Content uploaded by Annika Wille
Author content
All content in this area was uploaded by Annika Wille on Jan 18, 2024
Content may be subject to copyright.
Representation of numbers and variables in Austrian Sign Language
Flavio Angeloni1, Annika M. Wille1 and Christian Hausch2
1Leibniz Universität Hannover, Germany; angeloni@idmp.uni-hannover.de
2Universität Klagenfurt, Austria;
Sign languages provide various possibilities to sign about elementary algebra. Essential in this
context are representations of numbers and variables in the sign language space. The study presented
here investigates how sign language space is used in Austrian Sign Language. In particular, it
addresses how numbers and variables are located on multiple axes with different signs.
Keywords: Sign language, variables, numbers, representations, classifiers.
Introduction
The paper presented here is part of a broader project on the topic of sign language and mathematics
education. Since language has an essential role in mathematics learning (Fleming, 2008), it is
worthwhile to look at how mathematics is signed about in sign language (Krause, 2019;
Wille & Schreiber, 2019; Krause & Wille, 2021). The project focuses on elementary algebra
(Angeloni et al., 2022; Angeloni, in press). The part of the study presented here deals specifically
with the representation of numbers and variables in the sign language space.
Theoretical framework
There is not only one sign language in the world, but many, which are different from each other. The
examples below are in Austrian Sign Language (ÖGS). Sign languages are languages with their own
structure, which is different from the structure of spoken languages. They are realized in a four-
dimensional space (three spatial dimensions plus time) and consist of several components (called
manual and non-manual). There are different types of signs. In particular, three of them are relevant
for the study presented here: direct pointing signs, buoys and classifiers. Direct pointing signs are
signs that point to something. Their handshape is a raised pointing index finger in the direction of the
referred. Buoys are signs (typically of the non-dominant hand) that remain fixed while the dominant
hand continues to sign. Classifiers (CL) or depicting signs (Lackner, 2019) are often defined in
different ways (Oviedo, 2005). Here, a classifier is understood as a sign that represents entities (of
whatever kind) based on their characteristic features. A classifier “classifies” objects or processes
based on their common features and represents them with the same hand shape (e.g., the index finger
can represent a person or a pencil) (Zwitserlood, 2012). Classifiers can be signed to refer to a
characteristic property, the position, and the movement of the signifier (Beecken et al., 2014;
Angeloni et al., 2022).
An important feature of sign languages is their simultaneity. This means that in the (visual) modality
of sign language it is possible to articulate meaningful units simultaneously. Simultaneity is preferred
in sign languages to the linear arrangement of morphological elements (typical for spoken languages).
This depends in particular on the type of relation between the various terms or concepts that occur in
a phrase: If they are organized in a syntagmatic relation, then they are signed simultaneously and this
is typical for sign languages (Grote, 2010). In the syntagmatic relation the terms are related to each
Proceedings of CERME13
4385
other as in a sentence. When a term is named, it is connected, for example, with what properties it
has, what it does, or where it is. In a paradigmatic relation one associates the term with everything
that fits into a category to which this term belongs, or into a subcategory of this noun. In this case,
one does not think in terms of sentences, but in terms of single nouns. For mathematics education,
this can have implications for the organization of content and how it is addressed or explained. These
considerations are also relevant in the analysis of sign language signs, whether already known or not.
Research interest
Our study is concerned with communication about elementary algebra in sign languages. In the paper
presented here, we focus on the topic of signing about numbers and variables in the Austrian Sign
Language and, specifically, on the following research question:
• RQ: How can numbers and variables be represented in the sign language space of (Austrian)
Sign Language?
Learning environment
The learning environment that was developed for the study is based on the “Knack-die-Box” (crack
the box) model (Affolter et al., 2011). Similar to the learning environment in Angeloni et al. (2022),
the participants have to solve three tasks according to the dialogic principle “think, pair, and share”
(Green & Green, 2018). First, each student deals with the task alone. In this phase, the students do
not have to find a solution to the problem. Then, the students (in pairs or very small groups) share
with each other their ideas about a possible (way to a) solution. In the last phase, possible solutions
are discussed in the class. In such a setting, the participants have the chance to become active in their
language ÖGS. The purpose is to sign about variables without signing some artificial signs for the
word “variable”, which are not naturally signed by native signers.
Fig. 1a: Task 1 Fig. 1b: Task 2
Figure 1: Pictures for the three tasks (the colour names are not in the original learning environment)
The learning environment includes three tasks with blue and red boxes and matchsticks, which
were represented in a stylized form like the two examples in Figure 1. At the beginning, we do not
know how many sticks are in the boxes, but we do know that the two arrangements must have the
same total number of sticks. The tasks are presented to the participants in ÖGS, i.e., no German
language is used. First, they get the information that two boxes that have the same colour contain the
same number of sticks. Then the two arrangements of a task are signed (Figure 1a) with the
information that the two arrangements must have the same total number of sticks. The participants
are asked to find out how many sticks are in the blue and respectively in the red boxes. When they
have found and discussed a solution, the next task is given (e.g., Figure 1b).
Methods
The research took place in June 2022 in form of two 60-minutes sessions. The participants were seven
deaf adults (age range: 30 to 70) with different mathematical backgrounds divided in two groups (one
Proceedings of CERME13
4386
per session) of three and four persons. Their basic language was ÖGS or regional variants thereof (cf.
Angeloni et al., 2022). The sessions were moderated in ÖGS by a teacher (first author). He took care
of the technical tools and moderated the share-phase. All the sessions were video recorded. For the
annotation of sign languages there are different systems. Frames of the videos which are relevant for
the research questions were annotated according to a simplified form of the annotation system of
Lackner (2019) for ÖGS explained above. The focus of the research question is on the sign language
space. A fundamental component of the sign language annotation is the gloss. A gloss names a single
sign that does not necessarily correspond to the meaning of the sign. For example, the gloss
CONTENT (glosses are written in capital letters) corresponds to a certain sign that does not
necessarily mean “content” or something similar. Furthermore, it is not a translation. Glosses are
annotated in capital letters. Below you will find the legend of the annotations used here:
• Signs for numbers are annotated with No:GLOSS – e.g., No:THIRTEEN.
• The annotation of the pointing sign is IX:GLOSS with the referred in the referent line – e.g.
IX:THIRTEEN.
• For classifiers the annotation CL:HANDSHAPE.description-of-action is used – e.g. CL:IX.It-
moves-diagonal-upwards means that one hand with the hand shape of an outstretched index
finger (IX) moves diagonally upwards, and this sign is a classifier.
• A buoy is annotated with H:GLOSS – e.g. H:CL:B means that the classifier with the hand
shape (for the letter) B (CL:B) is fixed.
Other components of sign language, like movements, are annotated on a separate line directly under
the gloss to which they refer. The following is the annotation legend for the components (from the
examples presented below):
• The direction of the movement is annotated in the line “Mov. direct.” (e.g., “downwards”).
• A movement has also a quality that is annotated in the line “Mov. quality” (e.g., “slowly”).
• The movements of the mouth are annotated in the line “Mov. mouth” (e.g., “blow-strong,
long, broad” means that the air comes out of the mount for a (relatively) long time, the mouth
is almost closed, so that the lips could also vibrate).
Not every line is reproduced in the transcripts below. If a line is empty, it is omitted (e.g., if there is
no movement, the line with the quality of the movement is omitted here). Therefore, not only the
annotations, but especially the frames of the videos are used for the analysis of the material. An initial
screening of the video frames and transcript identified scenes in which participants were signing about
numbers and variables. A first interpretation of these scenes was discussed with a native signer (third
author). Based on this preliminary evaluation, one-to-one interviews with the participants were
planned. As of today, two of these have been carried out. The others will be implemented in the near
future. In the interviews, uncertain interpretations of the first screening were addressed. In addition,
other questions were asked, such as: “Why did you sign it like that?”, “Do you know how it could
have been signed differently?” and “What is the difference between these variants?”. The interviews
were video recorded and interpreted similarly to the learning sessions.
Proceedings of CERME13
4387
Findings
Regarding the research question of how numbers and variables can be represented in the sign
language space of (Austrian) sign language, we can notice different representations that occur in the
learning environment. In general, we observed that signs in relation to numbers and variables were
signed on different sign axes. We can distinguish here between representations on a vertical and on a
horizontal plane.
The vertical plane
We call the plane in the sign language space parallel to the signer’s body vertical plane. If the
participants signed about numbers or variables on this plane, then they mostly signed three different
types of signs: the classifier in Figure 2b, the (specific) number signs such as in Figure 2c (signed
here on the left due to the dominant hand of the signer) or, in some cases, the general sign NUMBER.
Regarding the sign axes, here is what we could observe.
Fig. 2a: The vertical axis Fig. 2b: A classifier for a number Fig. 2c: Sign for the number 1
Figure 2: The vertical axis
Fig. 3a: The diagonal axes Fig. 3b: Number “13” Fig. 3c: Number “5” Fig. 3d: Number “1” Fig. 3e
Figure 3: The diagonal axes
The vertical axis in Figure 2a is normally oriented upwards, so that positive numbers are represented
by the signers in the upper part of the axis and negative ones in the lower part. (This was confirmed
in the interview.) Participants also sometimes set a mark for zero with one hand, while the other hand
signed the classifier for the number or for the variable above or below the mark. Similarly, specific
numbers are signed in relation to a mark. The mark is thus a buoy. If the mark is not set, then one
does not know whether the value of the sign is positive or negative. The participants commonly signed
the information “positive” or “negative” in advance (an example appears in Table 1).
Table 1: Annotation of an excerpt from the video no. 4 of the problem solving event
00:25v4 Participant 1: MINUS NUMBER H:CL:B CL:IX.It-points-downwoards
Proceedings of CERME13
4388
Mov. direct.: downwards downwards
Translation: The negative numbers are deeper and deeper.
In this example there is no mark for zero (or another number), but the sign of the numbers (in this
case “minus”: MINUS NUMBER) is signed at the beginning. So now one knows that the classifier
“CL:IX.It-points-downwoards” represents numbers that become smaller and smaller. In the following
example, the participant has signed the mark (H:CL:IX.upwards), but she does not give the
information about its value.
Table 2: Annotation of an excerpt from the video no. 4 of the problem-solving event
01:10v4 Participant 1: H:CL:IX.upwards PLUS NUMBER MINUS NUMBER
Mov. direct.: upwards downwards
Mov. quality: slowly slowly
Translation: The positive numbers are in the upper part and the negative numbers are in
the lower part.
More specifically, with the classifier in Figure 2b, the numbers or the variables are represented on
the vertical axis relative to their value. That means, a classifier that refers to a number or variable
near zero is signed approximately, as in Figure 2b. If the referred number or variable is very big, then
the arm is stretched upwards and the classifier has a big height. If the value of a variable is unknown,
the classifier in Figure 2b is repeatedly moved up and down. Moreover, if the participants have
represented a specific number or value of a variable (e.g., of value 13) with the classifier, then they
previously signed the specific ÖGS-sign for the referred number (e.g., THIRTEEN). In the case of
specific values, the participants also signed only the number with the number signs on the
corresponding height. Two examples:
Table 3: Annotation of an excerpt from the video no. 2 of the problem-solving event
11:33v2 Particip. 1: H:IX:RED H:IX:RED IX IX EQUAL
Referred: box on the left box on the right box on the left box on the right
Translation: The red boxes on the left and on the right, the two boxes have
COLOR NOT NUMBER IT-DOES-NOT-MATTER No:TWO
Translation: the same color. There is any (fixed) number, it does not matter if two,
No:FOUR No:SIX NUMBER NUMBER
Mov. direc.: upwards
Mov. qual.: from slowly to fast
Translation: four, six and so on.
Proceedings of CERME13
4389
In the example above (11:33v2), the ÖGS-signs No:TWO, No:FOUR and No:SIX are signed with
increasing height. The sign NUMBER is then signed upwards.
Figure 3a shows two diagonal axes. On those axes, the signers have signed only numbers that are
fixed and form an ordered sequence (such as Figures 3b to 3d). In the example in Figure 3, the
numbers 13, 5 and 1 are signed, with the height of the sign becoming lower. If the numbers in the
sequence become greater, then the number signs were signed with increasing height on a diagonal
axis. An example from the interview:
Table 4: Annotation of an excerpt from the video (no. 1) of the interview
01:03:12i1 Part. 2: TO-SIGN ALWAYS CL:B.It-moves-diagonal-upwards
Translation: It is always signed diagonally upwards
We could additionally see in the interviews that there is no difference between the two diagonal axes
in Figure 3a. The interviewee explained that the diagonal axes and the vertical axis in Figure 2a can
be used equivalently. However, if there is an image that is referred to by signing (e.g., numbers are
written on the black board or they refer to objects shown on a picture), the image influences which
axis is chosen. If a number was not fixed and could become arbitrarily big, this was signed on the
diagonal axis as in Figure 3e. The case of an arbitrarily small number did not occur.
The horizontal plane
Fig. 4a Fig. 4b Fig. 4c Fig. 4d Fig. 4e
Figure 4: Axes on the horizontal plane
We call the plane in the sign language space that is in front of the signer’s body and parallel to the
floor, horizontal plane. When the participants signed about numbers and variables, they also signed
on different axes on the horizontal plane. They signed on the axis in Figure 4a with an orientation
that does not necessarily correspond to the usual orientation with the positive numbers on the right
side with respect to an origin. It also happened that the signers signed according to an orientation with
the negative numbers on the right side. On this axis, the classifier from Figure 2b was not signed, but
the specific number signs for numbers or the general sign NUMBER. Two additional axes away from
the signer’s body occurred in the sessions: Figures 4b (diagonally) and 4d (normal to the signer’s
body). On these axes, the signers have represented a number or a variable that is not fixed and can
01:03:15i1 Part. 2: NUMBER ALWAYS CL:IX.It-moves-diagonal-upwards
Translation: The numbers increase diagonally upwards
Proceedings of CERME13
4390
become arbitrarily big. This representation is realized with a variation of the classifier in Figure 2b:
the hand shape is basically the same, but the fingers vibrate as if they were playing the harp (Figure
4c). In addition, the participant in the interview signed that it is possible to set a mark on the axis in
Figure 4d for the first number. For example, the mark on the normal axis in Figure 4d is set to 100:
Table 5: Annotation of an excerpt from the video (no. 1) of the interview
The participant himself confirmed that the concrete value does not matter. Analogical to the
diagonal axis in Figure 3, the case of an arbitrarily small number did not occur. In single cases, it
was signed in the direction behind the signer’s body (Figure 4e). For example, if someone signed
about negative numbers or numbers that are behind a fixed mark and become arbitrary small.
Discussion
Numbers and variables can be represented in various ways in sign language. As discussed above, the
sign language space and in particular vertical, diagonal and horizontal planes and axes can be used.
Three types of signs could be identified in this study for numbers or variables: Signs for concrete
numbers, the general sign for number, and classifiers. Thereby, the classifier acts like a bridge
between numbers and variables, since it can be signed for both. For example, in Figure 2b, the
classifier represents a number located at a certain height on the vertical axis. This classifier, when
moved (up and down), can also represent a variable that stands for an unknown or unspecific number.
These observations are consistent with the syntagmatic relations of sign language phrases: A sign that
refers to a number or variable is usually signed with the associated property (e.g., “large” or
“unknown”) or additional information is given about what to do with it. This is possible due to its
simultaneity and occurs with syntagmatic relations. The results suggest that when teaching
mathematics a simple translation from spoken to sign language of mathematical terms may not be
sufficient and that further aspects of these terms should be considered. Additionally, of particular
interest, is how numbers and variables are set in relation to each other along the axes. This will be
part of our subsequent research.
References
Affolter W., Amstad, H., Beerli, G., Doebeli, M., Hurschler H., Jaggi, B., Jundt, W., Krummenacher,
R., Nydegger, A., Wälti, B., & Wieland, G. (2011). Das Mathematikbuch 7 [The mathematics
book 7]. Ernst Klett Verlag.
Angeloni, F., Wille, A. M., & Hausch, C. (2022). Signing about elementary algebra in Austrian Sign
Language: What signs of the notion of variable can represent. In J. Hodgen, E. Geraniou, G.
11:00i1 Participant 2: No:HUNDRED CL:B.Palm-to-the-body-and-fingers-vibrate
Mov. direction: forwards
Mov. quality: slowly
Mov. mouth: Blow-strong, long, broad
Translation: An (arbitrary) number greater than or equal to 100
Proceedings of CERME13
4391
Bolondi, & F. Ferretti. (Eds.), Proceedings of the Twelfth Congress of European Research Society
in Mathematics Teaching (CERME12) (pp. 4218–4225). Free University of Bozen-Bolzano and
ERME. https://hal.science/hal-03765017
Angeloni, F. (2023). Gebärden über Variablen unter dem Gegenstandsaspekt [Signs about variables
under the object aspect]. Beiträge zum Mathematikunterricht 2022. WTM Verlag.
https://doi.org/10.17877/DE290R-23545
Beecken, A., Keller, J., Prillwitz, S., & Zienert, H. (2014). Grundkurs Deutsche Gebärdensprache:
Stufe I; Arbeitsbuch [Basic course German Sign Language: Level I; Workbook].
Gebärdensprachlehre: Vol. 3 (4th ed.). Signum Verlag.
Fleming, M. (2008). Languages of schooling within a European framework for languages of
education: Learning, teaching, assessment. Intergovernmental Conference. Prague.
Green, N., & Green, K. (2018). Kooperatives Lernen im Klassenraum und im Kollegium: Das
Trainingsbuch [Cooperative Learning in the Classroom and in the collegium: The Training Book]
(8th ed.). Klett/Kallmeyer.
Grote, K. (2010). Denken Gehörlose anders? Auswirkungen der gestisch-visuellen Gebärdensprache
auf die Begriffsbildung. Das Zeichen. Zeitschrift für Sprache und Kultur Gehörloser (85).
Krause, C. M. (2019). What you see is what you get? – Sign language in the mathematics classroom.
Journal for Research in Mathematics Education, 50, 84–97.
https://doi.org/10.5951/jresematheduc.50.1.0084
Krause, C. M., & Wille, A. M. (2021). Sign Language in Light of Mathematics Education: An
exploration within semiotic and embodiment theories of learning mathematics. American Annals
of the Deaf, 166, 358–383. https://doi.org/10.1353/aad.2021.0025
Lackner, A., Graf, I., Raffer, L., Scharfetter, E., Riemer-Kankkonen, N., Stalzer, C., Hausch, C.,
Unterberger, N., & Bergmeister, E. (2019). Austrian Sign Language (ÖGS) Corpus Annotation:
Annotation des ÖGS-Korpus. Veröffentlichungen des Zentrums für Gebärdensprache und
Hörbehindertenkommunikation: Bd. 25. Eigenverlag.
Oviedo, A. (2005). Classifiers in Venezuelan Sign Language. Signum Seedorf.
Wille, A. M., & Schreiber, C. (2019). Explaining geometrical concepts in sign language and in spoken
language – a comparison. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.),
Proceedings of the Eleventh Congress of the European Society for Research in Mathematics
Education. Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
https://hal.science/hal-02435340
Zwitserlood, I. (2012). Classifiers. In R. Pfau, M. Steinbach, & B. Woll (Eds.), Handbooks of
Linguistics and Communication Science. Handbücher zur Sprach- und
Kommunikationswissenschaft: Vol. 37. Sign Language: An International Handbook (pp. 158–
187). Walter de Gruyter GmbH. https://doi.org/10.1515/9783110261325.158
Proceedings of CERME13
4392