Learning Number with TouchCounts: The Roleof Emotions and the Body in MathematicalCommunication

Nathalie Sinclair • Einat Heyd-Metzuyanim

� Springer Science+Business Media Dordrecht 2014

Abstract In this paper we describe a touchscreen application called TouchCounts, which

is designed to support the development of number sense in the early years. We first provide

an a priori analysis of its affordances. Then, using Sfard’s communicational approach,

augmented by a focus both on the role of the body—particularly the fingers and hands—

and emotions in the mathematical communication of a child, a teacher and a touchscreen

device, we show how two 5-year-old girls learn about counting and adding.

Keywords Emotions � Tools � Touchscreen � Gestures � Goals �Number sense � Communicational approach � Early years

1 Introduction

The touchscreen is a novel technological affordance in mathematics education. Through its

direct mediation, it offers new opportunities for mathematical expressivity. Touchscreen

devices enable children to produce and transform objects with fingers and gestures, instead

of through a keyboard or mouse. This makes it easier for many users to interact with these

devices, but it also opens the way for new forms of mathematical communication that are

haptic and tangible. Touchscreen devices such as iPads are also compelling, becoming for

many ‘‘evocative objects’’, a term Turkle (2011) uses to underscore the inseparability of

thought and feeling in our relationships to things. Various researchers have noted the high

level of interest that children seem to have when working with iPads (e.g., Lange and

Meaney 2013). Indeed, since Papert (1980), researchers have pointed to—often

N. Sinclair (&)Simon Fraser University, 8888 University Drive, Burnaby, BC V5A1S6, Canadae-mail: nathsinc@sfu.ca

E. Heyd-MetzuyanimDepartment of Education in Science and Technology, Technion-Israel Institute of Technology,3200003 Haifa, Israel

123

Tech Know LearnDOI 10.1007/s10758-014-9212-x

anecdotally—the emotional impact of expressive tools on learners, particularly including

learners’ apparent enthusiasm, interest and satisfaction. But little research has been done

on how exactly the emotional dimension of learning interacts with the cognitive one and,

more specifically, on how the emotional dimension of learning relates to emerging theories

of embodiment. We are thus interested in the question of how learning is mediated by the

affective and embodied (tapping and gesturing, but also swaying and rhythmically moving)

actions and interactions of the child/teacher/tool.

In this paper, we will present a new application called TouchCounts (Sinclair and

Jackiw 2011), whose design was motivated by the new affordances of touchscreen tech-

nology. Unlike many ‘‘educational’’ applications that can be found for the iPad, Touch-

Counts is open-ended and exploratory, rather than practice- and level-driven. It aims to

support the development of number sense by offering modes of interaction with these

mathematical concepts that feature the fingers and hands. Our goal is to describe the design

of the TouchCounts application and to show how its particular affordances support the

development of number sense. We also aim to show how the body and emotions play a

central role in this development. We do this by analysing several different small teaching

experiments that were undertaken with the dual purpose of studying how children used the

application and how they learned about number as they undertook particular tasks with the

help of a teacher. We begin by presenting our theoretical framework so that we can

describe the design of the application as well as the research literature in terms of its

central constructs. We then explain the principles guiding the design of TouchCounts and,

in so doing, relate particular design decisions to research findings on young children’s

development of number sense (particularly in terms of ordinality and cardinality) and

describe our hypotheses about the way the TouchCounts functionalities might motivate this

development. After that, we present several excerpts involving two 5-year-old children.

Each excerpt focuses on different affordances of the application.

2 Theoretical Perspectives

Broadly speaking, we take a participationist, non-dualistic perspective on thinking and

learning. This means that we do not make a priori distinctions between body and mind or

between thinking and communicating. We understand communication as something that

involves not just spoken or written words, but also gestures, facial expressions and

exclamations. We thus take our starting point in Sfard’s (2008) communicational approach,

which is a well-developed theoretical framework for explaining how learning occurs

through changes in discourse, that is, through changes in the way people talk. We will then

complement this theory with ones that more adequately account for the particular forms of

communication involving emotions and the body.

In an attempt to describe what makes the mathematical discourse distinct from others,

Sfard highlights the following four characteristics: word use, routines, visual mediators and

endorsed narratives. Each one could help someone identify, for example, if a particular

discourse was mathematical or not. In this study, we will be focusing on word use and

routines. The latter are defined as the collection of meta-level rules characterising repet-

itive patterns in discourse. The routines for identifying shapes are different in mathematics

than they are in everyday discourse. Further, learning mathematics involves changing one’s

routines for, say, identifying shapes. Word use refers to the particular meanings that certain

words have in a mathematical discourse as well as the particular ways in which certain

words are used. Again, as children learn, the way in which they use particular words

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changes. For example, they may initially use the word ‘‘four’’ as a number in the sequence

of counting. However, they will eventually also begin to use the word ‘‘four’’ to describe

the number of objects in a set.

Sfard also identifies three main discursive processes through which mathematical

objects emerge: saming, reifying and encapsulating. Only the first two were relevant in our

study. Reifying involves replacing talk about processes (such as the process of counting)

and actions with talk about states and objects (such as ‘the count’). Reifying can result in a

change in word use because, using the example from above, moving from ‘‘four’’ as a

process of counting to ‘‘four’’ as an artifact of a set involves replacing talk about process

with talk about object. Saming involves assigning one name to two (or more) things that

have not previously been considered to be ‘the same’. A simple example, described in

Sinclair and Moss (2012), is when the word ‘triangle’ is used to describe both the pro-

totypical equilateral-shaped three-sided figure and, say, a very long and skinny three-sided

figure. In terms of a number discourse, saming might occur when the routine of counting is

seen as the same as the routine of subitising.

Drawing on Vygotzkian theory, Sfard conceptualizes learning as a process of individ-

ualizing the discourse of experts into a discourse for oneself, which, in its internal form, is

what we talk about as ‘‘thinking’’. This process necessarily includes an expert from whom

the child can learn. The expert discourse might be offered by a teacher, but it might also be

offered by other resources, including software tools. Indeed, insofar as the iPad ‘speaks’

and moves, in interaction with the user, it takes on an animate role in the interaction, which

affects the communication patterns of the users. As Sfard and McClain (2002) argue, tools

should not be thought of as ‘‘optional avatars of independently existing mathematical

ideas’’ (p. 155) because the tools and their use are inextricable ingredients of communi-

cation. Indeed, Nemirovsky et al. (2013) stress this point of view when they describe the

way in which expressive digital tools can be thought of as mathematical instruments that

are material and semiotic tools, together with a set of embodied practices for their use

within the discipline of mathematics. Fluent use of such an instrument, which involves the

‘‘systematic interpenetration of perceptual and motor aspects’’ (p. 372) of playing it, allows

learners to participate in a mathematical discourse.

In conceptualizing the process by which children individualize mathematical routines,

Sfard distinguished between two kinds of participation modes: the first is ritual partici-

pation (Heyd-Metzuyanim 2013; Sfard and Lavie 2005) which is participation for the sake

of connecting with other people. In ritual participation, the child is mainly imitating the

grownup or expert in performing the routines, without being able to explain why these

routines produce the outcomes they do. According to Sfard (2008), this form of partici-

pation is an essential step towards individualizing new routines. However, in order for the

child to master the new discourse, she has to turn it into a discourse for herself, where she

is able to flexibly and independently manipulate the new discursive objects. For instance,

in the case of dealing with numbers, this would mean that the child is able to talk about

number and verbally explore number relationships (such as what number is the sum of 3

and 4 or what number are smaller than 10) without the assistance of an expert.

Given our interest in the ways that learning with TouchCounts may enhance explorative

goals, we thus extend Sfard’s basic framework to account for embodied and affective

forms of communication. In terms of the former, since we see discourse as not only verbal,

we place as much emphasis on the broad and varied ways of communicating involved in

mathematical activity—including gestures, bodily movements, tone of voice, gaze, etc.—

as we do on language. This is in accord with principles of embodied cognition, which posit

that cognitive functions are ‘‘directly and indirectly related to a large range of sensorimotor

Learning Number with TouchCounts

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functions expressed through the organism’s movement, tactility, sound reception and

production, perception, etc.’’ (Radford 2012). Although not all approaches to embodied

cognition adopt a non-dualist perspective, we follow those that do (including Radford, but

also de Freitas and Sinclair 2013; Nemirovsky et al. 2013; Roth 2011).

Finally, we follow Nemirovsky (2011) and Roth and Radford (2011) in stressing the

importance of emotion in the process of learning. However, in line with our communi-

cational approach, we take a somewhat different view than theirs in that we concentrate on

the emotional aspects of the teacher-child-tool communication, while taking into account

the interaction of the two human subjects with the non-human one (the iPad). Specifically,

we are interested in emotional expressions for two reasons. One focuses on how the teacher

uses emotional scaffolding—emotional expressions that accompany the teaching–learning

interaction and assist in the process of learning. Such emotional scaffolding may stir the

student into attending to a certain property of the task (by using expressions of interest or

alarm), signal success (using praise and enthusiasm) or failure (by expressing disap-

pointment or disaffection). The second function of looking at emotional expressions is to

make inferences about the goals of the child in performing certain actions with Touch-

Counts. Expressions such as those that communicate interest, strong engagement or

wonder indicate for us goals that are internal to the mathematical activity (participating in

the discourse for oneself) whereas looking up at the teacher for reassurance, or smiling and

clapping together with her signal goals that have to do with making connections with the

teacher (discourse for someone else).

3 The TouchCounts Application

TouchCounts is intended to offer an expressive environment in which learners could create

and relate mathematical objects directly with their fingers and hands.1 The initial moti-

vation for TouchCounts arose out of a desire to help young children perform the often-

challenging task of coordinating their finger pointing with their number counting. We were

interested in the iPad’s potential to provide a multimodal correspondence between finger

touching, numeral seeing and number-word hearing (a one-to-one-to-one correspondence

of touch, sight and sound). Fingers have long been known to play an important role in

children’s development of number sense, but recent neuroscientific research shows that the

use of fingers in counting should be encouraged and supported. Indeed, this research has

already shown that consistent use of fingers positively affects the formation of number

sense and thus also the development of calculation skills (Gracia-Baffaluy and Noel 2008).

In the context of counting we may think of fingers unfurling, pointing and touching, but

these actions can also be thought of as gestures. Indeed, on the iPad, fingers are used to

accomplish a range of ‘‘gestures’’ such as swiping and pinching. These gestures differ

somewhat from the kinds that have been studied by researchers such as McNeill (1992) in

that they both involve contact with the screen and perform particular actions. In addition,

for the most part, they function more like speech (similar to ‘‘iconic’’ gestures like ‘thumbs

up’) than like the more spontaneous, unplanned gestures that function ideationally, gen-

erating new conceptual understandings (Nunez 2003; Sinclair et al. 2013). Nevertheless,

the touchscreen gestures offer a new potential for communication and, as we will describe

below, new opportunities to communicate mathematically.

1 A similar project has been undertaken by Ladel and Kortenkamp (2011), who have developed a multi-touch-table environment in which children can place tokens on the table using their fingers.

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TouchCounts can be thought of as a microworld for exploring number. At present, it

offers two different environments, one called Counting (1, 2, 3, …) and the other Adding(1 1 2 1 3 1 _). We will describe each in turn, providing a rationale for specific design

choices as well as research hypotheses.

3.1 Counting (1, 2, 3, …)

Counting starts almost blank, except for a horizontal bar representing a shelf (Fig. 1a). In

this world, a learner taps her fingers on the screen to summon numbered discs—or objects

representing numbers. The first tap produces a new yellow disc on which the numeral ‘‘1’’

appears. Subsequent taps produce sequentially higher numbered discs. As each tap sum-

mons a new numbered disc, TouchCounts audibly speaks the name of its number (‘‘one,’’

‘‘two’’). As long as the learner’s finger remains on the glass, it holds the numbered disc, but

as soon as she ‘‘lets go’’ (by lifting her finger) the disc falls to and ‘‘off’’ the bottom of the

screen, captured by some virtual gravity. If the learner releases her numbered disc above the

shelf, or ‘‘flicks’’ it above the shelf on release, it falls only to the shelf, and comes to rest

there, visibly and permanently on screen, rather than vanishing out of sight ‘‘below.’’ (Thus

Fig. 1b describes a situation in which there have been four taps below the shelf—these

numbered discs were falling—and then a ‘‘5’’ was placed above the shelf.) Since each time a

finger is placed on the screen, a new numbered disc is created, one cannot ‘‘catch’’ or

reposition an existing numbered disc by retapping it.

Fingers can be placed on the screen one at a time, or simultaneously. Thus, with five

subsequent taps, a learner sees five numbered discs appear sequentially on the screen and

hears these numbers counted one by one. However, if she places two fingers on the screen

simultaneously, she sees two numbered discs appear simultaneously but only hears the

higher-numbered shape explicitly named (‘‘two,’’ if these are the first two taps). Thus

repeatedly tapping two fingers on the screen produces the number names of ‘‘two, four, six,

eight, ….’’.

Finally, the virtual gravity—which pulls all released numbers down to the shelf or off

the screen—can be disabled in a teacher-accessible Preferences panel. In ‘‘no gravity’’

mode, the shelf disappears, and all numbered discs remain on the screen where they were

Fig. 1 a Counting (initial state); b after four taps below the shelf and a fifth tap above the shelf (arrowsindicate falling numbered discs); c ‘‘no gravity’’ numbered discs (objects remain where placed)

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created (see Fig. 1c). And of course the entire ‘‘world’’ can be reset, clearing it of all

numbered discs and returning the ‘‘count’’ of the next created numbered disc to one.

Having described the use of Counting, we now formulate some hypotheses that we have

about its learning potential, taking into account related research on early number sense

development. We being with the five principles of counting identified by Gelman and Meck

(1983): (1) the importance of assigning only one counting tag to each counted objects in the

array (the principle of one-to-one correspondence); (2) that number words should be pro-

vided in a constant order when counting; (3) that repeated counting of the same set must

always produce the same final number word (the cardinality principle) (see Sfard (2014) on

how this reformulation of Gelman & Meck’s cardinality principle is more in keeping with a

discursive perspective); (4) that it doesn’t matter in which order objects are counted; and, (5)

that it doesn’t matter whether the items in the set are identical. In terms of the first principle,

the research has found that when attempting to count a set of physically-enumerable objects

(like their own fingers), children between the ages of three and five often fail to coordinate

verbal and indexical counts precisely, skipping either physical objects or named numbers in

their respective sequences, or stating number names while indicating no corresponding

physical object (Fuson 1988). Using Sfard’s language, these children engage in a kind of

‘‘rote counting’’ routine of number recitation (as if they were singing a song), while also

engaging in a ‘‘finger pointing’’ routine of sequential digital identification—but these

routines are not coordinated. In TouchCounts, every finger tap produces a numbered object

as well as a spoken word in a constant (increasing) order. In doing so, it unifies the first two

of Gelman and Meck’s principles of counting into a one-to-one-to-one correspondence of

the physical, auditory, and symbolic presentation of numbers.

TouchCounts’s ‘‘shelf’’ enables certain numbers to be extracted from the sequence of

numbers by isolating them from preceding numbers. In other words, by asking a learner to

place ‘‘just five’’ on the shelf, the learner would have to tap four times below the shelf first

and then move her hand to tap above it. We hypothesise that this activity would help

learners begin to treat numbers (talk about and act on them) as things that have certain

properties rather than just as words in a recitation. For example, by placing five on the

shelf, the learner communicates that she knows that five is the number after four. This may

support the process of reification since the number five becomes an object as it gains

certain properties.

In the ‘‘no-gravity’’ world, the numbered discs do not fall away and instead all remain

on the screen. That means that number of taps (made sequentially or simultaneously) is

also the number of discs on the screen, which can reinforce the cardinality principle since

the last number ‘‘counted’’ (spoken out loud by TouchCounts) is exactly ‘‘how many’’

numbered objects there are. Furthermore, it is also the last number spoken by Touch-

Counts. Research shows that even after children have counted a set of objects (counting up

to five, say), when they are asked ‘‘how many’’ objects are in a given set, they will often

count the objects again (Baroody and Wilkins 1999). In other words, the ‘‘how many’’

question provokes a routine of sequential counting (Sfard and Lavie 2005). This processual

routine, in which numbers are seen as a sequence of words, eventually becomes reified so

that numbers are seen as attributes of a group of objects. In TouchCounts, the child is

engaged in a somewhat different routine—rather than counting a given set, she is actively

producing that set, and elements of that set count themselves (both aurally and symboli-

cally) as they are summoned into existence.

In summary, Counting focuses on sequential counting (ordinality) but also, through the

‘‘shelf’’, aims to help children begin to develop a more reified discourse around number.

Furthermore, certain tasks in Counting can also invite the use of numbers as attributes of

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objects (relevant to the cardinality principle)—for example, asking children to make five

‘‘all at once’’ rather than sequentially. However, it is in the Adding portion of Touch-

Counts, which we describe next, that numbers are used as objects on which learners can

operate.

3.2 Adding (1 1 2 1 3 1 …)

While tapping on the screen in Counting creates sequentially increasing numbered objects,

tapping on the screen in this new world creates autonomous sets of numbers. The user’s

number creation choreography starts by placing fingers on the screen, thus producing

digital counters, either simultaneously or sequentially (keeping each finger on the screen as

new ones alight). These fingers are then lassoed into a herd (Fig. 2a) until they are lifted

from the screen, at which point the encompassing lasso draws the digital counters into a

regular arrangement of counters, the sum of which is displayed numerically at their centre

and also spoken aloud (Fig. 2b). After two or more such arrangements have been produced

(as in Fig. 2c) they can be pinched together. The herds join, dynamically becoming one

herd that contains the digital counters from each herd, thus Adding them together. The new

herd is labelled with the associated sum (Fig. 2d), which TouchCounts announces aloud.

As evident in the description below, producing and then adding numbers simply

requires some tapping and then a pinching gesture. In other words, children can create and

pinch herds together without planning to or even knowing that they are adding. However,

given that research shows that even toddlers can develop simple ideas of addition and

subtraction (Aubrey 1997; Fuson 1992; Groen and Resnick 1977; Siegler 1996), the finger

choreography involved in using Adding provides a situation in which gathering herds

together into one makes sense. Indeed, the pinching gesture draws on one of the four

grounding metaphors for addition, that of object collection (see Lakoff and Nunez 2000). It

expresses the very idea of adding, and does so in a symmetric way so that, unlike symbolic

expressions of adding (3 ? 4 or three plus four), it does not imply a particular order.

Adding offers children the action of operating on numbers without necessarily requiring

them to calculate the sum of the numbers being added. Unlike with the calculator, which

can also perform addition, TouchCounts first requires the production of the digital counters

that will be labelled by a number (indicating ‘‘how many’’ are in the herd) and then enacts

the gathering mechanism in which the two herds join both visually and temporally.

Unlike other researchers, who link the understanding of cardinality with the ability to

identify the last number in a counting sequence as the number of objects in the set being

counted, Vergnaud (2008) argues that cardinality involves being able to use numbers as

objects, which can be operated on. This can be seen, for example, when children ‘count

on’. In a similar way, Sfard (2008) focuses not only on reification, through which processes

become objects, but on the recursive nature of the mathematical discourse, in which objects

Fig. 2 a Creating a herd of digital counters; b the rearranged herd; c pinching two herds together; d the sumof two herds

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themselves become parts of new processes. In TouchCounts, when children pinch two

herds, they are acting on these objects (which contain a certain number of digital counters

as well as their numerical attribute). They can do this, obviously, without having an

objectified discourse and without knowing that the transformation that occurs is the

operation of addition. In this sense, TouchCounts invites the children into a gesturally-

mediated activity with number that we hypothesised might support the development of

their processual or even reified discourse. By virtue of acting on the herds and observing

the effect of the pinching gesture, and by working on certain tasks in which target numbers

have to be produced, children will be effectively producing the same result as that which a

‘counting on’ routine produces. When doing so, their attention will not be focused on the

underlying routine of counting, since TouchCounts calculates the sum, but on the result of

the addition (or ‘‘gathering’’) operation. We note that while a teacher might introduce the

word ‘adding’ to the task, it does not appear on the screen. As such, words such as

‘‘making,’’ ‘‘putting together,’’ ‘‘joining’’ can all be used to describe the action of pinching

herds together.

4 Three TouchCounts Encounters

In this section, we present and analyse three different episodes. The first two involve the

same 5 year old kindergarten child named Katy and focus both on how Katy evolves in her

use of TouchCounts as well as on her changing understanding of ordinal numbers in the

default mode (with gravity) Counting World. The third episode involves another 5 year old

kindergarten child named Chloe and her interactions in the Adding World. Chloe and Katy

were each interviewed separately, by the first author, in a room close to their classroom.

When describing the actions and speech of the interviewer, the pronoun ‘I’ is used; when

commenting on these actions and speech, the interviewer is referred to by name.

4.1 First Steps: Mastering Touch and Count

The session began in the Counting world, with me saying ‘‘Let’s start with number.’’

1. Without any instruction, Katy started by placing her right index finger on the screen

and swiping it downward (Fig. 3a). She did this slowly, repeating the numbers (which

the iPad was producing orally) as she went (saying some of them out loud, like two,

three and mouthing the others).

2. After nine, Katy put her head down to look more closely at the screen, tapped a

number, then repeated ‘‘ten’’ out loud.

Fig. 3 a First swipes to create numbers; b making numbers above 10; c making numbers without looking;d making numbers above the shelf

N. Sinclair, E. Heyd-Metzuyanim

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3. Once the iPad went on to ‘‘eleven’’ and ‘‘twelve’’, Katy stopped uttering the numbers

out loud and only swiped her finger to make the numbers. When she arrived at

fourteen, she somehow touched the screen twice quickly. The iPad vocalised ‘‘four’’

and then ‘‘fifteen’’ (since the ‘‘-teen’’ of the 14 was overlapped by the next tap that

produces ‘‘fifteen’’). Katy lifted her head a bit toward Nathalie and asked ‘‘how do you

get this game?’’ Nathalie didn’t answer and Katy went on swiping and making

numbers silently.

4. At seventeen, Katy put several fingers on the screen at once so that the next number the

iPad vocalised, after ‘‘seventeen’’, was ‘‘twenty-one’’. Nathalie gasped and smiled at

her. Katy did not look up at Nathalie but paused and smiled.

5. Katy continued with her index finger to make numbers. After twenty-two she started

saying the numbers out loud in a rhythmic fashion. Her head went up and down and up

and her uttering of the numbers became more pronounced. At twenty-eight, she looked

up and said the numbers while tapping, squinting at some point in front of her, saying

the numbers at the same time as TouchCounts. (Fig. 3c). She did this for twenty-eight

and twenty-nine.

At this point, Katy had automated her number-making, swiping the screen at a constant

rhythm without having to look. She seemed to be pleased with this new skill as she made a

rather theatrical face, which could mean ‘‘look, I can count without looking at the screen’’.

(Fig. 3c).

All this activity ([1]–[5]) was completely self-initiated by Katy. She was the one who

decided on what tempo, where and how many times to tap the screen. Her leaning toward

the iPad (3b) shows deep engagement with the activity and she did not look at Nathalie

even once for reassurance. We thus conclude that the goal of her actions was mastering the

tool, and in that sense, it was exploratory. Another goal, which emerged in [5] was

harnessing the tool for mastering counting. Katy seemed to be pleased with being able to

count up into the high twenties, as can be seen by her strengthening voice and distinct

pronunciation of ‘‘twenty-eight, twenty-nine’’ and her choice to look up while counting and

swiping the iPad (3c). Yet the pleasure Katy derived from counting seems to have brought

to a certain stagnation of the activity and did not lead to new discoveries. Katy did not see

that swiping was not needed to create new numbers, it’s enough to simply tap. Neither did

she discover by herself that touching above the shelf would get the numbers to stay on the

screen. Of course, given enough time, she might have discovered these things by herself.

However, I decided to intervene by asking ‘‘What happens when you put it above the

line?’’. This turned Katy’s attention to the upper part of the screen. She made an attempt to

touch that part of the screen but because she continued swiping down, the number did not

stay on the shelf. I thus strengthened my assistance by motor (pointing to the space above

the shelf, Fig. 4a) and verbal scaffolding (‘‘Touch in the space, not on the line’’). Now

Katy put her finger above the shelf but did not lift it, so I assisted again: ‘‘Touch and let go.

Yep. Let go.’’).

When Katy lifted her finger, the number was left on the shelf. This seemed to puzzle

her. She pulled her hand back and said ‘‘It stops the number. Why?’’ and then immediately

initiated a new pointing gesture (Fig. 4b) accompanying it with the utterance ‘‘pop!’’.

She continued putting numbers above the shelf, saying ‘‘pop’’ each time, and lining them

up in a row from right to left. She stopped at forty-seven, sat back and smiled a bit, looking at

me (Fig. 4c). By this, Katy was signalling not only that she had enough of this particular

activity, but perhaps also that she had completed my request to see ‘‘what happens when you

put it above the line’’ and was waiting for new directions. I asked Katy to press the Reset

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button and pointed to it. Katy tried to tap it but her finger landed somewhat underneath. This

produced more numbers. Katy tried tapping very fast, seemingly enjoying the speed of the

numbers. I said ‘‘Wait, wait. Reset it. Wait, wait’’ and laughed. After Katy succeeded in

pressing the button, I asked ‘‘I would like you to do something for me which is to put just 5

here’’. While saying that, I held her hand to stop her from going on and tapping the iPad

(Fig. 4d). At that point, Katy had a mischievous smile,2 which might be seen as expressing the

momentary conflict between her own goal (to go on playfully tapping the iPad) and my goal

(to perform specific tasks). She pulled and wiggled away from my holding hand and started to

tap the screen almost together with the end of my verbal request. The iPad vocalized ‘‘one’’.

Joining her mischievous expression, I said in a playful voice ‘‘no I don’t wanna see one!’’ and

then, after another tap, ‘‘I don’t wanna see two!’’. Katy smiled then tapped two more times

(three and four). When she arrived at five she tapped below the shelf while saying ‘‘five’’. She

immediately saw her mistake and uttered ‘‘no!… arghh!’’ when she saw the ‘5’ dropped down

the screen. By this expression of frustration, Katy was signalling her full emotional

engagement with the task. The goal to do what I had asked her (to please me), and her own

desire to master the tool were now indiscernible.

After hearing/seeing five fall down, Katy reset (without being asked) and laughed. She

then put one above the shelf, then two, three and four below the shelf, and then five above.

She sat back and smiled, signalling that she saw her task as completed. I then prompted her

to put just five on the shelf. Katy reset on her own, put one, two, three, four below the shelf,

and said ‘‘four’’ out loud, just after the iPad. She paused, then put five above the shelf, also

saying five out loud along with the iPad. I responded with a smiling ‘‘Good job!’’. Katy sat

back a bit and looked down.

During this interaction, Katy used her right index finger almost exclusively. However, the

way she touched the screen changed from a slow swipe when she first made new numbers, to a

quick tapping (‘‘pop’’) when she was on her third try at placing ‘‘just five’’ above the shelf. On

this last try, the quick rhythmic tapping of one, two, three and four suggested that she wanted

to get four taps below the shelf and put the fifth tap above. Such rhythmic tapping could

provide a basis for the emergence of a reified discourse on number in which both the number

five is separated from its predecessors and one, two, three and four are bound together as a

unit. Interestingly, the motoric aspect of the activity simulated the verbal reification process

of moving from talking about process to labelling it with one word—indeed, in her last

attempt Katy spoke the numbers as she tapped the screen. This emergent discourse was

developed through repeated attempts at touching/seeing/hearing the sequence of numbers

from one to five being produced. It occurred through a back and forth movement between

activity that was aimed at satisfying herself and activity that was aimed at satisfying Nathalie.

Fig. 4 a Pointing to the space above the shelf; b Katy’s new pointing gesture; c Katy looking at Nathalie;d Nathalie holding Katy’s hand

2 We are well aware of the fact that ‘‘mischievous smile’’ is a highly interpretative way of describing anemotional expression. However, this was the best emotion-word that we could find to describe the particularexpression that we saw on Katy’s face.

N. Sinclair, E. Heyd-Metzuyanim

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The self- satisfying activity seemed to be mostly motivated by the specific characteristics of

the iPad, and the pleasure Katy derived from hearing it vocalise numbers and produce ‘‘falling

numbers’’ (especially in quick tempos). In contrast, the satisfying -Nathalie activity, or doing

as Nathalie asked, was probably aimed at connecting with a grown-up and receiving positive

feedback about herself as a student.

What could be seen in this episode, is that even the teacher- satisfying activity quickly

turned into self- satisfying activity, as Katy expressed emotional engagement with the tool

and frustration when it did not work as she wished it to. In the next episode, we show how

this back and forth movement between self- satisfying and teacher- satisfying activity

shaped the learning process.

4.2 Objectifying Number: Exploring the Neighbourhood of 5, 6, 7, 8, 9 and 10

After Katy successfully put ‘‘just five’’ on the shelf, I then asked her ‘‘could you just put

five and then ten up there?’’ Katy started putting numbers below the shelf. She succeeded

in putting 5 above the shelf, then tapped ‘‘six’’ below the shelf and looked up at me.

Speaker What is said What is done

75 N No, you can keep going. Yea, Idon’t wanna see 6 up there

76 Katy Taps below the shelf

77 iPad Six

78 Katy Looks up at Nathalie

79 N Just wanna see ten next

80 Katy Taps twice

81 iPad Seven.. eight

82 Katy Hesitates, her finger in the air

83 N I don’t wanna see eight!

84 Katy Smiles, taps once and sits back

85 iPad Nine

86 N I don’t wanna see eight! Smiling, looking at Katy

87 Katy

88 N I don’t wanna see nine! I don’t wanna see othernumbers

With a higher pitch

89 iPad Ten eleven, six- eight- nine- twenty- twenty-twenty- twenty-

90 Interviewer I just wanna see five and ten

91 iPad Twenty- twenty- thirty- thirty- Katy keeps on tapping

92 N Let’s try again. Press reset.

93 Katy No! Keeps on tapping, smilingmischievously

94 iPad Forty- forty- forty

95 N Imagine five and ten are your best friends

96 Katy Sits back, looks at Nathalie

97 N And they’re the only ones you want to havecome over to your house

Learning Number with TouchCounts

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As we shall see in the next episode, at the heart of the Katy’s difficulty with this task is

her inability to predict reliably the order of numbers above 6 [107]. Therefore, she had to

seek my assistance, which she did with her questioning gazes and hesitating taps ([78],

[82], [84]). However, my comments did not help. They might have even interfered since in

[85] I say ‘‘I don’t wanna see eight’’ though the iPad had already produced the ‘‘nine’’.

Having no idea what to expect after nine, Katy seemed to lean on my repeated ‘‘I don’t

wanna see…’’ as a signal that she should go on tapping. She tapped a few times, missing

the ten completely. At that point, Katy seemed to give up. She turned from the teacher-

satisfying goal (to put 10 on the shelf) to her own self- satisfying goal (making as many

numbers as quickly as possible) [89],[91]. This goal was so strong that she resisted out loud

my attempts to ‘‘try again’’ [93].

Apparently, ‘‘satisfying Nathalie’’ was no longer a strong enough goal for Katy to

continue with this challenging task. This is why it is interesting that introducing an

imaginary ‘‘best friends’’ immediately halted Katy’s repeated tapping and grabbed her

attention [96]. Personifying the numbers into ‘‘friends’’ made it relevant enough for her to

stop her self-initiated activity and give another try to the challenging task that had already

proved itself as potentially frustrating.

Katy started tapping the iPad again, producing two, three and four below the shelf, then

5 above the shelf and then 6 below.

107 Katy What kind of number is going to come after?

108 N After 6? What do you think?

109 Katy Don’t know. One, two, three, four, five, six, seven! [Taps 7 down and then

looks up]

110 iP Seven

111 Katy Eight. Does he go there?

110 N He’s not your friend, just ten.

Katy tapped eight below the shelf, said ‘‘nine’’ and looked at me. I said that nine was

‘‘not your friend’’. She asked ‘‘is nine going to come after?’’ and when I responded ‘‘you

just did 8, what do you think?’’ she began to count starting from one. She counted fast,

skipping some numbers, and went all the way to nineteen, made a grimace and said ‘‘no!’’.

She then tapped below the shelf (for nine) and looked at me.

120 N (looking at Katy) and after nine?

121 Katy (taps below the shelf)

122 N Oh! Ten was our friend! We wanted to have him

123 Katy (looks back) One and an o (looks at screen, smiles, raises index finger).

After this moment of disappointment, Katy started over again. She tapped four times

below the shelf, looked up at me and then tapped the five above the shelf. She tapped six,

seven and eight below the shelf and looked up at me, eventually deciding to tap the nine below

the shelf. She then paused, looked up at me, and was about to place her finger on the shelf

when I said that she should make sure to place it above the shelf if she wanted it to stay. Katy

tapped above the shelf. I exclaimed happily ‘‘Yeah! good job! You got both of your friends!’’

(Fig. 5a) and started clapping, Katy turned to face me and started clapping too (Fig. 5b).

Based on our communicational framework, we wish to point to significant tool-based

and emotional interactions involved in Katy’s learning to put ‘‘just 5 and 10’’ on the shelf.

We begin with the observation that in putting ‘‘just five’’ above the shelf, Katy changed

from tapping her finger five times, to tapping it four times below the shelf and once above.

N. Sinclair, E. Heyd-Metzuyanim

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She also stopped repeating the numbers aloud, though she said them to herself as she

tapped, apparently ignoring the voice of the iPad. She learned to do this thanks in part to

the visual and auditory feedback of TouchCounts, both available only temporally, since the

numbers fell away and the number was said aloud only once. Her use of the cardinality

principle, which involved reifying five (five was no longer just the endpoint of a process of

counting, but had become the number that follows four, and could stand alone on the shelf

without the previous numbers) had an important rhythmic nature entirely situated within

TouchCounts—in other words, Katy’s reification was expressed with her finger tapping on

the screen.

[107] and [113] show that Katy was deciding where to tap based on which number

comes after the one that was just created (and said aloud). She had to use the counting song

to determine what comes after both six and eight, and was evidently not prepared for the

arrival of ten. Indeed, when Katy ‘‘dropped’’ the ten [121–123], Nathalie’s emotional

expression (‘‘Oh! Ten was our friend!’’), actually preceded Katy’s expression of alarm.

Obviously, Nathalie was much quicker to pick on the ‘‘failure’’ of Katy to put ten on the

shelf. Her emotional expression thus signalled to Katy both that she should be doing

something else (not tapping so fast) and that she should be disappointed with this

momentarily failure. This emotional scaffolding was enacted not only through facial

expressions and tone of voice, but also through Nathalie’s use of first person plural form

(‘‘ten was our friend; we wanted to have him’’) that framed the task as a mutual effort.

On the next time around, Katy did not need to think about what follows seven and eight.

While it may be that she had remembered the successors of seven and eight, it is more

likely that Katy was either mimicking the rhythmic tapping that she had used so often to

get ‘‘just five,’’ or remembering that several taps were needed before 10 came along. The

mutual excitement about Katy’s achievement at the end also involves important emotional

communication. In fact, a frame-by-frame examination of the video reveals that Nathalie’s

excitement—and emphatic ‘‘Yeah! Good job!— slightly preceded Katy’s more tentative,

but growing emotional expression. By her emotional expressions, Nathalie was conveying

to Katy information concerning the mathematical aspect of the activity (signalling ten as

important) and also the self-related aspect (signalling that Katy should be proud of herself).

4.3 Objectifying Number Patterns: From Skip Counting to Adding

Chloe, who was in the same kindergarten class as Katy, had started out using the Counting

World very differently than Katy. For example, she immediately put several fingers and

Fig. 5 a, b You got both of your friends!

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even both hands down. She made many, many numbers and, at one point, upon lifting her

hands and hearing the iPad, she exclaimed ‘‘two hundred and forty-seven.’’ She was also

able to place five and ten above the shelf quite easily, without verbal counting, suggesting

that she had already reified these specific numbers. So I switched to the Adding World. I

showed her how to pinch two discs together and let her try making numbers. After she

pinched together a few numbers, I asked Chloe if she could ‘‘make a five’’. She did that

very easily, both by tapping all her five fingers together, and by tapping with her index

finger five times and then trying to pinch the separate ‘ones’ into a group of five. While

doing that, she accidentally made a group of seven. I said ‘‘oh, you made a group of seven.

Can you make another group of seven?’’ Chloe said ‘‘hmm.. do I have to add five and

two?’’ and easily pinched a herd of five and a herd of two to make seven. This exemplified

her already objectified way of talking about numbers (‘‘add five and two’’). Chloe spon-

taneously used a ‘‘subitising gestures’’ to create the groups, always extending her fingers

out first, and then placing them carefully on the screen.

After this, I asked Chloe if there was another way she could make a seven. Chloe

suggested ‘‘five and three’’, then halted and looked in the distance. She asked ‘‘would that

work?’’ and I answered ‘‘let’s try’’. Chloe made a group of five all-at-once and then

attempted to make a group of three but accidentally made a four by touching the screen

with her folded finger (Fig. 6a). I assisted her in getting rid of the four group and she

attempted again to ‘‘make a three’’ by carefully touching the screen with three of her

fingers. Now she pinched the five and three, which produced an eight. At that point Chloe

expressed her surprise by opening her mouth wide and saying ‘‘what?!’’ (Fig. 6b). I said

‘‘five and three gave you eight’’. Chloe suggested ‘‘five and one’’ but then, after a second of

thinking, she exclaimed in a lower voice ‘‘that will make four’’. She continued thinking for

a few seconds, then said ‘‘six, actually’’, slightly smiling at me. I started saying ‘‘let’s try’’

and pointed to the screen when Chloe exclaimed excitedly ‘‘four and two!’’ I suggested that

she try. I reset the screen for her and she started by tapping four fingers. She halted, thought

a bit, then tried to tap two fingers on the screen, but her thumb touched the screen

producing three. Chloe reset and configured her fingers again above the screen. She started

by stretching out three but then folded one of them and tapped with two. Her choice to start

with two, not four, indicates a preliminary understanding of the commutative law of

addition, another signal of an objectified discourse about numbers.

After making the four and two groups, Chloe carefully pinched them together. When

she saw the number that was obtained, she said in a thoughtful voice ‘‘that’s six’’, squinting

up to a far away point. Then she exclaimed ‘‘THREE and four!’’ She reset and tried again,

this time putting three fingers down. Now she tried to make a four gesture but her fingers

accidentally made a three. She uttered an ‘‘ah, its’’ and looked at her fingers, but decided to

Fig. 6 a Chloe accidentally making 4, even though she only has three extended fingers; b Chloe saying‘‘What’’? after seeing that five and three make eight; c Chloe pleased with having made seven by puttingthree, three and one together

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fix the error by tapping another one with her pinkie and then pinching the three and one

together. Now that she had four and three, she pinched them together. At the sight of seven

she threw her head back, smiling (Fig. 6c).

It seemed that the ‘‘making seven’’ episode took a substantial toll on Chloe. She

stretched back, took a big breath and even remarked shyly ‘‘I’m stretching’’. This fact

makes the following episode even more remarkable. Not only did Chloe not give up after

her challenging task, she actually initiated the next one.

After stretching and resting for a few seconds, Chloe leaned again toward the iPad,

uttered an ‘umm..’ and thought for a few seconds, her gaze wandering around the iPad. I

asked ‘‘what are you thinking about?’’ and Chloe responded ‘‘I’m thinking about making,

adding ten and ten. I wonder what that makes’’. I showed her a ‘‘trick’’ that could make it

easier for her to do that. I put my 5 fingers on the screen, then added five ‘ones’ with my

index finger in the big ‘five’ disc. This added the numbers straight into the group without

the need to pinch them in. Now I let Chloe try it.

Chloe Places right hand on screen (Fig. 7a)

I Yeah, so you’re at five now

Chloe […] make it huge (looking down at her fingers and spreading them out so that the

lasso becomes bigger)

I Now you have six actually

Chloe Uses left hand index finger to tap four times (Fig. 7b)

I Let go

Chloe Lifts her hand from the screen

iPad 10

Chloe Places right hand on screen then taps five times with left index finger and then lifts

her hand from the screen

iPad 10

I Very nice

Chloe Pinches the two herds together with her right hand and then lifts her hand from

the screen (Fig. 7c). Twenty?

I Twenty

Chloe That’s why they say five, ten [short pause] fifteen, twenty (looks at interviewer,

Fig. 7d).

Chloe continued this way, making herds of ten and pinching them to her every-growing

herd of 20, 30, 40, … 90, until she reached 100. Chloe seemed to have known that she

would be able to use the pinching gesture to create a large number like 100. Further, unlike

Katy, Chloe had no physical problems making the pinching gesture and, furthermore, was

willing to pinch two herds of size ten together (Katy only ever pinched herds when one of

them was of size one). However, as evident in her questioning tone when the pinch gesture

created a herd of twenty, Chloe had not anticipated this result (though presumably she new

that the result of the pinching would produce a bigger number). Once she saw this new

group (which had twenty discs in it as well as the numeral ‘20’, whose size was confirmed

by the interviewer3) Chloe looked up, with a smile, and uttered a statement that connected

the routine of skip counting by five, which is processual, to the routine of successively

adding five, which is a reified discourse. In other words, she was ‘‘saming’’ a discourse that

she had heard (‘‘they say’’) with the action she had just made of adding ten and ten to get

3 In this earlier version of TouchCounts, the sum of pinched herds was not given orally. However, this is nolonger the case.

Learning Number with TouchCounts

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twenty. Apparently, the subjective experience of this saming was emotionally exciting for

Chloe, as evidenced by her heightened tone of voice, her wide-open eyes and her smile.

Chloe’s interaction with TouchCounts was much more fluent than Katy’s. This was in

part because of her physical skill at making complex gestures (using 3, 4 and 5 fingers;

pinching; co-ordinating her right and left hands) and partially because her discourse on

numbers was more objectified, making many more activities in TouchCounts meaningful

for her. The interesting emotional correlate of this numerical skill could be seen in the

explorative and self-initiated type of activity that characterised Chloe’s interaction with

Nathalie and with TouchCounts. In fact, the whole second episode (adding 10 and 10)

which, as we saw, was a very productive episode for Chloe, was initiated by her, including

stating the goal and working towards it. In so doing, Chloe’s goals were geared towards

expanding her discourse on numbers, not just mastering the tool and producing pleasure

from ‘‘making big numbers’’ (which was a goal at the beginning for her too, but was

quickly set aside after the initial episode).

5 Conclusion

In all three episodes we have presented (two with Katy and one with Chloe), we argued

that there was a shift in discourse from the processual to the reified. In the first episode,

Katy shifted towards a reified discourse about five while in the second episode, she

extended this discourse to numbers between five and ten. This reified discourse was

communicated in her finger actions on the screen (e.g., placing five on top of the shelf) and

also through words (e.g., when she asks ‘‘what comes after six?’’). Her rhythmic sequence

of tapping differentiated numbers below n from n - 1 (for n smaller or equal to ten). We

see Katy and the iPad in communication inasmuch as TouchCounts becomes a participant

in the conversation (making utterances as well), telling Katy what she has done, but also

through its constraints and affordances. For example: ‘‘gravity’’ sometimes makes it dif-

ficult for Katy to remember what number she has created, so that she must tap anew in

order to figure out where she’s at; the shelf gives numbers tapped above it a special status

in that they’ve been extracted from the sequence of numbers.

For Chloe, the shift in discourse was from the processual skip-counting to the reified

addition operation. More significantly, in the sense that Chloe had already been working

within a reified discourse while ‘‘making seven’’, she managed a ‘saming’ of two routines

that had previously been separate for her. Again, TouchCounts also became a participant in

the conversation, announcing the group number that Chloe had created, while also taking

care of calculating the sum of the two pinched herds, which allowed Chloe to focus on the

action of gathering the herds rather than on having to figure out the result.

Fig. 7 a Chloe making five; b using her left index to add to the group of 5; c Pinching the two groups of tentogether; d Sharing her discovery with Nathalie

N. Sinclair, E. Heyd-Metzuyanim

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However, we have underscored the way in which these discursive successes of the two

girls cannot be adequately explained or understood without also accounting for their

emotional communication, which includes their interactions with the teacher. For Katy, the

interaction with the teacher was pivotal in helping her successfully advance towards a

reification of number. While TouchCounts invites action and attention, the emotional

scaffolding with the teacher enabled Katy to focus on what is mathematically interesting

and/or important (in the eyes of the teacher), such as being able to predict what comes after

nine. The back-and-forth movement between self-satisfying and teacher-satisfying activity

was also central to Katy’s discursive success. Katy’s initial goal of mastering the tool and

hearing/making/seeing bigger and bigger numbers drove her initial interactions. It is then

through teacher-satisfying activity that Katy worked through, several times, the sequence

of numbers between five and ten. Just as Katy learned a new discourse, she also learned,

through her interaction with the teacher, when to be satisfied about her achievement. As

such, Katy not only learned about herself as a capable learner (something that some would

call her ‘self-efficacy’ beliefs (Zimmerman 2000) or her ‘identity’ (Sfard and Prusak 2005)

but also about what is valued in the mathematical discourse (often called ‘socio-mathe-

matical norms’ (Yackel and Cobb 1996)).

For Chloe, the activity was almost all self initiated. Chloe set a goal for herself (of

making one hundred) and in the course of trying to achieve that goal, encountered a new

idea. Thurston’s (1990) recollection of ‘‘an amazing (to me) realization’’ (p. 847) that the

answer to 134 divided by 29 is 134/29, evokes the power of the pleasure of saming, which

Chloe seemed to have experienced. The meaning of the ‘‘saming’’ for Chloe is entangled

with the setting, in which she found self-satisfying activity, and the supporting role of the

teacher, who was a partner for her excitement. The difference that self-initiated and

exploratory activity has on the level of learning (deep/conceptual), in contrast to externally

motivated activity, (ritual/processual) has been documented in several works (Meece et al.

2006; Roth and Lee 2007). Here, we see that Chloe’s emotional expressions were quite

different than those of Katy. While Chloe definitely displayed positive emotions such as

satisfaction (Fig. 6c) and wonder (Fig. 7d), these emotional expressions were not directed

at Nathalie. In fact, she hardly looked at Nathalie throughout the whole activity. In con-

trast, Katy’s expressions of happiness were all part of the ‘‘cheering’’ ritual that she enacted

together with Nathalie. In that sense, the emotional expressions are indicative just as they

are constitutive of the ‘‘ritualness’’ or the ‘‘explorativeness’’ of the girls’ activity. Since, as

Roth and Lee (2007) state, ‘‘emotions are always tied to the motives and goals of learn-

ing’’, examining the girls’ emotional expressions not only tells us something about their

possible subjective experience, but also about the goals of their activity.

In addition to the differences in emotions, there was a very significant motoric differ-

ence between Katy and Chloe’s use of TouchCounts. While Katy was a one-finger, one-

handed player, Chloe involved both her hands and all her fingers. Right from the begin-

ning, Chloe made five all-at-once in the Counting World and made a group of five in

Adding. Her gestural subisiting suggests that she had already reified numbers from one to

five. In contrast, Katy preferred to use only her index finger, and never extended her fingers

out before placing them on the screen. Of course, if the teacher had asked Katy to extend

three fingers and put them on the screen, she physically could have done so. We are thus

interested in whether explicit requests to place several fingers on the screen all-at-once

might be an effective way of developing gestural subitising, which would support a shift in

discourse form the processual to the reified. In other words, a question for future research is

whether a new discourse can be prompted by a new way of acting on the screen (see

Sinclair 2013, for preliminary results).

Learning Number with TouchCounts

123

We believe that by concentrating on the cognitive as well as the affective aspects of the

child-teacher-iPad interaction, we have been able to show in this study the precise

mechanisms by which TouchCounts turns out to be such an engaging and productive tool

for learning. The multi-touch affordances of the iPad also enabled us to explore a different

aspect of number that previous research has not considered, that of one-to-one-to-one

corresponding in which touch, count and object are coordinated in the finger tapping action

on the screen. Also present, though perhaps subordinated in the examples discussed here,

are the written numerals on the objects. In addition, TouchCounts offeres the teacher and

the child a new set of tasks that would not be possible in other environments either because

of the body syntonicity of the interaction—in which your fingers are the makers of

numbers—or the multimodality of the communication. For teacher and learner alike,

TouchCounts can become an evocative object that acts as a catalyst for a fusion of emotion

and mathematical thinking.

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Learning Number with TouchCounts

123