Language, arithmetic and young children's interpretations.

Throughout the last two decades the language of mathematics has been acknowledged as being important to the development of mathematics understanding. It is believed that one of the main reasons children have difficulties with mathematics is the learning of the vocabulary. It seems that children are required to attach new meaning and learn new noun or verb forms to words that already exist in the learner's everyday speech. Halliday (1979) referred to this phenomenon as dealing with the mathematical register, a separate international language system with its own symbols, vocabulary, syntax, grammar and semantics. Children have to become aware that the meaning of words in the mathematics classroom is more constrained or precise as compared with words used in everyday speech (Gal, 1999).

The goal of this paper is to contribute to a reflection on young children's understanding of words commonly used in addition and subtraction, the meaning of the vocabulary, the relationships they see between the words, and the conceptualization they use in delineating this connectedness. In times where the focus is on reading and writing, few studies have explored how young children comprehend the language of mathematics. The literature often refers to this as children's metalinguistic awareness of mathematical words (MacGregor & Stacey, 1999), that is, the linguistic ability that allows the language user to reflect on and analyze the spoken or written word. Metalinguistic awareness is important for understanding algebraic notation (MacGregor & Price, 1999). Past research has tended to examine language proficiency with English as a second language (Adler, 1995; Bums, 1992; Lopez Real, 1997) or investigated children's knowledge of the meanings of words, their familiarity with patterns of discourse used in the classroom, and their ability to comprehend word problems.

As our attention turns to a social constructivist world and the learning of algebra in the early years, the role of language increases in importance. A vast amount of recent research has focused on the processes of communication in classrooms and the emergence of shared meaning. The underlying belief is that when people communicate through all sorts of signs (both idiosyncratic and conventional) knowledge emerges, with the individual continually interpreting and reinterpreting these signs (Peirce, 1960). However, MacGregor (1993) suggests research has shown that group activities involving sharing ideas, discussing procedures and writing reports may not necessarily be helpful to all students. These discussions themselves involve group language that participants may not feel comfortable engaging in as they may not understand the explanations of others and may not be able to communicate their ideas verbally. Nevertheless Arzarello (1998) believes that natural language is crucial to developing an algebraic way of thinking. When solving problems the verbal code activates an inner language that serves to aid the communication of knowledge and solutions to problems. Signs and tools are used by the learner to mediate the learning activity (Vygotsky, 1978, 1986), and words and dialogue play an important role in both external and internal processes of learning. From this premise the meanings of words used by young children become of great interest.

The epistemological stance taken in this analysis is the science of semiotics, a means of addressing signs, their connections and meanings. In this instance signs refer to external representations. Presmeg (1997) suggests that when one recognizes the structure of the system he or she engages in, explains this structure to others by such means as encoding it in a diagram or applying some overarching framework then mathematics exists. So while semiotics is commonly used to construct links between cultural and historical practices and mathematics (Presmeg, 1997; Radford, 1997) it also assists us to understand classroom discourse in mathematics (Saenz-Ludlow, 2001), and children's understanding of the relationship between words. Radford (2001) claims that signs play a dual role in cognition. On one hand they are a means of dealing with the object of knowledge while on the other they are social where 'we find a niche for meaning'. In semiotics neither the cognitive domain of the individual nor the social interaction is primary. Both coexist and support the evolving construction of meaning. Signs go beyond mirroring the inner cognitive processes. They also can be tools of actions as determined by the contextual demands in which individuals are located (Radford, 2001). Sign interpretation is a personal process with some students being unable to go beyond the physical characteristics of the sign (the external representation). Peirce (1960) believes that the sign relation is inherently triadic, linking an object, a representation and an interpretation so that the object determines the representation and in turn determines the interpretation. Thus semiosis involves the process of going beyond particular signs to more and more complex representations incorporating new signs and generalizations. The signs explored in this paper are the words commonly associated with addition and subtraction problems, and how young children interpret these signs to produce new signs and generalizations.

The specific aim of this paper is to delineate young children's understanding and interpretation of words commonly used in addition and subtraction. This paper also investigates how children in the study related words to each other and the overarching framework they applied when encoding these words, their representations and interpretations.

Participants and Procedures

The sample was comprised of 87 children from four elementary schools in low to medium socio-economic areas in Australia. The children are all participants in a three year longitudinal study investigating early literacy and numeracy development. The average age of the sample was 8 years and 6 months and all had completed the first three years of formal education.

The methodology chosen for this research is based on the work of Luria (1981). Berenson (1997, p. 2) claims that "Luria's method of defining, comparing, differentiating and classifying words provides powerful instruments for assessing students' understanding." In this process children are given a selection of words and are asked to sort them into groups, give one name to each group, and communicate the connections between the words and why they belong in the group. The children are encouraged to sort the words into groups up to three times, each time creating fewer groups. This provides ample opportunity to explore the multiple meanings children have for the 'words'.

A scan of syllabus documents and common classroom texts elicited fifteen words that are commonly used in the early years with addition and subtraction. An expert, who was asked to identify all the words he believed children commonly experienced when learning addition and subtraction in the early years, verified the appropriateness of the selection. Figure 1 delineates the fifteen words chosen for this study. Each was represented on a card and all fifteen cards were given to the children to sort.

The card sorting activity occurred in a one to one interview situation. Each child was asked to read out loud the words on the cards and were questioned if they had heard these words before. They were then asked: Can you sort the cards into groups? What name will you give each of your groups? Why does each word belong in your group? Can you sort the cards into fewer groups? Given the age of the children (8 to 9 years), most had difficulty sorting the cards more than twice. The interviews were audiotaped and transcribed for analysis.

The children were also asked to complete a written test. This test consisted of seven typical addition and subtraction problem. These problems involved either translating words into symbols or providing a story for problems written in symbols. This was believed to give insights into children's capability of changing from the word registrar to a symbol registrar and vice versa. One of the word problems was Chris bought two presents. They cost $15.75 and $9.15. How much did they cost altogether? Also a symbol to language problem was Tell me a story about 9-5. These tests were marked and each child was assigned a score out of seven. All the problems chosen for the test were selected from either published diagnostic tools or state wide testing programs used to ascertain children's understanding of arithmetic (e.g., Booker, 1994; Queensland Year 3 test: Numeracy, 1999).

Data Collection and Analysis

Peirce's triad of object, representation and interpretation was used to guide the analysis of the data. In this instance, the objects were the words on the cards; the representations were the names attached to each group of cards; and, the interpretations were the reasons given for the inclusion of particular cards in particular groups. The results discussed here relate to the final card sort that the children performed. Initially, many experienced difficulty sorting the cards. It became evident that the more they sorted the cards the more the children became familiar with both the cards and the task itself. The resorting helped them clarify what each card meant and how it could relate to other cards in the mathematical registrar. This was greatly assisted by the request to justify each card's inclusion in each group, forcing the child to go beyond the signs and identify new signs, representations, and interpretations.

The children tended to sort the cards into three common groups. As they sorted the cards children were asked to give a name to each group. They were reminded that the name had to reflect the meanings that they assigned to the words in each group. Initially some volunteered names such as more, away and altogether. As they sorted their cards for a second time they not only refined the groups they made but also the names they assigned to each. Commonly they tended to settle on three 'new signs' to represent each group. These included add or plus, away or subtract, and equal or total. For the purpose of analysis the names ADD/PLUS, TAKE AWAY/SUBTRACT, and EQUAL/TOTAL were assigned to these three groups.

Trends in the ADD/PLUS group

Forty six children identified an ADD/PLUS group. There was a wide range of cards included in this group with the most common three being add, more, and many more. Thirty-two children included these three cards in this group. Some typical arguments for the inclusion of these three were:

A: Add means bigger and when you add you get more.... When it's many more you are going to get many more. Many more means lots more. It is bigger.

B: Add, more and many more all give higher.

E: Many more is at the end of an add. There are 14 birds and 20 more came along [using her hands to imitate a joining process] then how many more are there.

M: 173 add 253 [using a calculator] equals 522 and that is many more.

For young children it seemed that a common understanding of ADD is that it results in a 'bigger' number. No one identified many more with comparison situations. Even though Ella (E) articulated such a problem, the joining idea of addition seemed to remain prevalent in her thinking. Altogether caused confusion for many of the children. Eleven included it in the ADD/PLUS group, and twelve in the EQUAL/TOTAL group. Some reasons given were:

C: I included altogether [in the add group] because it usually is a bigger number. It is always a bigger number when you are adding.... There are 2 lollipops and 4 balloons how many are there altogether. [Hence its inclusion in the ADD/PLUS group].

M: Total and altogether mean equals. When you do the answer it is altogether. [Hence its placement in the EQUAL/TOTAL group.]

One child confused altogether with all together and thus saw it as belonging to ADD

Well there are 2 birds and they fly away and then they might find other birds and then they would be all together [the idea of joining two groups].

Trends in the TAKE AWAY/SUBTRACT group

Seventy children identified a takeaway or subtract group in their card sorting. Their initial representation of this group included words such as away, subtraction, left over and different size. The two most common cards placed in this group were take and takeaway (sixty-two children), with a common reason given "they both start with the same word". Forty-four of these also included gives away and twenty-five children included subtract.

D: Take, takeaway, and gives away mean that you take away a number. It is the smaller group you get less.

The three words that caused most difficulties for the children were difference, many left and compare. While five children included difference in the TAKE AWAY group, many refused to include it anywhere. For some, difference equated with different. For example, "Difference means that the two numbers are different".

Only five children included many left in TAKEAWAY/SUBTRACT group. A common reason given was "Many left is at the end of a takeaway. In a takeaway when you are at the end of it you have to see how many are left". Three included many left in the EQUAL/TOTAL group 'because it is at the end of a sum just like equal, total and altogether. The introduction of trading in takeaway problems seemed to interfere with one child's understanding of many more. "Many more is a takeaway because when you have 54 and takeaway 29 you can always go and get many more [pointing to the 5 and indicating the process of trading].

The confusion with compare seemed to stem from classroom activities and actions used to explore what is meant by compare.

C: Compare is like there is one group over here and another group over there and comparing is like comparing the groups. It is like adding them together.

Mi: Compare is in the PLUS group because when you compare a number it has more.

S: When you compare groups you put them in pairs or something--you join them.

B: 23 and 52 and compare them together you get 76.

V: When you have two groups with say ten things in each, you compare them to each other, you put them together (link them up). Comparing and adding are the same.

Again the action of joining or aligning groups seemed to override compare as involving an understanding of subtraction or ascertaining how many more there are in the larger group. The aligning of the groups seemed to visually evoke the action of joining and hence the common interpretation of compare as addition.

Trends in the EQUAL/TOTAL group

Thirty four children identified this group. The two most common cards placed in this group were equal and total (29 children) followed by altogether (19). A typical reason given was "these three make what it all equals--altogether is when you do the answer it is altogether". The words seemed to trigger an inadequate conceptualization commonly expressed in language as 'find the answer'.

While many children in the same recognized three general classes or representations "strands of the same object" (Parmentier, 1985, p. 28), their interpretations of these representations tended to be very different. These differences were not only delineated in the cards they included in each group but also by the differing reasons they gave for their inclusion. For example, the inclusion of altogether in the ADD/PLUS group was because 'it usually is a bigger number' compared with 'the answer is altogether'. It is the interaction between the sign, representation and interpretation that begins to give us some insights into how young children construct their understanding of mathematical words. In these understands it seemed that for many both the end product and the action associated with modeling the operations greatly influenced these understandings. For example, addition gives a bigger group and subtraction gives a smaller group, and when you model two groups the action is usually joining rather than comparing. The next section explores the commonalities and differences in representations and interpretations of five high achieving students.

Relationships between card sorting activity and successfully solving addition and subtraction problems

Five children scored 7 out of 7 on the written test. According to these results, these children are considered to be 'good' at mathematics as all items on the written test were selected from a state-wide testing program or a well recognized diagnostic numeracy tool. The results of their card sorts provided some further insights into the struggle that young children experience when interpreting the words commonly used in addition and subtraction. Not only did the names of their groups vary but also the words they included in each. All sorted the cards into two common groups, ADD/PLUS and TAKEAWAY/SUBTRACT. Four children identified a third group, an EQUAL or TOGETHER group. Table 1 summarizes the words each included for the ADD/PLUS group.

All five children used either ADD or PLUS to represent their group and included add and many more in ADD/PLUS group. When asked why many more fitted into this group Liam commented

  well if Sarah had 5 lollies and Sam had 3 then how many more lollies
  did Sarah have than Sam. That's a plus.... Oops no it is a take away
  [but still left the card in the ADD group].

By contrast, Elizabeth said

  When you plus you get many more. Like six plus one gives many more
  than what you had before.

Mitchell and Ben suggested in their interviews that "when you add you get a lot more--many more". It seemed that nearly all of these students possessed an inadequate conceptualization of many more and had interpreted it as addition rather than many more suggesting the action of comparison. This could also reflect the fact that the words were given out of mathematical context and the interpretation the children tended to give was literal. While this could be seen as a limitation of this study, its contribution lies in the fact that the context in which the sign is commonly situated seems to influence its representation and interpretation. Thus as we assist young children in linking signs to representations and interpretation we need to pay explicit attention to the context in which the signs are situated, especially when the context varies.

Three included altogether in the ADD/PLUS group. When asked why they had put it in the group Elizabeth said, "it might be in the equals group as well" and this is the group that Ben included it in. "Altogether is just the same as total. You are adding them together". The other two students reiterated Ben's interpretation and said "when you plus you get something altogether". Charlotte's comments about sum, equal and total were interesting. "Maths is to do with sums and adding sums together to make totals--what it equals". She was already beginning to articulate a very narrow view of mathematics, suggesting that this was the most common type of classroom activity that she was experiencing.

The TAKEAWAY group was another common group delineated by the five children. Table 2 summarizes the words each child included in this group.

Four children used TAKEAWAY to represent this group. The other one represented the group by SUBTRACT. Liam was able to clearly articulate reasons for his choices in what he referred to as the subtract group.

  When you have say 7 and 14 then the difference is say 14-7 equals
  something and if you compare 7 and 14 it would be a difference of 7.
  For gives away, if Byron had 5 lollies and he had 4 friends and if
  each friend had to have the same amount of lollies as Byron and each
  other how many would Byron have to give away. He would give away 4
  lollies. If Sam had 15 marbles and he lost 5 how many would he have
  left. And if Sam had 15 marbles and Adrian had 10--how many less
  marbles did Sam have?

Ben's conceptualization of the problem seemed to activate a rule that had more to do with the setting out of the sum rather than subtraction or take away.

  Less is usually--it is underneath it. There is a bigger number in a
  sum and there is usually a number that is less than the bigger one at
  the bottom of the sum [reflecting the vertical format for subtraction]

Mitchell called this group the LESS group.

The placement of compare varied from child to child and reflected many of the difficulties that the whole group was experiencing with this word. Elizabeth placed it in the EQUAL group along with equal, total, altogether and many left saying "you compare numbers like 15 and 16 and you compare them and that would be 31. These words are all used at the end of sums--they are what it equals". For Mitchell and Liam, compare belonged in the TAKE AWAY group but both experienced difficulty in explaining why they felt it belonged in this group. Ben grouped compare with total and altogether and called this group the TOGETHER group offering, "when you are comparing you are comparing two numbers together", indicating confusion between altogether and together.

The final group was the EQUAL or TOGETHER group. Table 3 summarizes the cards included in this group

Liam, Ben and Elizabeth all interpreted equals in terms of an action to find the answer. Their responses included "It's what it equals you need to find what it equals", "A sum always has an equals. You need to find the answer" and "The answer equals the answer".

As compared to the rest of the group, the performance of these five children on this task seemed different in terms of their initial engagement in the task and the ease in which they grouped the cards, represented these groups with an overarching name, and gave reasons (interpreted) why particular cards (signs) belonged in each. Four of the five children's initial sorting of the cards resulted in three distinct groups. While other children also initially sorted the cards into three groups, their groups tended to only consist of two or three cards with the rest remaining unsorted. When these five children were asked to resort their groups, for most the original groups remained the same. Charlotte collapsed her initial three groups into two. The differences between the five children lay in the cards they included in each group and the reasons they gave for their inclusion. While all could articulate for them 'valid' interpretations, these interpretations mirrored many of the misconceptions held by the whole group.

In spite of all the differences in interpretation and explanations that the five children gave for the card sorting activity, they managed to successfully navigate all seven problems in the written test. This could reflect the types of problems chosen for the written test. While all the problems were selected from either published diagnostic tests or state wide testing programs, there was limited use of the words used in the word sorting activity. For example, two problems had altogether as end words and compare, difference and equal did not appear in any of the problems.

Discussion and Conclusion

The objectification of the signs appears to be accomplished in three different ways. First, attention to the linguistic intention of the words played a key role in producing and explaining 'new signs' and generalizations. For example, many children explained that take and take away or more and many more belonged together because of the commonality of take or more. They seemed to fail to understand the nuances used in relation to mathematics especially when more than one word contributes to the meaning. Some children also exhibited confusion between difference and different, and altogether and all together. While the situations they used to describe these words had elements of the mathematical registrar, (e.g., you have 5 birds here and 6 over there) their linguistic interpretation of these words resulted in a conceptualization that seemed to miss the subtleties of the words (i.e., these groups are different rather than the difference between these two groups is 1).

Second, from the results it is conjectured that the position of the word in routine addition and subtraction word problems also seemed to bring to mind a different reason for producing new signs. In this instance some children 'objectified' some signs as being 'end words', representing an action of finding the answer. This is illustrated by the responses given by Liam, Ben and Elizabeth to why they included equal in the EQUAL or TOGETHER group. Also three children grouped many left with equal, total and altogether "because they are all at the end of a sum". Another common misconception that supported this generalization was "These words are at the end of the story so they go together (equal, altogether, and total)". It seems that some children have experienced an array of classroom experiences that place equal as an end word and thus its relationship to total. The concern is that for algebraic reasoning children also need to see equal as equivalent. The challenge is to develop a conceptualization that calls to mind equal associated with equivalent situations. At present for these children the predominant context appears to be equal related to addition and the common action of joining and finding the total. The difficulties that young children experience with the equal sign have been well documented (e.g., Saenz-Ludlow & Walgamuth, 1998). Young children generally believe that equal means "carry out the operation". It has been conjectured that this could reflect classroom symbolic activities that commonly have only one number after the equal sign (Carpenter & Levi, 2000; Warren, 2001) and when presented with a problem such as 2 + 3 = 3 + 2 children believed that the statement is false as there should only be one number after the equal sign.

This present research indicates that understanding equal as equality is also influenced by the interpretation that children associate with this sign. Many children gave interpretations where equal was at the end of the sentence and required the action of finding the answer. Many texts and word problems reinforce this understanding. While there is agreement that mathematics can only present decontextualized problems after the student has been sufficiently immersed in similar problems in context (Bickmore Brand, 1990), we must question the types of contextualized problems the children are experiencing. With the case of equality, how difficult is it to talk about a variety of contextual situations or does contextualizing of some situations such as 2 + 8 =? + 5 add to the complexity of the concept and in fact make the problem inaccessible? Tsamir and Bazzinni (2001) contend that past research has paid little attention to students' conceptions of inequality. It could be through the exploration of inequality and related signs, representations and interpretations that meaning for equality is found.

Third, the individuals' actions commonly associated with some words also seemed an important source for assigning meaning. In the early years, children's arithmetic experiences generally begin with using concrete materials to represent actions in realistic contexts. These experiences are commonly accompanied by teacher directed discourse to describe the situation under consideration. For example, 'there are 2 birds in the tree and 3 more join them' is accompanied by the action of initially placing 2 blocks in an imaginary tree and then another 3 blocks to represent the birds that have come to join the original 2. 'How many are there altogether' is linked to counting the 5 blocks. Children are required to abstract the isomorphic structure between the actions and the words and 'objectify' add as a joining idea. While these processes are believed to assist understanding, as indicated by this study, many children have difficulties in linking all these ideas, and sometimes perhaps the ideas become so powerful they seem to interfere with further learning. For example, Ella's interpretation of many more included two groups but her hands movement indicated that when you have two groups you join them (not compare them). Thus the powerful joining action of addition seemed to preclude a conceptualization of many more in a comparison context. Elizabeth's (one of the high achieving students) comment about compare also supports this conjecture "you compare numbers like 15 and 16 and then that would be 31".

Each child seems to bring different understandings to the signs they meet and these understandings seem embedded in past experiences. For many, the action associated with addition seems to activate an interpretation that links addition with bigger and the action of joining. Compare also evokes an action of 'joining'. A common classroom practice for compare is 'aligning' the two groups in one to one matching and counting the unmatched entities. Many children seemed confused about this subtle difference and the action of joining seemed to be so overwhelming that their interpretation suggested that compare was also about addition.

Sign interpretation is a personal process. In some instances it seems that children are unable to go beyond the written mark, the literal interpretation. The inherent triadic nature of sign relations (object, representations and interpretation) are exhibited in this research. The words presented in this research induce an interaction between these three dimensions. The meanings of words and symbols inevitably grows, and abstraction is believed to be the process of leaving behind the concrete mathematical contexts (Saenz-Ludlow, 2001). Our role as teachers is to respond to these interactions, ensuring that both the classroom discourse and mathematical contexts are rich and representative of the full range of understanding so that the original intended meanings are reached by children participating in the dialogue. Saenz-Ludlow (2001) refers to this as engaging in interpreting games, continually presenting different mathematical contexts in which the same words present different meanings (e.g., the use of equals as a command to find the answer, a clue to find the missing number, or represent equivalent situations), and using language and questioning to group common words with similar meanings (e.g., difference, subtraction, take away, gives away and compare).

Add   Subtract  Take away   Difference  Many left
More  Less      Compare     Take        Gives away
Sum   Total     Altogether  Equal       Many more

Figure 1. The fifteen words used in the interview.

TABLE 1 Words included in the ADD/PLUS group

Child      Add      Many more  More     Altogether

Liam       [check]  [check]    [check]  [check]
Elizabeth  [check]  [check]    [check]
Ben        [check]  [check]    [check]
Mitchell   [check]  [check]    [check]  [check]
Charlotte  [check]  [check]             [check]

Child      Sum      Equal    Total

Liam
Elizabeth
Ben
Mitchell
Charlotte  [check]  [check]  [check]

TABLE 2 Words included in the TAKEAWAY group

Child      Take     Takeaway  Less     Givesaway  Subtract

Liam       [check]  [check]   [check]  [check]    [check]
Elizabeth  [check]  [check]   [check]  [check]    [check]
Ben        [check]  [check]   [check]
Mitchell   [check]  [check]   [check]  [check]    [check]
Charlotte  [check]  [check]   [check]  [check]    [check]

Child      Many left  Difference  Compare

Liam       [check]    [check]     [check]
Elizabeth
Ben        [check]
Mitchell   [check]                [check]
Charlotte  [check]

TABLE 3 Words included in the EQUAL TOTAL group

Child      Total    Altogether  Compare   Equal    Many left  Sum

Liam       [check]                        [check]
Elizabeth  [check]  [check]     [check]   [check]  [check]
Ben        [check]  [check]     [check]
Mitchell   [check]                        [check]             [check]

REFERENCES

Adler, J. (1995). Dilemmas and a Paradox--Secondary Mathematics Teachers' Knowledge of Their Teaching in Multilingual Classrooms. Teaching & Teacher Education, 11(3), 263 274.

Arzarello, F. (1998). The role of natural language in prealgebraic and algebraic thinking. In H. Steinbring, M. Bussi & A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 249-261). Reston, Virginia: National Council of Teachers of Mathematics.

Berenson, S. (1997). Language, diversity, and assessment in mathematics learning. Focus on learning problems in mathematics. 19(4), 1-10.

Bickmore Brand, J. (1990). Implications from recent research in language arts for mathematical teaching. In J. Bickmore Brand (Ed.), Language in mathematics. Carlton South, Victoria: Australian Reading Association Inc.

Booker, G. (1994). Booker profiles in mathematics: Numeration and computation. Melbourne, Vic: The Australian Council for Educational Research.

Bums, A. (1992). Teacher beliefs and their influence on classroom practice. Prospect, 7(3), 57-65.

Carpenter, T.P., & Levi, L. (2000). "Developing Conceptions of Algebraic Reasoning in the Primary Grades." Paper presented at the annual meeting of the American Educational Research Association, Montreal.

Gal, I. (1999). Links between literacy and numeracy.. In D. Wagner & R. Venezky & B. Street (Eds.), Literacy: An international handbook (pp. 227-231). Colarado, USA: Westview Press.

Halliday, M.A.K. (1979). Language as social semiotic: The social interpretation of language and meaning. London: Edward Arnold Publishing.

Lopez Real, F. (1997). Effect of different syntactic structure of English and Chinese in simple algebraic problems. In F. Biddulph & K. Carr (Eds.), Proceedings of the 20th annual conference of the Mathematics Education Research Group of Australasia. (pp. 317-323). Rotorua: MERGA.

Luria, A. (1981). Language and cognition. Washington, DC: V.H. Winston.

MacGregor, M. (1993). Interaction of language competence and mathematics learning. In M. Stephens & A. Waywood & D. Clarke & J. Izard (Eds.), Communicating mathematics: Perspectives from classroom practice and current research (pp. 51-59). Hawthorn, Victoria: The Australian Council for Educational Research.

MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. Journal for Research in Mathematics Education, 30(4), 449.

MacGregor, M., & Stacey, K. (1999). A Flying start to algebra, Teaching Children Mathematics, 6(2), 78-85.

Parmentier, R. (1985). Signs place in medias res: Peirce's concept of semiotic mediation. In E. Mertz & R. Parmentier (Eds.), Semiotic Mediation: Structural and psychological perspectives. Orlando: Academic Press.

Peirce, C.S. (1960). Collected papers. Cambridge, Mass: Harvard University Press.

Presmeg, N. (1997). A semiotic framework for linking cultural practice and classroom mathematics. Ed 425-257.

Queensland School Curriculum Council (1999). Queensland Year 3 test: Numeracy. Brisbane, Qld: Government Printers.

Radford, L. (1997). On psychology, historical epistemology, and the teaching of mathematics: Towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17(1), 26-33.

Radford, L. (2001). On the relevance of semiotics in mathematics education. Paper presented at the annual meeting of the Psychology of Mathematics Education, Utrecht, Netherlands.

Saenz-Ludlow, A. (2001). Classroom mathematics discourse as an evolving interpreting game presented. Paper presented at the annual meeting of the Psychology of Mathematics Education, Utrecht, Netherlands.

Saenz-Ludlow, A., & Walgamuth, C. (1998). Third graders interpretation of equality and the equal symbol. Educational Studies in Mathematics. 35(2), 153 187.

Tsamir, P., & Bazzini, L. (2001). Investigation the influence of the intuitive rules in Israel an din Taiwan. In M. Heuvel Pahhuizen, (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education. Utrecht University: Freudenthal Institute.

Vygostky, L. (1978). Mind in society. Cambridge, MA: Harvard University.

Vygostky, L. (1986). Thought and language. Cambridge, MA: MIT.

Warren, E. (2001). Algebraic understanding and the importance of operation sense. In M. van den Heuvel Penhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education, 4, 399-406. Utrech, Netherlands.

Elizabeth Warren

Australian Catholic University