is the
core *learning issue* of this analytic process LF is to master. To
these ends, the order/no-order ambiguity is designed into the
mathematical objects precisely so as to foster an opportunity for the
student to discover that this property of order is a pertinent semiotic
means of warranting the initial intuitive judgment (i.e., that 2g-2b
would be the most common draw). Also, Int. was expecting LF to build 16
cards the complete sample space that includes all the
permutations and so Int. interpreted the 5 cards not only as
representing the 5 possible events (combinations) but also as an
incomplete set of outcomes (permutations). However, as the ensuing
conversation reveals, LF sees the cards as events, because his
communicative goal (representational intention) is to objectify the
presymbolic notion that corresponds to events, not outcomes. For LF to
attend to order, the dyad must first acknowledge their differing
*orientations of view* (Stevens & Hall, 1998) toward the cards,
objectify and name these orientations, and then negotiate whether one of
the orientations is more advantageous toward achieving the common goal.
Yet, this goal - applying combinatorial analysis to determine expected
experimental outcomes (i.e., showing 'what we could get' so as to
determine 'how often we will get it') - itself is emergent for LF.
For him, perhaps, the process of building the five cards appears as no
more than an opportunity to reiterate his earlier statement, using
semiotic objects conducive to deixis. That is, LF does not initially
know that by attending to order he can create a set of objects, the
sample space, that would collectively index his presymbolic notion. The
complete mathematical pertinence of the combinatorial analysis procedure
is thus temporarily suspended - the procedure will be
*instrumentalized* (Vérillon & Rabardel, 1995) only once LF sees
the sample space (the product of this procedure) in its entirety as
indexing his presymbolic notion; only once LF *objectifies*
(Radford, 2003) the combinations tower as resonant with his intuitive
judgment; only once he *blends* (Fauconnier & Turner, 2002;
Hutchins, 2005) the unarticulated sense of frequency into the 16 cards;
only once he *synthesizes* the intuitive and formal (Schön, 1981).
Thus the goal of the activity of coloring the cards is telescopically
enfolded into their 'hidden' property of order.

Interactions such as LF and Int. experienced could be interpreted as creating confusion and hence injurious to the learning trajectory. Yet, I propose, these moments can be regarded alternatively as necessary transitional psychological states most amenable to the appropriation of cultural artifacts. That is, given a viable semiotic means of elaborating on the five cards such that they connote the mathematically normative presymbolic qualitative distribution, LF is likely to appropriate those means. One could imagine a variety of such semiotic means, e.g., using a red marker to indicate upon the five cards the 'intensity' of their respective felt likelihoods, yet only combinatorial analysis offers a semiotic means in line with the cultural practice of mathematical argumentation; once constructed, the columns of the combinations tower anchor the felt frequencies as five vertical projections and, in so doing, render the qualitative notion quantitatively explicit. Moreover, I propose, LF possibly needed to objectify the sample space piecemeal - first the objects (combinations), then their properties (frequency) - in order to render the phenomenological object– property compound semiotic. Indeed, through interacting with the interviewer, LF will come to regard the sample space as a means of recording his felt sense of relative frequency. Yet, en route to understanding, as we shall see, LF himself is affected reflexively by the emerging meanings of the mathematical object he created - LF enters a transitional phase of generative confusion, as follows.

Gazing at the five cards he had just
created, LF retracts his initial intuitive judgment, stating that the
five events are equally likely to occur [sic!]. To the extent that
LF's representational intention had been to objectify frequency
distribution, he has concretized the 'what' (the five events) but
not their 'how often' (frequency) - the *objects* he had
created are now laid out for his inspection but their *properties*
had not been inscribed. Perhaps LF still entertained those properties
for a fleeting moment, blending them into the cards, but then, as in a
Stroop effect of sorts, the immediate perceptuality of the cards,
semiotically undifferentiated as they were one from another with respect
to their expected frequencies, superceded the presymbolic notion - in
the absence of any such qualification, the cards appeared equal. Just as
the 4-block structure of the cards compelled an ordered display of an
orderless notion, so now the cards' lack of affordance for marking
frequency engendered the apprehension of a flat distribution. It is as
though LF has unburdened the onus of information onto the cards and now,
gazing afresh, trusts the cards to re-evoke that presymbolic notion.
Thus, crucial aspects of a presymbolic notion are attenuated once the
notion is mirrored through a person's constrained fluency with a
representational system, breeding conflict between the intuitive and the
inscribed. How can this conflict be resolved? What can a teacher do to
mediate conflict resolution?

Int.'s objective becomes to enable LF to regain and sustain his presymbolic sense of the distribution by appropriating semiotic means of objectification in keeping with mathematical convention and validity. He asks LF to create the permutations on the five combinations. Somewhat reluctantly, LF builds the remaining 11 cards, yet he objects that the permutations are not pertinent to the analysis, because the initial question was about events, not outcomes (see Figure 2). Then, upon further observation, he sees the middle column as implying a 6/16 chance for that 2g-2g event. It is as though LF has "shifted gears" and is now building an analytic argument in a mental space that temporarily cordons off the event-based intuition that had been so robust. Yet as his gaze shifts to other parts of the combinations tower, specifically to the 4g single-outcome event, LF reasserts his flat-distribution hypothesis. In response, the interviewer prompts LF to re-consider the marbles box, thus re-evoking for LF the activity's initial context. LF experiences resonance between the sample space and his initial intuition: he sees the columns' relative heights or counts as indexing the anticipated plurality of 2g-2b (see Abrahamson, Bryant, Howison, & Relaford-Doyle, 2008, for further case analyses).

LF's passage from intuition to inscription, though
ostensibly complete, was a bumpy passage that may have left scars on the
traveler. Though a logical, procedural, perceptual string does connect
LF's presymbolic intuition to the interpreted sample space, this
string is quite threadbare, perhaps leaving LF in a state that Wilensky
(1997) calls *epistemological anxiety*, i.e., knowing that you know
how to work something but not knowing why it works the way it works. A
radical implication of the above would be that students are to be given
all the leisure, latitude, and learning materials needed to create their
own idiosyncratic representations of their intuitive notions - to create
their personal paths *from intuition to inscription* (Abrahamson,
2007a), from phenomenology to semiosis. Yet, the necessity of shared
semiosis (not to mention the assessment-based exigencies of the school
context) appears to deem such a position impractical. Indeed, a common
misunderstanding of radical constructivism is that children are left as
"free range agents in the learning environment (von Glasersfeld,
1992). The question evoked by the LF data, then, is how to foster
students' reflective negotiation between their presymbolic notions and
the target semiotic systems they are to learn, e.g., how to ethically
establish a sample space as the *epistemic form* for predicting
random distribution; and combinatorial analysis as the *epistemic
game* (Collins & Ferguson, 1993) one must play to construct this
form. The key, I believe, reinterprets the constructivist caveat of
departing from *where the students are* - we are to grasp what the
students are *trying to express* as they engage in problem solving.
Such insight into students' presymbolic notions can be enhanced by
attending not only to their utterance and inscription, but to their
entire multimodal, multimedia, multi-system embodied behavior. Doing so,
we may be better able to determine what semiotic means of
objectification students need that we are to design. Following a
theoretical elaboration on the above, we will return to the data.

# 4. Close Listening to Gesture: An Embodied-Design Perspective on Mathematical Reasoning

I depart from a premise, grounded in phenomenological philosophy and cognitive-science research, that embodied mechanisms - including kinesthesia, visuo-spatial images, audiated sound, proprioceptive motorics, etc. - are constitutive of reasoning, i.e. they are not mere epiphenomena of some would-be proposition-based symbol-processing reasoning (e.g., Barsalou, 1999; Goldin, 1987; Merleau-Ponty, 1992; Rizzolatti & Arbib, 1998; Varela, Thompson, & Rosch, 1991). One window onto such embodied mechanisms is gesture. Gesture - aspects of hand/arm motion that do not physically manipulate utensils or the environment - is associated with dedicated aspects of reasoning that may not be communicated through verbal utterance (e.g., Kendon, 2004; McNeill, 1992; Radford, 2003; Schegloff, 1984). I focus on gesture in the context of mathematical learning and practice.

Gestures people perform as they reason about
mathematical problems are informative of the multimodal resources they
are bringing to bear (e.g., Alibali, Bassok, Olseth, Syc, &
Goldin-Meadow, 1999). In particular, gesture acts as a unique window
onto reasoning associated with artifacts, such as tools, e.g., an
abacus, or inscribed representations, e.g., an equation (Alibali,
Flevares, & Goldin-Meadow, 1997), whether or not these artifacts are
physically present. That is, gesturing, like interaction with physically
present mathematical artifacts, transpires within the spatial medium,
and so gestures associated with mathematical artifacts may evince
spatial features of these interactions, revealing aspects of the
artifacts that the person is attending to as significant for a problem
at hand. It follows that some aspects of mathematical reasoning are
grounded as manipulations of imaged artifacts (Presmeg, 2006). Thus, the
routinization of practice with physical mathematical artifacts renders
them *embodied* constructions (Hatano, Miyake, & Binks, 1977;
Verillon & Rabardel, 1995; Vygotsky, 1978/1930).

In the case of artifacts that are new to a problem solver - a typical scenario in design-based research studies - gesture may uniquely reveal conceptual construction in action, such as when a student interacting with the artifacts builds connections among personal resources (Case & Okamoto, 1996). Thus, in studying learners' gesture-based interactions with mathematical artifacts under development, I investigate the nature of multimodal resources these artifacts afford. What developmental trajectories do these artifacts enable toward appropriation of normative mathematical forms? What limitations might the artifacts be imposing on students' expressivity? What design-theory principles do the data suggest in terms of developing effective learning tools?

Earlier, the LF data served as
context for explaining ontological imperialism and its pedagogical
problematics. The dilemma concerned the question of how to preempt such
imperialism yet shepherd students into convergence with normative
mathematics. To do so, I am suggesting, we should listen closely to
students' embodied presymbolic ontology and create cognitively
ergonomic mathematical learning tools, i.e., artifacts aligned with, and
hence affording, students' embodied reasoning. At the same time, these
tools should "surreptitiously include mathematical features
inviting elaboration. Following, the MK data will serve as context to
demonstrate this method, which I call *embodied design* (cf. Van
Rompay & Hekkert, 2001).

# 5. A Case of Coping With the Obduracy of the World (and How a Designer Tried to Improve This World

MK, a senior statistics major, has been working with
*4-Blocks*, a computer-based simulation of the marbles-scooper
experiment, with the *p* value set at .5 (see Applet 1, below). An
on-screen histogram, which tracks the accumulation of outcomes, has
converged, over thousands of trials, closely upon the expected 1-4-6-4-1
distribution.

[The applet requires Java 1.4.1 or higher. It will not run on Windows 95 or Mac OS 8 or 9. Mac users must have OS X 10.2.6 or higher and use a browser that supports Java 1.4. (Safari works, IE does not. Mac OS X comes with Safari. Open Safari and set it as your default web browser under Safari/Preferences/General.) On other operating systems, you may obtain the latest Java plugin from Sun's Java site

*Applet 1.* The
4-Blocks simulation of the marbles-box probability experiment NetLogo model. To
interact with other ProbLab models not discussed in the paper, go here.

Int. asks MK what the distribution might be for *p* values
higher than the default .5. MK replies:

It would be... If it was more likely to be green, it would be skewed (see Figure 3a)... This [right-side histogram columns in the 4-Blocks simulation] would get bigger, this [left-side column] would get smaller...

a | b | c | d | e | f |

*Figure 3*. Negotiating media constraints
on image expressivity, MK: (a) manipulates the on-screen histogram
"hands off" to indicate the expected histogram for a *p* value
of .6; (b) manipulates the on-screen histogram "hands on"; (c)
considers pen and paper, but declines; (d) manipulates the on-screen
histogram "hands on" remotely, using the combinations tower; (e)
shifts up the right-most single card to show the expected shape (compare
to Figure 3b); but (f) returns the card because the shift violated
constraints of the representational form.

Recognizing that she cannot manipulate the on-screen histogram directly, MK remote-manipulates it. First, she frames the histogram - her gaze peering through her hands toward the screen yet not focusing on the transparent plain subtended by her hands and tilts the histogram to the right (see Figure 3a). Next, she touches the screen itself, calibrating histogram columns up and down to adjust it (see Figure 3b). Unsatisfied, she then turns away from the screen, and her right hand hovers momentarily over a pen (see Figure 3c), as though she means to sketch the expected histogram upon the available sheet of paper. But she abandons that medium and turns to the combinations tower on the desk, saying: "but it would shift, like...," (see Figure 3d). Moving her hands in opposite vertical directions - left hand down, right hand up - she has actually made the 1-column shorter and the 3-column taller. Now she turns to the single card on the right of the tower, wishing to show that this 4-green column, too, would become taller. She lays her right hand on the card and pushes it "up to the desired height (see Figure 3e). Yet, once up there, this single card no longer aligns with the bottom of the tower so MK returns the card to its original location (Figure 3f), stating that the card medium ultimately limits her in expressing her mathematical reasoning: "You can't really do it on these cool things [cards], but it would be more like that.""

It appears, thus, that the semiotic means of objectification available
to MK constrained her expressivity. These observations of student
behavior directly informed subsequent design. Namely, I created
*Histo-Blocks* (see below) to enable student expression of expected
outcome distributions for literally any *p* value as transformation
on the sample space, the very transformation that MK attempted yet, as
we saw, turns out to be either imprecise or impossible in the physical
space.

# 6. Histo-Blocks: From an Obdurate Physical World to an Accommodating Electronic World

*Histo-Blocks* (see Figure
4) is a computer-based interactive visualization designed to foster deep
understanding of the binomial function by enabling electronically the
gestures students were observed to attempt physically on the sample
space. The model also enhances the expressivity, vividness, and
precision of the transformations. The binomial function expresses the
probability of sampling a particular combination, for instance any
4-block with exactly three green squares, for given chances of getting a
green square, say .6 (as when 60% of the marbles are green). In order to
help students understand the binomial formula, its two
factors - combinatorics and chance - have been conceptually decoupled,
as in the binomial function, and distributed over unique yet interlinked
interface elements (see, in Figure 4, three monitors on the bottom
left - the bottom one displays the product of the two above it).

*Figure 4*. Interface of ProbLab model
Histo-Blocks, built in NetLogo. In this screen-shot, the function,
histogram, and monitors all express properties of the middle column of
the combinations tower for *p*=.6. The middle column in the
stratified histogram directly above the tower is equal in height
to the column immediately adjacent to its right, because the properties
of these two columns are

6 * 0.6 * 0.6 * 0.4 * 0.4 = ~.346

4 * 0.6 * 0.6 * 0.6 * 0.4 = ~.346

respectively (see in
the monitors; note how the '4' and '6' compensate for each
other).

*Applet 2.* The
Histo-Blocks theoretical-probabilty NetLogo model (not an empirical simulation).

Manipulating the "*p*" slider (see in Applet 2,
under the tower) directly redistributes the total area of the expected
outcome distribution in the histogram (above the tower), just as MK
attempted to do with the cards (the physical embodiment of the
combinations-tower). Yet, unlike the cards, the elements of each column
in the Histo-Blocks histogram are given to simultaneous and precise
stretching and shrinking—uniform within columns, variable between
columns—in response to a single manipulation of the *p*
slider. Pertinently, note how the *entire* 4-green column in the
histogram is stretched up without losing its base as it did for MK.
Thus, Histo-Blocks enables learners to manipulate the sample space (the
combinations tower) so as to express the outcome
distribution they anticipate to recieve in actual experiments. In the later interviews of our
study, we had opportunities to engage several students in pilot
activities in which they successfully coordinated the physical
combinations tower with elements of the Histo-Blocks simulation
(Abrahamson, in 2008a).

# 7. Summary Remarks: Deep Learning as Creative Response to Breakdown

A major challenge of
mathematical learning is to develop fluency with its notational system.
Leading scholars operating in the contexts of embodied reasoning,
techo-science, and cultural semiotics have pointed to the
*rupture* (Radford, 2003) students experience as they ford the
*topological-typological* hiatus (Lemke, 1998) between embodied
presymbolic notions and normative forms of inscribed expression. Yet,
whereas cultures have evolved artifacts that mind the
embodied-symbolical gap (e.g., the clock, Collins & Ferguson, 1993;
Hutchins, 2005), mathematics-education designers - who cannot wait
millennia for the survival of the fittest instructional materials - need
design principles to help students ground mathematical notation today.
The electronic medium offers a potentially powerful space for students
to develop such coordination through manipulating interlinked spatial
and symbolical representations (e.g., Kaput & West, 1994), and operating
within the design-based research paradigm has enabled researchers to
carefully select or create interactive media tailored to the
domain-specific understandings of the target clients. In this paper I
have proposed that an embodied-cognition perspective on mathematical
reasoning may offer a powerful lens on video data of students engaging
in situated problem solving—a lens that is complementary to
traditional focus on verbal utterance only. In particular, identifying
moments of students' truncated embodied expression can alert designers
to the absence of appropriate semiotic means of objectification, which
they can then create and make available to learners.

However, a
designer's quest to streamline the learning environment such that it
afford students tacit appropriation of representational forms may
compete with a pedagogical principle that students should reflect on the
procedures they are adopting. In fact, two classical
constructs - Heidegger's 'breakdown' and Piaget's 'reflective
abstraction' - are akin inasmuch as they both focus on moments of
disruption in the flow of action as engendering constructive
consideration of the equipment at hand, be it mechanical or epistemic
(see Abrahamson in-press; cf. Koschmann, Kuuti, & Hickman, 1998). I
identify a continuum between these mundane moments of reflection and
monumental moments of creativity. Thus, designing content-targeted
learning tools that both trigger students' experiences of the obduracy
of the world and structure their appropriation of prepared solutions may
enable learners to progress within the discipline of mathematics and
practice creative problem-solving. Namely, usability per se, such as
through avoidance of ambiguity, need not necessarily be the golden
standard in educational design as it is in industrial design (cf.
Norman, 2002). Rather, educational design is to some extent a craft of
tradeoffs between usability and pedagogy. For example, by virtue of
facilitating students electronic manipulation of the
combinations-tower columns, I am liable to deny students opportunities
to reflect on the media and content. Perhaps it was important that MK
manipulated the obdurate cards physically before working with the
computer-based model - perhaps she learned by ensuring that the middle
column becomes shorter so as to compensate for the growth of the columns
to its right?^{1}

^{1} I wish to thank Joey
Relaford-Doyle for our ongoing dialogue that helps me understand these
challenging data.

# References

Abrahamson, D. (2004). *Keeping meaning in proportion: The
multiplication table as a case of pedagogical bridging tools*.
Unpublished doctoral dissertation. Northwestern University, Evanston,
IL.

Abrahamson, D. (2006a). Learning chance: Lessons from a learning-axes and
bridging-tools perspective. In A. Rossman & B. Chance (Eds.),
*Proceedings of the Seventh International Conference on Teaching of
Statistics*.
Salvador, Bahia, Brazil.

Abrahamson, D. (2006b). The shape of things to come: The
computational pictograph as a bridge from combinatorial space to outcome
distribution. *International Journal of Computers for Mathematical
Learning, 11*(1),
137-146.

Abrahamson, D. (2007a). *From
intuition to inscription: Emerging design principles for mathematics
education*. Paper
presented at the annual meeting of the International Society for Design
and Development in Education (ISDDE), Berkeley, CA, September 17 -
21.

Abrahamson, D. (2007b). *The real world as a trick question: Undergraduate
statistics majors' construction-based modeling of
probability.*
Paper presented at the annual
meeting of the American Education Research Association, Chicago,
IL.

Abrahamson, D. (2007c). Handling problems: Embodied reasoning in situated
mathematics. In T. Lamberg & L. Wiest (Eds.).
*Proceedings of the Twenty Ninth
Annual Meeting of the North American Chapter of the International Group
for the Psychology of Mathematics Education* (pp. 219-226). Stateline
(Lake Tahoe), NV: University of Nevada, Reno.

Abrahamson, D. (in 2008a). Embodied design: Constructing means for constructing meaning. *Educational Studies in Mathematics.*
.

Abrahamson, D. (in 2008b).
Probability by design: Synthesizing event-based intuition and
outcome-based analysis. *Cognition and Instruction*
. *(Manuscript under
revision)*
.

Abrahamson, D. (in-press). Bridging theory: A case study of an
11-year-old student engaged in activities designed to support the
grounding of outcome-based combinatorial analysis in event-based
intuitive judgment. In M. Borovcnik & D. Pratt (Eds. of Topic Study
Group 13, Research and Development in the Teaching and Learning of
Probability), in the *Proceedings of the International Congress on
Mathematical Education (ICME 11)*. Monterrey, Mexico:
ICME.

Abrahamson, D., Bryant, M. J.,
Howison, M. L., & Relaford-Doyle, J. J. (2008). *Toward a
phenomenology of mathematical artifacts: A circumspective deconstruction
of a design for the binomial*
. Paper presented at the annual
conference of the American Education Research Association, New York,
March 24-28.

Abrahamson, D., & Cendak, R. M.
(2006). The odds of understanding the law of large numbers: A design for grounding intuitive probability in combinatorial analysis. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Ed.), *Proceedings of the Thirtieth Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 1 - 8). Prague, Czech Republic: Charles University.

Abrahamson, D., & Wilensky, U.
(2002). *ProbLab*
. Northwestern University,
Evanston, IL: The Center for Connected Learning and Computer-Based
Modeling, Northwestern University.

Abrahamson, D., & Wilensky, U.
(2007). Learning axes and bridging tools
in a technology-based design for statistics. *International
Journal of Computers for Mathematical Learning, 12*
(1), 23-55.

Alibali, M. W., Bassok, M., Olseth,
K. L., Syc, S. E., & Goldin-Meadow, S. (1999). Illuminating mental
representations through speech and gesture. *Psychological Science,
10*,
327-333.

Alibali, M. W., Flevares, L. M.,
& Goldin-Meadow, S. (1997). Assessing knowledge conveyed in gesture:
Do teachers have the upper hand? *Journal of Educational Psychology,
89*(1),
183-193.

Bamberger, J., & diSessa, A. A.
(2003). Music as embodied mathematics: A study of a mutually informing
affinity. *International Journal of Computers for Mathematical
Learning, 8*(2),
123-160.

Barab, S., Zuiker, S., Warren, S.,
Hickey, D., Ingram-Goble, A., Kwon, E.-J., et al. (2007). Situationally
embodied curriculum: Relating formalisms and contexts. *Sci Ed (Wiley
InterScience), 91*,
750-782.

Barsalou, L. W. (1999). Perceptual
symbol systems. *Behavioral and brain sciences, 22, 577-660.*

Brown, A. (1992). Design
experiments: Theoretical and methodological challenges in creating
complex interventions in classroom settings. *Journal of the Learning
Sciences, 2*(2),
141-178.

Case, R., & Okamoto, Y. (1996).
The role of central conceptual structures in the development of
children's thought. In *Monographs of the society for research in
child development: Serial No. 246*
(Vol. 61). Chicago: University of
Chicago Press.

Cobb, P., & Bauersfeld, H.
(Eds.). (1995). *The emergence of mathematical meaning: Interaction in
classroom cultures*
. Hillsdale, NJ: Lawrence
Erlbaum.

Cobb, P., Confrey, J., diSessa, A.,
Lehrer, R., & Schauble, L. (2003). Design experiments in educational
research. *Educational Researcher, 32(1), 9-13.*

Collins, A. (1992). Towards a
design science of education. In E. Scanlon & T. O'shea (Eds.),
*New directions in educational technology*
(pp. 15-22). Berlin:
Springer.

Collins, A., & Ferguson, W.
(1993). Epistemic forms and epistemic games: Structures and strategies
to guide inquiry. *Educational Psychologist,* 28
(1), 25-42.

Confrey, J. (1991). Learning to
listen: A student's understanding of powers of ten. In E. v.
Glassersfeld (Ed.), *Radical constructivism in mathematics
education *(pp.
111-138). Dordrecht, The Netherlands: Kluwer Academic
Publishers.

Confrey, J. (2005). The evolution
of design studies as methodology. In R. K. Sawyer (Ed.), *The
Cambridge handbook of the learning sciences*
(pp. 135-151). Cambridge, MA:
Cambridge University Press.

Ernest, P. (1988). *Social
constructivism as a philosophy of mathematics* Albany, NY: SUNY
Press.

Fauconnier, G., & Turner, M.
(2002). *The way we think: Conceptual blending and the mind's hidden
complexities*. New
York: Basic Books.

Gee, J. P. (1996). *Social
linguistics and literacies: Ideology in discourses (2nd
ed.)*. London:
Taylor & Francis.

Ginsburg, H. P. (1997). *Entering
the child's mind*.
New York: Cambridge University Press.

Goodwin, C. (1994). Professional
vision. *American Anthropologist, 96*
(3), 603-633.

Hatano, G., Miyake, Y., &
Binks, M. (1977). Performance of expert abacus operators. *Cognition,
5*,
57-71.

Hutchins, E. (2005). Material
anchors for conceptual blends. *Journal of Pragmatics,
37*(10),
1555-1577.

Jones, G. A., Langrall, C. W.,
& Mooney, E. S. (2007). Research in probability: Responding to
classroom realities. In F. Lester (Ed.), *Second handbook of research
on mathematics teaching and learning*
(pp. 909 - 955). Charlotte, NC:
Information Age Publishing.

Kaput, J., & West, M. M.
(1994). Missing-value proportional reasoning problems: Factors affecting
informal reasoning patterns. In G. Harel & J. Confrey (Eds.), *The
development of multiplicative reasoning in the learning of
mathematics* (pp.
237-287). Albany, NY: SUNY.

Kendon, A. (2004). *Gesture:
Visible action as utterance *
Cambridge, UK: Cambridge University
Press.

Kirschner, P. A., Sweller, J.,
& Clark, R. E. (2006). Why minimal guidance during instruction does
not work: An analysis of the failure of constructivist, discovery,
problem-based, experiential, and inquiry-based teaching. *Educational
Psychologist, 41*
(2), 75-86.

Koschmann, T., Kuuti, K., &
Hickman, L. (1998). The concept of breakdown in Heidegger, Leont'ev, and
Dewey and Its implications for education. *Mind, Culture, and
Activity, 5*(1),
25-41.

Lemke, J. L. (1998). Multiplying
meaning: Visual and verbal semiotics in scientific text. In J. R. Martin
& R. Veel (Eds.), *Reading science: Critical and functional
perspectives on discourses of science*
(pp. 87-113). London:
Routledge.

McNeill, D. (1992). *Hand and
mind: What gestures reveal about thought* Chicago: University of Chicago
Press.

Merleau-Ponty, M. (1992).
*Phenomenology of perception*
(C. Smith, Trans.). New York:
Routlidge.

Norman, D. A. (2002). *The design
of everyday things*
. New York: Basic Books.

Olson, D. R. (1994). *The world
on paper*.
Cambridge, UK: Cambridge University Press.

Presmeg, N. (2006). Research on
visualization in learning and teaching mathematics: Emergence from
psychology. In A. Gutierrez; P. Boero (Eds.), *Handbook of
research on the psychology of mathematics education: Past, present, and
future* (pp.
205-235). Rotterdam: Sense Publishers.

Radford, L. (2003). Gestures,
speech, and the sprouting of signs: A semiotic-cultural approach to
students' types of generalization. *Mathematical Thinking and
Learning, 5*(1),
37-70.

Rizzolatti, G., & Arbib, M. A.
(1998). Language within our grasp. *Trends in Neurosciences,
21*(5),
188-194.

Rotman, B. (2000). *Mathematics
as sign: Writing, imagining, counting*
style='font-size:10.0pt; color:black'>. Stanford, CA: Stanford
University Press.

Sawyer, R. K. (2007). *Group
genius: The creative power of collaboration*
style='font-size:10.0pt; color:black'>. New York: Perseus Books
Group.

Saxe, G. B. (1981). Body parts as
numerals: A developmental analysis of numeration among the Oksapmin in
Papua New Guinea. *Child Development, 52*
(1), 306-331.

Schegloff, E. A. (1984). On some
gestures' relation to talk In J. M. Atkinson & E. J. Heritage
(Eds.), *Structures of Social Action: Studies in Conversation
Analysis* (pp.
266-296). Cambridge: Cambridge University Press.

Schön, D. A. (1981). *Intuitive
thinking? A metaphor underlying some ideas of educational reform
(Working Paper 8)*:
Division for Study and Research, M.I.T.

Siegler, R. S., & Crowley, K.
(1991). The microgenetic method: a direct means for studying cognitive
development. *American Psychologist, 46*
(6), 606-620.

Stevens, R., & Hall, R. (1998).
Disciplined perception: Learning to see in technoscience. In M. Lampert
& M. L. Blunk (Eds.), *Talking mathematics in school: Studies of
teaching and learning*
(pp. 107-149). New York: Cambridge
University Press.

Van Rompay, T., & Hekkert, P.
(2001). Embodied design: on the role of bodily experiences in product
design. In *Proceedings of the International Conference on Affective
Human Factors Design*
(pp. 39-46). Singapore.

Varela, F. J., Thompson, E., &
Rosch, E. (1991). *The embodied mind: Cognitive science and human
experience*.
Cambridge, MA: M.I.T. Press.

Vérillon, P., & Rabardel, P.
(1995). Cognition and artifacts: A contribution to the study of thought
in relation to instrumented activity. *European Journal of Psychology
of Education, 10*
(1), 77-101.

von Glasersfeld, E. (1992).
*Aspects of radical constructivism and its educational recommendations
(Working Group #4).*
Paper presented at the Seventh
International Congress on Mathematics Education (ICME7),
Quebec.

Vygotsky, L. S. (1978/1930).
*Mind in society: The development of higher psychological
processes*.
Cambridge: Harvard University Press.

Wilensky, U. (1997). What is normal
anyway?: Therapy for epistemological anxiety. *Educational Studies in
Mathematics, 33*
(2), 171-202.

Wilensky, U. (1999).
*NetLogo*.
Northwestern University, Evanston, IL: The Center for Connected Learning
and Computer-Based Modeling
.

Wittgenstein, L. (1956). *Remarks
on the foundations of mathematics*
. UK: Basil Blackwell.