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Watson, 2009; Zieffler & Garfield, 2009), and students’ preconceptions of the terms related to statistics (Kaplan, Fisher, & Rogness, 2009). A literature search of the titles, keywords and abstracts of all papers in the Journal of Statistics Education and the Statistics Education Research Journal suggest that none had a primary focus on investigating and exploring students’ use and understanding of the symbolic system of statistics although one paper did draw upon the onto-semiotic tradition to describe student errors related to representations of the mean and median (Mayen, Diaz, Batanero, 2009).

The research on students’ conceptual understanding of statistical concepts has, thus far, avoided discussion of the importance of representation. Yet, the onto-semiotic research tradition proposes that “Representations cannot be understood on their own. An equation or specific formula, a particular graph in a Cartesian system only acquires meaning as part of a larger system with established meanings and conventions” (Font, Godino, & D’Amore, 2007, p. 6). The implication is that, without considering the effects of the different representations of the concepts under study, researchers are being, to use Font, Godino and D’Amore’s term, naïve in their study

of students’ thinking. In particular, they argue:

In the onto-semiotic approach, the introduction of the unitary-systemic duality in the analysis of the representations enables us to reformulate the naïve vision that there is one ‘same’ object with different representations. What there is, is a complex system of 15TH Annual Conference on Research in Undergraduate Mathematics Education 607 practices in which each one of the different pairs object/representation (without segregating them) makes possible a subset of the set of practices that are considered to be the meaning of the object (p. 7).

Within the realm of statistics, even when the object under consideration seems relatively simple, ∑ x ) which such as the mean, there are often multiple symbolic representations (such as x = n are interchangeably used, by statisticians without consideration of any other type of representation (graphical, verbal etc). When moving to a more complex idea such as the standard deviation of a sample mean, the individual paired relationships between object and € representations become even more complex due to a layering of representations. These different possible pairs can arguably convey entirely different meanings of the same object.

Arcavi’s (1994) seminal article on symbol sense in mathematics, while not explicitly situated within the tradition of onto-semiotic adopted the position that symbolic understanding and fluency was an important component in knowing and doing algebra. That is, fluency with particular types of mathematics required fluency with a broad range or presentations, including

the symbolic. In particular, Arcavi claimed that students should, at minimum:

• Know how and when symbols can and should be used in order to display relationships

• Have a feeling for when to abandon symbols in favor of other approaches

• Have an ability to select a representation and, if necessary, change it

• Understand “the constant need to check symbol meanings while solving a problem, and to compare and contrast those meanings with one’s one intuitions or with the expected outcomes of that problem” (p. 31).

It is certainly true that algebraic skills do support students’ ability to do and understand statistical concepts (Lunsford & Poppin, 2011). As a result, we argue that there are reasonable analogues to Arcavi’s habits and skills in the realm of probability and statistics that are important to consider.

1. Research Aims This theoretical report aligns itself with Arcavi’s (1994) work and the tradition of ontosemiotic research in mathematics education (Font, Godino, & D’Amore, 2007) and is situated in

the context of statistics education. This report will:

• articulate a notion of symbol sense in statistics

• explain the importance to student understanding of the development of symbol sense.

The goal of this work is to guide both research and curriculum design efforts for introductory undergraduate statistics courses.

2. Theoretical Perspective Symbolic representations are regarded as particularly critical due to Hewitt’s (1999, 2001a, 2001b) distinction between arbitrary and necessary elements of the mathematical system.

Hewitt notes that names, symbols and other aspects of a representation system are culturally agreed upon conventions and, while many may feel sensible, when a member of a community of practice has an understanding of the culture, “names and labels can feel arbitrary for students, in the sense that there does not appear to be any reason why something has to be called that particular name. Indeed, there is no reason why something has to be given a particular name” (1999, p. 3). Hewitt continues by differentiating between those aspects of a concept used by a community of practice which can only be learned by being told and then memorizing, which he labels arbitrary, and those which can be learned or understood through exploration and practice, 608 15TH Annual Conference on Research in Undergraduate Mathematics Education which he labels necessary. Additionally he notes that for students to become proficient at communicating with established members of the community of practice, they must both memorize the arbitrary elements and correctly associate them with appropriate understandings of the necessary elements.

Eco (1976) gave the term semiotic function to describe the dependence between a text and its components and between the components. The semiotic function relates the antecedent (that which is being signified) and the consequent sign (or that which symbolizes the antecedent) (Noth, 1995). When considering the statistical community and the representation system in use within that community, have defined a complex web of semiotic functions and shared concepts that “take into account the essentially relational nature of mathematics and generalize the notion of representation: the role of representation is not totally undertaken by language (oral, written, gestures, …)” (Font, Godino, & D’Amore, 2007, p. 4). Throughout this paper, we recognize the inherent arbitrary nature of much of the symbolic system of statistics and draw on the notion of semiotic function as a means of linking a particular representation with the relevant concept. In doing so, we articulate specific linkages that students should be developing and describe some of the difficulties and potential pitfalls of the symbolic system.

3. A Notion of Statistical Symbol Sense.

Many of the habits and skills that Arcavi (1994) described have a natural analog in statistics. Most important of these is knowing how and when symbols can and should be used.

In mathematics a symbol typically represents an unknown or is defined to represent a single mathematical concept; however, in statistics symbols often carry multiple layers of meaning.

For example, both x and µ are well defined as an arithmetic mean; however, each has a second layer definition defining what type of data set the arithmetic mean comes from; x is the mean of a set of sample data and µ is the mean of a set of population data. This additional layer of information is crucial in displaying relationships, that is, should be encoded in the semiotic € function linking the representation (symbol) and the concept of mean, and should be a part of a € student’s statistical skill set at the end of a course.

Arcavi also recommends knowing when to abandon symbols in favor of other approaches. This has a non-mathematical application to statistics. While statistical procedure revolves around the relationship between symbols and their relationship to a sample and a population the practical use of statistics is much less technical. In many instances statistics is the tool used to explain or reason about something in a different discipline such as psychology or biology; disciplines that are not necessarily rooted in mathematics. It is important to be able to abandon descriptive symbols in favor of concise statements such that a hypothesis or a conclusion can be interpreted without understanding what a symbol represents. A student should not only be able to abandon formal symbol representation, but be able to “translate” symbolic statements into something easily understood by all.

Finally, Arcavi states that a constant check of symbol meanings during problem solving is needed. In statistics, the multi-layered meaning of symbols makes this important.

Additionally, there are general mathematical symbols that are mathematical operators; however, in statistics it is a general rule that a Greek symbol represents a population summary and a Roman symbol represents a sample summary, but there are times when Greek and Roman symbols are nearly indistinguishable such as with Nu. A student might see N = 25, and not understand why one is to use capital N for a population and lower-case N for a sample while a statistician might be surprised that the student does not recognize Nu! Thus, from the different perspectives, a symbol might be completely reasonable or seemingly arbitrary. This continues 15TH Annual Conference on Research in Undergraduate Mathematics Education 609 with inclusion of symbols such as “∑” as operators, rather than conveying information about a population, will sometimes confuse students and makes these general rules less clear than intended.

3.1 An expansion of Arcavi’s list.

The following section will briefly outline a few ideas that might be understood as forming part of a statistical symbol sense. It is important that students have a clear understanding of relevant terms and be able to correctly associate each term with the most

appropriate symbol. Beyond that, students should:

• Understand, in the context of a given problem, which symbols represent constants (even if unknowable) and which represent values that can vary.

• Understand that symbols which are constant for a given problem can also be understood as varying across problem contexts.

• Possess a feeling for when symbols should be used to display relationships and when visual representations better convey appropriate information.

• Demonstrate an ability to read symbolic expressions for meaning, both in the context of the problem, while also connecting them to their abstracted.

• Consistently check the meaning of the symbols against the problem and with their own intuition.

• Possess an understanding of the difference between different symbols that represent the same basic concept (such as a sample mean versus a population mean).

To illustrate these, we will use the standard error of a sample mean. In explaining how this case illustrates aspects of a statistical symbol sense, we will concentrate on two of the bullet points above; the need to understand constants and variables, as well as the ability to read expressions for meaning.

Because the standard error of a sample mean requires the creation of a sampling distribution, it would be helpful if students had a dynamic image in their heads of samples being created from the original population, each sample being of size n. Then, for each sample, the sample mean is computed and the distribution is created. This distribution also has a fixed mean and standard deviation. The mean of the sampling distribution is at the same value as the mean of the original distribution, that is, a subtle point that is too often glossed over. The mean of all possible sample means is the same as the mean of the original population. The standard deviation of the sample means is measuring the spread of the sample means of size n from their mean. That is, this formula is meant as a measure of how spread out a population of sample means is. In order to make sense of this formula, it requires the students to have constructed a mental landscape with the ability to operate on at least two levels of abstraction; one is relatively low and is the original distribution, while the second is relatively high and asks students to contemplate the distribution of all possible sample means of size n where the individual samples are drawn from the original distribution. Let’s now describe some of the reasons that understanding this formula may be problematic for students, and, how a statistical symbol sense would help.

3.2 Reading of symbols.

When students confront the equation σx = σ, one of their first realizations should be n the formula mixes notation for populations and samples (σ and n, respectively). As a result, students need to have a decision rule that allows them to understand what is being described; is this formula describing a sample? A population? In fact, this formula is describing an entirely € new distribution, one that is distinct from the original population, demands consideration of a 610 15TH Annual Conference on Research in Undergraduate Mathematics Education sample of size n, and is based upon the old distribution. In order to realize this the students need to recognize that when elements related to a population and sample are mixed, the students need to realize that the new symbol must be describing a sampling distribution.

The students should also look at the equation and read in terms of how the standard deviation of the sample mean compares with the original standard deviation. Students should ask themselves what division by the square root of n does, especially as n varies. Students should ask, what happens when n is 1? Students should understand that this would recapitulate the original distribution, both because each ‘sample’ would be exactly one individual (meaning that each individual in the population is then in exactly one sample) and because the symbols show that the square root of 1 is 1, and then the standard deviation of the sample means is the same as the standard deviation of the population because of division by 1. Then, the students should be able to explain how the value of the standard deviation of the sample means will change as the sample size increases by nothing that sigma is a constant and, then, division by an increasing value will cause a corresponding decrease in the final result. The students should imagine the distribution (the graphical representative) collapsing about the mean in a dynamic way.

Insert Diagram 1a: A normal distribution and the distribution of sample means from samples of size 2, 10 and 100.

Insert Diagram 1b: A normal distribution and the distribution of sample means (n = 2, 10 and

100) scaled towards the parent distribution

3.3 Reading symbols for meaning related to the problem.

A student must be able to answer “What can vary?” and “What’s constant, even if unknown?” to fully understand a problem. In the context of the formula above, students should be asking themselves these questions. Yet, the answers require a non-trivial ability to negotiate between contextualized and generalized understandings. At the most general, both σ and n can be understood as varying, the formula is applicable to all distributions, and, therefore, any sigma.

But, in most situations that the students encounter, they should be thinking in terms of a specific underlying distribution, which means that σ is fixed; although, it may be unknown (which the students should be able to discern). Yet, we want the students to understand that once the population, and thereby σ is fixed, that by changing sample sizes they create a large number of different sampling distributions. That requires students to understand the sample size n as able to vary and we should teach them to think this way.

To liken this to an element of algebra, when students consider quadratic functions, they should understand that f (x) = ax 2 gives rise to a quadratic, and, that for a particular instance, a is fixed, but we also want them to understand that a can vary and what that variation does to the function. Yet, they also need to be able to proceed into further contextualized problems where n has also been fixed and they, then, need to be able to picture the shape of the distribution and € describe what effect n has on the shape of the distribution. Students might do this by drawing an appropriate picture of the distribution with ranges variation, as described by differences from the mean, marked.

The example of the standard error of a sample mean is an example of a concept that, when understood, makes understanding expected results straightforward. It is this concept of what is expected that is a building block of statistical inference. Students often dive into 15TH Annual Conference on Research in Undergraduate Mathematics Education 611 inference without conceptual understanding of what “should” happen under the premises provided. The ability to read expressions for meaning is a skill we should expect of statistics students. If a student has information about σ, then that student should have the ability to infer what outcomes for the sample mean are most common, and how they vary. This skill, directly leads to the concept of “unlikely events” and a student can then infer what is likely versus what is unlikely by only understanding what the premise of the problem.

3.4 On visualization and selection of the display.

One of the challenges for students in understanding the sampling distribution is making sense of what individuals represent. They typically begin a statistics class by exploring data where an individual is a single measurement from one member of the population under study.

This might be a heartbeat, count of siblings, or Likert scale rating, but, each number could be understood as describing one individual and often a person. That is, a single thing that could be visualized. When students start to consider a sampling distribution, the individual members of the population are now samples, and the measurement of each individual that we are considering is a mean. That is, we have asked the students to operate on, as an individual, this concept that was originally introduced as a collection of individuals.

When we talk about visualizations of distributions, we might want students to visualize the individuals in the original distribution being selected into the sample. Then, they need to see the sample mean becoming an individual in the sampling distribution. Let us look at a diagram that might depict these ideas.

Insert Diagram 2a: A normal distribution with a sample of 13 plotted and the meanof that sample identified.

Insert Diagram 2b: The distribution of all sample means (of size 13) from a normal distribution with the sample mean of the 13 points from Diagram 2a shown.

4.0 Pointing towards more advanced statistical concepts Finally, we will to undergird this discussion, with a few extensions, outlined here and to be discussed in more detail in the presentation and subsequent papers. We first note the complications in understanding that result from estimated constants, and, we’ll discuss the role of the standard error in the Central Limit Theorem (CLT) and how the coordinated understandings that we have described above are essential to understanding the CLT.

Oversimplifying a bit, the CLT is a weak convergence theorem that states the conditions under which a sampling distribution approaches normality. It is one of the most important results in probability and lies at the heart of much of the inferential statistics taught in an introductory course. For the purposes of an introductory statistics course, Moore offers the following statement, “as we take more and more observations at random from any population, the distribution of the mean of these observations eventually gets close to a normal distribution.

(There are some technical qualifications to this big fact, but in practice we can ignore them.)” (2001, p. 488). The normal distribution of sample means that is being defined has the standard error as one of its two parameters.

There are instances when a sampling distribution for the sample mean is desired, but the parent distribution is unknown. In this situation one is able to apply the central limit theorem and obtain an estimated sampling distribution. Without access to information from the parent distribution, one must estimate the constants based on information gathered in the sample. These 612 15TH Annual Conference on Research in Undergraduate Mathematics Education estimated constants are denoted by a “hat” and are able to vary. Understanding when a sampling distribution is reported verses when an estimated sampling distribution is reported and the differences between them is an important skill that is confusing due to similar looking parameters as both are denoted with marks above the symbol.

A further generalization is the standard error’s importance to a student’s understanding of the Central Limit Theorem (CLT). A primary goal in an introductory statistics course is to convey an understanding of the CLT. The relationship between a distribution’s variance and the standard error that defines the distribution of its sample means is a fundamental component in inference and CLT. A student must be able to ascertain if the true standard error is being used or an estimate and be able to understand and convey how that affects any inferential conclusions.

5.0 Summary The list of behaviors and understandings (including semiotic functions) proposed above is knowingly incomplete. It is meant as a beginning description of the significant difficulties that students face in coming to know statistics. We believe it helpful as a first step for researchers in statistics education in that it can set the direction for future research. It reminds us that understanding is multi-faceted and that symbol reading, recognition, and use is intimately tied to students’ conceptual development. For instructors, we believe that this description can raise awareness of the issues, emphasizes the difficulties for students, and argues for more targeted teaching and explicit descriptions of the codes carried by the symbols (perhaps explanations of why a particular symbol was chosen to represent a particular concept). Finally, we note the overall inadequacy of merely cataloguing and argue that significantly more work is needed in this field to further explore the types of needs that learners have, the means by which people develop appropriate (and inappropriate) semiotic functions and symbol sense, and the development of instructional sequences that support students’ learning.

15TH Annual Conference on Research in Undergraduate Mathematics Education 613


Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics, For the Learning of Mathematics 14(3), 24-35.

Eco, U. (1976). A Theory of Semiotics. Bloomington: Indiana University Press.

Font, V., Godino, J., & D’Amore, B. (2007). An onto-semiotic approach to representation in mathematics education. For the learning of mathematics, 27(2), 2-7,14.

Hewitt, D. (1999). Arbitrary and necessary: Part 1 a way of viewing the mathematics curriculum. For the learning of mathematics, 19(3), 2-9.

Hewitt, D. (2001a). Arbitrary and necessary: Part 2 assisting memory. For the learning of mathematics, 21(1), 44-51.

Hewitt, D. (2001b). Arbitrary and necessary: Part 3 educating awareness. For the learning of mathematics, 21(2), 47-59.

Kaplan, J., Fisher, D., & Rogness, N., (2009). Lexical ambiguity in statistics: How students use and define the words: association, average, confidence, random and spread. Journal of statistics education, 18(1). http://www.amstat.org/publications/jse/v18n2/kaplan.pdf Lunsford, M., & Poppin, P. (2011). From research to practice: Basic mathematical skills and success in introductory statistics. Journal of Statistics Education, 19(1).

http://www.amstat.org/publications/jse/v19n1/lunsford.pdf Mayen, S., Diaz, C., Batanero, C. (2009). Students’ semiotic conflicts in the concept of median.

Statistics education research journal, 8(2), 74-93.

Moore, D. (2001). Statistics; Concepts and controversies (5th ed). New York, NY: W.H.


Noth, W. (1995). Handbook of semiotics (Advances in semiotics). Bloomington, IN: Indiana University Press.

Peters, S. Robust understanding of statistical variation. Statistics education research journal, 10(1), 52-88.

Watson, J., (2009). The influence of variation and expectation on the developing awareness of distribution. Statistics education research journal, 8(1), 32-61.

Watier, N., Lamontagne, C., & Chartier, S., (2011). What does the mean mean? Journal of statistics education, 19(2). http://www.amstat.org/publications/jse/v19n2/watier.pdf Zieffler, A., & Garfield, J. (2009). Modeling the growth of students’ covariational reasoning during an introductory statistics course. Statistics education research journal, 8(1), 7-31.

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Robert Ely University of Idaho Iuliana Radu Rutgers University Abstract: Recent research about the thorny Tennis Ball Problem has revealed that students respond in different ways depending on the type of properties they generalize from the finite steps to the envisioned final state of the infinite process. These generalizations, in turn, depend on the features of the finite steps their attention is directed toward. Undergraduate students who attend to the labeling of objects, rather than simply counting the objects, are using object-based reasoning, which is crucial to their ability to understand Cantorian set theory. We propose a sequence of tasks centered around the Tennis Ball Problem that our research has shown to help students build object-based reasoning.

Keywords: infinite process, encapsulation, transfer The Tennis Ball Problem and student reasoning about infinite sets In the last decade, the Tennis Ball Problem has been used to reveal how students reason about infinite processes (Dubinsky, Weller, McDonald, & Brown 2005; Dubinsky, Weller, Stenger, & Vidakovic 2008; Ely 2007, 2011; Mamolo & Zazkis 2008; Radu 2009; Radu & Weber 2011;

Weller, Brown, Dubinsky, McDonald, & Stenger 2004). Although this problem appeared at least as early as Littlewood (1953), we believe its appearance in education research was largely due to

Falk (1994). One variant of the problem is this:

Suppose you are given an infinite set of numbered tennis balls (1, 2, 3,...) and two bins of unlimited capacity, labeled A and B. At step 1 you place balls 1 and 2 in bin A and then move ball 1 to bin B. At step 2 you place balls 3 and 4 in bin A and then move ball 2 to bin B. At step 3 you place balls 5 and 6 in bin A and then move ball 3 to bin B. This process is continued in this manner ad infinitum. Now assume that all steps have been completed. What are the contents of the two bins at this point?

Undergraduate and graduate students typically produce three kinds of solutions. The most common is (i) the infinitely-many-balls answer: Bin A contains infinitely many balls, or Bin A contains “half of infinity” (e.g., Ely 2007, 2011; Mamolo & Zazkis 2009; Radu 2009; Radu & Weber 2011). A typical reason given is that after every step there is one more ball in each bin than there was after the previous step, so after infinitely many steps each bin will hold infinitely many balls. Another response that is less common is (ii) the empty-bin answer: Bin A contains no balls. The reason here is that on step 1 ball 1 is moved from Bin A to Bin B, on step 2 ball 2 is moved from Bin A to Bin B, and so on, so that after infinitely many steps all of the balls have 15TH Annual Conference on Research in Undergraduate Mathematics Education 617 been moved out of Bin A into Bin B. A third type of response is (iii) that you can’t answer the question because “you’re never done moving the balls.” Many of the students who produce this third answer are also willing to produce one of the other two answers as well; sometimes it helps if they hear a “time-sensitive” version of the problem in which the steps are performed at 1 minute till noon, ½ minute till noon, ¼ minute till noon, etc. (step k occurs at 1/2k minutes till noon), and then they are asked what the bins hold at noon (Radu 2009; Ely 2011). In addition, one could choose to ask what the “limit state” is rather than asking about the process being completed.

Some researchers consider the empty-bin answer to be unambiguously the correct one (Dubinsky, et al. 2005; Dubinsky et al. 2008; Mamolo & Zazkis 2008; Weller, et al. 2004), although research in philosophy indicates that this conclusion is far from obvious (Allis & Koetsier 1991, 1995; van Bendegem 1994). Our purpose is not to take a stance on this issue, but rather to discuss (a) what student responses to this problem indicate about student reasoning and (b) how this problem, and other carefully-designed problems involving infinite processes, can be used to promote what Radu & Weber (2011) call “object-based reasoning” (OBJ) among upperlevel mathematics students. As we explain later, our interest in helping students engage in object-based reasoning in the context of infinite processes stems from our belief that this type of reasoning is important in Cantorian set theory, particularly because it promotes the understanding of correspondences between infinite sets.

Researchers who use APOS (action, process, object, schema) theory to interpret student responses on this problem focus on how students who produce the infinitely-many-balls answer are unable to treat the infinite process as being encapsulated into a single object (Dubinsky, et al.

2005; Dubinsky et al. 2008; Mamolo & Zazkis 2008; Weller, et al. 2004). On the other hand, we have found students who can encapsulate the infinite process but who defend either the infinitely-many-balls answer or the empty-bin answer or both, based on which properties of the finite states they choose to generalize when envisioning a final state. Furthermore, these generalizations depend on the properties of the infinite process their attention is directed toward.

For instance, students who were asked "which balls" were in the bin (instead of "how many") were much more likely to generate and be able to explain the empty bin solution, even if they personally preferred the infinitely-many-balls solution more. When students provide the infinitely-many-balls solution, they generalize properties of count or cardinality from the finite states—they count the number in the bin at each step and generalize that this count is always growing. They ignore the labels on the balls entirely, and often claim that the problem would be exactly the same as the “odd-even” version (see below). On the other hand, when students provide the empty-bin answer, they instead attend to and generalize to the final state a pattern in the labeling of the objects in the finite states (Ely 2011). These properties that are generalized from the finite states to an envisioned final state have been termed infinite projections (Ely 2011).

By attending to the labeling on the balls, students demonstrate object-based reasoning (Radu & Weber 2011), rather than reasoning by counting or cardinality (or "rate", Mamolo & Zazkis 2009). In this context the term "object-based" is not meant to be in contrast with "processbased;" it has nothing to do with whether a student views the infinite process or its result as an object or a process. Rather, object-based reasoning (OBJ) indicates that the student primarily 618 15TH Annual Conference on Research in Undergraduate Mathematics Education attends to, and generalizes properties of, objects. The student focuses on objects and where they end up rather than on sets and a trend in their sizes. By focusing on the objects first, students who use this kind of reasoning attend to the way the objects are labeled, not just how many there are at finite states.

We argue that the ability to use OBJ is important in advanced mathematical thinking, and it is crucial to Cantorian set theory. In order to extend the notion of “size” from finite sets to infinite ones in Cantor’s way, it is not the count of the objects, but rather the way that they are indexed or labeled, that is important to attend to. This is counterintuitive—when we count a finite set the labeling, the way that we temporarily assign names to the objects in the set (“one”, “two”, “three”, …) is unimportant. The last name we say is what is important. When ascertaining the size of an infinite set, the idea of the last number loses all importance but the way that we index the set becomes crucial. The set’s size is determined by what kind of set suffices for indexing it.

For example, a problem that might appear in an upper-level mathematics course is to suppose Q={x1, x2,..., xn, …}, and let Bn = {x1, x2,..., xn} and An = Q − Bn. How many elements are in each An? What is the intersection of all the Ans? With object-based reasoning, the student is able to fix a given element and look at what happens to the element as n increases, rather than to consider only the sizes of the sets and what happens to those sizes. In fact, the explicit notion of the limit of a sequence of sets can be found in some courses, where an upper limit set (which contains all elements that are contained in infinitely many sets in the sequence), and a lower limit set (which contains all elements that are eventually in the sets of the sequence, and a notion of the convergence of a sequence of sets precisely if its lower and upper limit coincide (e.g., Hausdorff 1957). Such a situation requires OBJ, because the limit set contains each elements that ends up in all of the Ans from some point onward.

It is OBJ, particularly the attention to and generalization of labeling rather than count, that is indicated by a student’s ability to understand and justify the empty-bin answer to the Tennis Ball Problem. It is for this reason that we want students to be able to envision and to explore the implications of the empty-bin solution, not because we believe that this solution is uniquely and unambiguously correct. For this reason, we devised a sequence of activities with infinite processes that focus on developing students' object-based reasoning. Based on how undergraduate students' thinking developed with these activities in a teaching experiment, we propose a sequence of activities that could be used for developing object-based reasoning for mathematics majors (Radu 2009; Radu & Weber 2011).

Based on our research with these problems, one way students’ object-based reasoning was promoted was when they were asked to investigate features of an envisioned “final state,” even if they themselves were not willing to commit to the answer they were exploring the implications of (Ely 2011). Because it is ambiguous how to mathematically model the Tennis Ball Problem context using a sequence of sets with a specified metric for convergence, it is possible for the discourse to devolve into a debate about this mathematization process, which, while potentially worthwhile from a broader mathematical point of view, is unproductive for developing students’ object-based reasoning. By instead bringing focus to how the properties of the finite states are generalized or extended to the envisioned final state, particularly to the property of labeling, rather than counting, the instructor can help foster the development of object-based reasoning.

15TH Annual Conference on Research in Undergraduate Mathematics Education 619 The sequence of problems is in keeping with Wagner’s theory of transfer in pieces (2006).

According to this framework, transfer of knowledge is a complex process during which an initially topical set of principles is constantly refined to account for (and not ignore) the new contexts of the problems encountered as one progresses through a sequence of problems with a common mathematical core. Thus, the acquisition of abstract knowledge can be seen as a consequence of transfer and not a required initial component for it to happen. In our own work with students, we found students did not abstract general principles from one of these problem contexts and then apply them to another. Rather, as they worked through a class of related problems, cross-references between prior and current tasks were made based on perceived structural commonalities among the tasks, which often resulted in changes in students’ reasoning on one or more of the tasks involved in the comparison, and thus in the refinement and expansion of topical principles (Radu 2009; Radu & Weber 2011). Below we present a proposed sequence of tasks designed to help students envision object-based reasoning, accompanied by the rationale for each task.

A sequence of tasks that support object-based reasoning

1. The Tennis Ball Problem (described at the beginning of this paper) This can serve as an informal assessment of how the students react to an infinite process problem that challenges them to envision a limit (final) state, and what infinite projections they focus on (if any).

2. The Odd-Even Tennis Ball Problem This problem is a variation of the first problem: at step n, balls 2n and 2n-1 are placed in bin A,

then ball 2n-1 is moved from bin A to bin B. It can be used for two purposes:

i) with student(s) who cannot envision any limit state to the original Tennis Ball Problem. Since in the odd-even problem each individual ball is affected (moved) by exactly one step, students will likely have no difficulty in envisioning a limit state where bin A contains all even-numbered balls and bin B all odd-numbered balls.

ii) with students who could envision only an “infinitely-many-balls” limit state for bin A for the original Tennis Ball Problem. In the context of the Odd-Even version, once the student envisions the odd-even limit state, the facilitator can ask questions about specific balls (e.g., why is ball 5 in bin B?), thus helping the student reflect on the action of the steps of the process on a particular ball and how that affects the position of that particular ball with respect to the limit state. Finally, students can discuss the difference between this problem and the original Tennis Ball Problem.

In our experience, students who are reasoning according to count rather than using object-based reasoning will consider the two problems to be the same, but that in the Odd-Even version one is more certain about which balls remain in the bin.

3. The Vector Problem. Let. You are going to “edit” this vector step by step.

Step 1:

Step 2:

Step 3:

• …………………………………..

This process is continued ad infinitum. Now assume ALL steps have been completed.

620 15TH Annual Conference on Research in Undergraduate Mathematics Education Describe v at this point.

In a teaching experiment with four math majors, each student easily employed OBJ in the context of this problem. Furthermore, the discussion of the Vector problem evoked spontaneous references to the Tennis Ball Problem. In students who had previously produced only rate/cardinality reasoning to the Tennis Ball Problem, these back references to this problem resulted in students envisioning what OBJ reasoning may mean in that context for the first time.

In cases where the student had envisioned both OBJ and rate/cardinality reasoning when working on the Tennis Ball problem in a prior session, work on the Vector problem resulted in students’ revisiting of the Tennis Ball problem and ending up preferring OBJ over rate/cardinality arguments. We argue that the vector context of this particular problem encouraged students to focus on individual positions/objects and made it less likely that they would focus on cardinality issues (given that there’s no evident “growing set” in this process). For more detailed discussion of student episodes related to the Vector Problem see Radu (2009) and Radu and Weber (2011).

4. The 10-Marble Problem.

This problem is similar to the Tennis Ball Problem, except that at step n, marbles 10n - 9 through 10n are put in a bin, and then marble n is removed from the bin. In this problem the set of marbles in the bin “grows” by 9 marbles at each step, which may make it even more counterintuitive to students to envision using OBJ and claim that the limit state is the empty set.

The role of this task at this point in the sequence (after the likely OBJ-inducive Vector Problem) is to explore the students’ reaction to a task whose context strongly encourages a rate/cardinality approach. For detailed accounts of student reasoning on a timed version of this problem see Mamolo & Zazkis 2008.

5. The Writer Problem. Tristram Shandy, the hero of a novel by Laurence Sterne, starts writing his biography at age 40. He writes it so conscientiously that it takes him one week to lay down the events of one day. If he is to document each day of his life and the pace at which he writes remains constant, can you envision a situation in which his autobiography can be completed?

This task offers a significant change of context, in the sense that we are no longer adding and removing objects from a bin. Additionally, the time component may cause the students to bring in a number of real-life considerations while reasoning on this task. The role of the Writer Problem in the sequence is to explore the students’ reaction to change of context and influence of real-life surface features of the problem on the students’ reasoning.

For a detailed discussion of how two different groups of students progressed through this sequence see Radu (2009). While there were certain differences between the paths of the two groups, what can be said about both groups is that i) there were numerous instances in which the students referenced prior tasks, and often such back references resulted in the students’ refining their reasoning on one or more of the tasks involved in the comparison; and ii) The Vector problem elicited OBJ from all students involved and significantly influenced the students’ reasoning on the rest of the problems, both prior and subsequent.

The task sequence can be extended to include the case of “oscillating” objects (objects that belong to infinitely many of the intermediate states while also not belonging to other infinitely many intermediate states), as well as processes manipulating objects in an implicit topological space (see Radu 2009 for examples of both). We believe such tasks are of interest from the point of view of transfer theories, but less so from a standard set theory perspective.

15TH Annual Conference on Research in Undergraduate Mathematics Education 621 References Allis, V. & Koetsier, T. (1991). On some paradoxes of the infinite. British Journal for the Philosophy of Science, 42(2), 187-194.

Allis, V. & Koetsier, T. (1995). On some paradoxes of the infinite II. British Journal for the Philosophy of Science, 46, 235-247.

Dubinsky, E., Weller, K., McDonald, M., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: an APOS-based analysis, Part 1.

Educational Studies in Mathematics, 58(3), 335-359.

Dubinsky, E., Weller, K., Mcdonald, M. A., & Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An APOS Analysis: Part 2. Educational Studies in Mathematics 60, 253-266.

Dubinsky, E., Weller, K., Stenger, C., & Vidakovic, D. (2008). Infinite iterative processes: The Tennis Ball Problem. European Journal Of Pure And Applied Mathematics, 1(1), 99-121.

Ely, R. (2011). Envisioning the infinite by projecting finite properties. Journal of Mathematical Behavior, 30(1), 1-18.

Ely, R. (2007). Nonstandard models of arithmetic found in student conceptions of infinite processes. In T. Lamberg, & L. R. Weist (Eds.), Proceedings of the 29th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 116–119). Stateline (Lake Tahoe), NV: University of Nevada, Reno.

Ely, R. (2007). Student Obstacles and Historical Obstacles to Foundational Concepts of Calculus. Unpublished doctoral dissertation. Madison: University of Wisconsin.

Falk, R. (1994). Infinity: A cognitive challenge. Theory & Psychology, 4(1), 35-60.

Littlewood, J. E. (1953). A mathematician's miscellany. London: Methuen.

Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in Mathematics Education, 10(2), 167-182.

Radu, I. (2009) To infinity and beyond: Toward a local instruction theory for completed infinite iteration. Doctoral dissertation, Rutgers University, New Jersey, USA.


Radu, I. & Weber, K. (2011). Refinements in mathematics undergraduate students’ reasoning on completed infinite iterative processes. Educational Studies in Mathematics, 78(2), 165Tall, D. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11(3), 271-284.

Van Bendegem, J. P. (1994). Ross' paradox is an impossible super-task. British Journal for the Philosophy of Science 45(2), 743-748.

Wagner, J. F. (2006). Transfer in pieces. Cognition and Instruction, 24(1), 1–71.

Weller, K., Brown, A., Dubinsky, E. McDonald, M., & Stenger, C. (2004). Intimations of infinity. Notices of the American Mathematical Society, 51(3), 741-750.

–  –  –

This paper illustrates how mathematical symbols can have different, but related, meanings depending on the context in which they are used. In other words, it illustrates how mathematical symbols are polysemous. In particular, it explores how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol.

Key words: ambiguity; polysemy; symbols; addition

Polysemy is a form of lexical ambiguity. A polysemous word is one that has two or more different, but related, meanings. For example, the English word “tie” may refer to an article of clothing worn around the neck or to the action of making a Windsor knot. The ambiguity may be resolved by considering the context in which the word is used. In mathematics, a word may be polysemous if its mathematical meaning is different from its everyday, familiar meaning (Durkin and Shire, 1991), or if it has two related, but different, mathematical meanings (Zazkis, 1998).

In a mathematical discourse, symbols such as +, =, and 1, may also be considered ‘words’ – they have their own definitions and can be strung together to form coherent mathematical phrases, such as 1+1=2. As such, they may also be a cause of ‘lexical’ ambiguity. Ambiguity in mathematics is recognized as “an essential characteristic of the conceptual development of the subject” (Byers, 2007, p.77) and as a feature which “opens the door to new ideas, new insights, deeper understanding” (p.78). Gray and Tall (1994) first alerted readers to the inherent ambiguity of symbols, such as 5 + 4, which may be understood both as processes and concepts, which they termed procepts. They advocated for the importance of flexibly interpreting procepts, and suggested that “This ambiguous use of symbolism is at the root of powerful mathematical thinking” (Gray and Tall, 1994, p.125). A flexible interpretation of a symbol can go beyond process-concept duality to include other ambiguities relating to the diverse meanings of that symbol, which in turn may also be the source of powerful mathematical thinking and learning. In this paper I discuss cases of ambiguity connected to the context-dependent definitions of symbols, that is, the polysemy of symbols. In particular, I examine the polysemy of the symbol ‘+’ as it manifests in the context of modular arithmetic and transfinite arithmetic. I also present an argument that suggests that the challenges learners face when dealing with polysemous terms are also at hand when dealing with mathematical symbols, focusing on cases where acknowledging the ambiguity in symbolism and explicitly identifying the precise, contextspecific, meaning of that symbolism go hand-in-hand with understanding the ideas involved.

A familiar meaning: The context of natural numbers The first context in which one encounters the symbol ‘+’ is in natural number arithmetic. The familiar phrase 1+1=2 can be considered as the sum of two cardinalities that are associated with two disjoint sets, and which yields the cardinality of the union set. (However, it is not uncommon to hear children claim, as my title suggests, that 1+1 does not equal 2, but instead equals a window.) For the purposes of this paper, let us consider the sum 1+2. In the familiar context of

natural numbers, the meaning of 1+2 can be broken down as in Table 1:

15TH Annual Conference on Research in Undergraduate Mathematics Education 623 Symbol Meaning in context of natural numbers 1 Cardinality of a set containing a single element 2 Cardinality of a set containing exactly two elements 1+2 Cardinality of the union set + Binary operation over the set of natural numbers Table 1: Summary of familiar meaning in As a binary operation, addition, its definition and properties, depends necessarily upon the domain to which it is applied – and this fact underlies the polysemy of ‘+’. Building on the idea of addition as a domain-dependent binary operation, the following sections consider two other domains: (i) the set {0, 1, 2} and (ii) the class of (generalised) cardinal numbers. These domains are of interest since: (i) the extended meanings of symbols such as ‘a + b’ contribute to results that are inconsistent with the ‘familiar’, and (ii) they are items in undergraduate mathematics courses and also pre-service teacher mathematics education. It is useful for purposes of clarity to distinguish between different definitions of the addition symbol as they apply to different domains. The symbol +N will be used to represent addition over the set of natural numbers, +Z as addition over the set of integers, +3 as addition over the set {0, 1, 2} (i.e. modular arithmetic, base 3), and +∞ as addition over the class of cardinal numbers (i.e. transfinite arithmetic).

An extended meaning: The context of modular arithmetic Consider the group 3 – the set of elements {0, 1, 2} with associated operation addition modulo

3. Within group theory the meanings of symbols such as 0, 1, 2, +, and 1+2 are extended from the familiar in several ways. As an element of 3, the symbol 0 is short-hand notation for the congruence class of 0 modulo 3. That is, it is taken to mean the set consisting of all the integral multiples of 3. The symbols 1 and 2 are analogously defined, and the symbol ‘+’ is defined as addition modulo 3. As such, the familiar ‘1+2’ now carries with it meaning quite distinct from before: just as ‘1’ and ‘2’ were, ‘1+2’ is also a congruence class. Dummit and Foote (1999) define the sum of congruence classes by outlining its computation, e.g. 1+2 (modulo 3), is computed by taking any representative integer in the set {… -5, -2, 1, 4, 7, …} and any representative integer in the set {… -4, -1, 2, 5, 8,…}, and summing them in the ‘usual integer way’. Thus, recalling the notation introduced in the previous section, sample computations to satisfy this definition include: 1 +3 2 = (1 +Z 2) modulo 3 = (1 +Z 5) modulo 3 = (-2 +Z -1) modulo 3 all of which are equal to the congruence class 0. Table 2 below summarizes the meanings of

the symbols ‘1’, ‘2’, and ‘1+2’, and ‘+’ when considered within the context of 3:

Symbol Meaning in context of 3 1 Congruence class of 1 modulo 3: {… -5, -2, 1, 4, 7, …} 2 Congruence class of 2 modulo 3: {… -4, -1, 2, 5, 8,…} 1+2 Congruence class of (1+2) modulo 3: {…, -3, 0, 3, …} + Binary operation over set {0, 1, 2}; addition modulo 3 Table 2: Summary of extended meaning in 3 The process of adding congruence classes by adding their representatives is a special case of the more general group theoretic construction of a quotient and quotient group – central ideas in algebra, and ones which have been acknowledged as problematic for learners (e.g. Asiala et al., 1997; Dubinsky et al., 1994).These concepts are challenging and abstract, and are made no less accessible by opaque symbolism. As in the case with words, the extended meaning of a symbol can be interpreted as a metaphoric use of the symbol, and thus may evoke prior knowledge or 624 15TH Annual Conference on Research in Undergraduate Mathematics Education experience that is incompatible with the broadened use. In a related discussion, Pimm (1987) notes that “the required mental shifts involved [in extending meaning from everyday language to mathematics] can be extreme, and are often accompanied by great distress, particularly if pupils are unaware that the difficulties they are experiencing are not an inherent problem with the idea itself” (p.107) but instead are a consequence of inappropriately carrying over meaning. A similar situation arises as one must extend their understanding of a mathematical symbol – an important mental shift that is taken for granted when clarification of symbol polysemy remains tacit.

An extended meaning: The context of transfinite arithmetic Transfinite arithmetic may be thought of as an extension of natural number arithmetic – its addends represent cardinalities of finite or infinite sets and a sum is defined as the cardinality of the union of two disjoint sets. Transfinite arithmetic poses many challenges for learners, not the least of which involves appreciating the idea of ‘infinity’ in terms of cardinalities of sets (i.e. the transfinite numbers 0, 1, 2, …). In resonance with Pimm’s (1987) observation regarding negative and complex numbers, the concept of a transfinite number “involves a metaphoric broadening of the notion of number itself” (p.107). In this case, the broadening also includes accommodating properties which are unfamiliar and inconsistent with natural number arithmetic.

Consider a generic example: the sum 0 + 1. It is the cardinality associated with the union. In this context, the addends are elements of the (generalised) class1 of set ∪ {β}, where β cardinals, which includes transfinite cardinals. Between the sets ∪ {β} and there exists a bijection, which, in line with the definition (Cantor, 1915), guarantees that the two sets have the

same cardinality – that is, 0 + 1 = 0. Table 3 summarizes the meaning of these symbols:

Symbol Meaning in context of transfinite arithmetic 1 Cardinality of the set with a single element; class element Cardinality of ; transfinite number; ‘infinity’ Cardinality of the set ∪ β; equal to 0 +1 + Binary operation over the class of transfinite numbers Table 3: Summary of extended meaning in transfinite arithmetic Similarly, one can show that 0 = 0 + υ, for any υ, and that 0 + 0 = 0. Thus, whereas with ‘+N’ adding two numbers always results in a new (distinct) number, with ‘+∞’ there exist nonunique sums. A consequence of non-unique sums is the existence of indeterminate differences.

Explicitly, since 0 = 0 + υ, for any υ, then 0 - 0 has no unique resolution. As such, the familiar notion that ‘anything minus itself is zero’ does not extend to transfinite subtraction. This property is part and parcel to the concept of transfinite numbers. Identifying precisely the context-specific meaning of these symbols (‘+∞’ and ‘−∞’) can help solidify the concept of transfinite numbers, while also deflecting naïve conceptions of infinity as simply a ‘big unknown number’ by emphasizing that transfinite numbers are different from ‘big numbers’ since they have different properties and are operated upon (arithmetically) in different ways.

Concluding Remarks This paper illustrates how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning that is inconsistent with their use in ‘familiar’ contexts. It focused on cases where acknowledging ambiguity in symbolism and explicitly identifying the precise (extended) meaning of that symbolism is necessary for understanding. While the focus was on examples of how For distinction between set and class, see Levy (1979).

15TH Annual Conference on Research in Undergraduate Mathematics Education 625 distinguishing among the symbolic notation of +N, +3, and +∞ is fundamental to appreciating the subtle (and not-so-subtle) differences among the corresponding addends, this argument has broader application. Just as knowledge of language includes “learning a meaning of a word, learning more than one meaning, and learning how to choose the contextually supported meaning” (Mason et al., 1979, p.64), knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol. Attending to the polysemy of symbols, either as a learner, for a learner, or as a researcher, may expose confusion or inappropriate associations that could otherwise go unresolved. Research in literacy suggests that students “will choose a common meaning, violating the context, when they know one meaning very well” (Mason et al., 1979, p.63). Further research in mathematics education is needed to establish to what degree analogous observations apply as students begin to learn ‘+’ in new contexts.

References Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S. and Oktaç, A. (1997). Development of students’ understanding of cosets, normality, and quotient groups. Journal of Mathematical Behavior, 16(3), 241–309.

Byers, W. (2007). How Mathematicians Think: Using Ambiguity, Contradiction and Paradox to Create Mathematics. Princeton, NJ: Princeton University Press.

Cantor, G. (1915) Contributions to the Founding of the Theory of Transfinite Numbers (translated by P. Jourdain, reprinted 1955). New York: Dover Publications Inc.

Dubinsky, E., Dautermann, J., Leron, U., and Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27, 267-305.

Dummit, D. S. and Foote, R. M. (1999). Abstract Algebra. New York, NY: John Wiley & Sons Inc.

Durkin, K. and Shire, B. (1991). Lexical ambiguity in mathematical contexts. In Durkin, K. and Shire, B. (eds), Language in Mathematical Education: Research and Practice. Milton Keynes, Bucks: Open University Press, pp.71-84.

Gray, E. and Tall, D. (1994). Duality, ambiguity and flexibility: A ‘proceptual’ view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116-140.

Levy, A. (1979) Basic Set Theory, Berlin, New York, Springer-Verlag.

Mason, J.M., Kniseley, E., and Kendall, J. (1979). Effects of polysemous words on sentence comprehension. Reading Research Quarterly, 15(1), 49-65.

Pimm, D. (1987) Speaking Mathematically: Communication in Mathematics Classroom, London, Routledge.

Zazkis, R. (1998) ‘Divisors and quotients: Acknowledging polysemy’, For the Learning of Mathematics, 18(3), 27-30.

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