Introduction

In the province of Ontario, the curriculum for grades 1 to 12 (elementary and secondary levels) is mandated by the Ministry of Education. In mathematics, there are three separate curriculum documents: grades 1 to 8 (Ontario Ministry of Education 2005a), grades 9 and 10 (Ontario Ministry of Education 2005b), and grades 11 and 12 (Ontario Ministry of Education 2007). A new and significantly changed curriculum for elementary mathematics (grades 1 to 8) was initially rolled out in 1997, with two secondary level documents following on its heels over the next few years. While the current documents contain slight revisions from these original documents, there have been no fundamental changes to any of them since their initial publication.

All teaching resources in schools in Ontario must align with the curriculum Expectations, i.e., the detailed, mandated lists of learning outcomes. To be adopted for classroom use, all mathematics textbooks must be “Ministry approved,” i.e., must address sufficient portions of these Expectations. Hence, at some level, the curriculum Expectations drive the content of adopted textbooks, and, taken together, they shape classroom learning.

In secondary school mathematics in Ontario, different pathways and levels of courses are offered. Students considering proceeding to university programs involving mathematics or sciences are expected to take “academic” (grades 9 and 10), and “university level” (grades 11 and 12) mathematics courses in high school, and it is the progression along this pathway—from elementary, through high school, to university—that is the context for this study.

Although there is no common university mathematics curriculum, even a superficial examination (of textbooks, or course outlines) shows that first-year undergraduate mathematics courses in Ontario tend to assume a working knowledge of the material taught in grade 12 courses, in particular, the “university level” Advanced Functions (MHV4U) and Vectors and Calculus (MCV4U) courses. At most universities, students whose background is inadequate or rusty may be offered remedial opportunities, in the form of courses or out-of-classroom support.

Given the substantial changes in the school mathematics curriculum over the past 20 years, the authors have an ongoing interest in the effects of school mathematics preparation on students’ success in first-year university mathematics courses. In an earlier paper, Kajander and Lovric (2009) examined the longitudinal development of the concept of a line tangent to the graph of a function, and studied “gray zones,” namely the material, often narratives, that seem to be left vague, or that implicitly assume prior knowledge or results. Looking even earlier in the curriculum, they examined a variety of treatments of the circle (disk) area formula in textbooks (Kajander & Lovric, 2017), and this led to the identification of further “gray zones.” Based on this work, the authors decided to explore the curriculum underpinnings of the common ground, i.e., the development of concepts related to the limit (in the case of the circle area, a geometric limit), and in particular, infinity. We wanted to explore how certain mathematical ideas are initially introduced into the curriculum, and how they further develop and formalize over the elementary and secondary school mathematics grades, in order to search for possible lack of clarity or coherence which might contribute to creation of such “gray zones.” As mentioned, such understandings are an important transitional issue in terms of the preparation for university mathematics courses.

It is our hope that the results of this paper will inform future iterations of mathematics curricula, and close any identified gaps, at least as far as the pathways from the mathematical topics of rational numbers to asymptotes and infinity are concerned. If it is more generally true that there is very little or no curricular support for the development of concepts related to infinity (for example, Kajander and Lovric (2017) illustrate one instance of this point as related to the circle area formula development), we believe this could have a significant impact on teachers and their decisions as to if, when, and how to introduce or deal with the topic of infinity in their classrooms. Additionally, assumptions made by university instructors, namely that their students bring certain solid prerequisite knowledge from high school, might be better informed and, hopefully, modified by this work.

Background and Rationale

The theme discussed in this paper belongs to the broad context of studying learning of mathematics in transition from secondary to tertiary education. It is a continuation of our previous work on transition (Kajander and Lovric, 2005), and misconceptions involving algebraic and geometric limiting processes (Kajander and Lovric, 2009; 2017).

University students’ difficulties in conceptualizing and working with limits have been well documented (Fischbein 2001; Jones 2015). In fact, one only needs to look at university calculus textbooks to realize how poorly, and inadequately, the concept of infinity is treated in a number of different situations (Lovric, 2012; Burazin and Lovric, 2018). It could be that the roots of university instructors’ beliefs that infinity have been “covered” lie in the fact that the concepts developed in elementary and high school (such as circle area formula, division by zero, the limit of a sequence, and the sum of a geometric series, to name a few) require certain level of proficiency in understanding infinity and infinite processes. Additional challenges arise from the fact that there is no single, unique “infinity,” and that very little effort is usually made to explicitly address this (Lovric, 2012).

This paper is our attempt at identifying the root causes of this situation, by identifying the themes in the Ontario curriculum documents which lead to the development of concepts needed to understand limits and infinity. Our previous work suggests that the gaps and inconsistencies in student learning (i.e., their synthetic models; see next section), which are typical to first-year university students, form and grow during their (much) earlier school experiences. In the case on infinity, we claim, they go all the way back to elementary grades teaching of fractions, and division by zero. The key component of building an understanding around these issues is a thorough examination of the creation, and the development, of students’ cognitive models. We believe that important components of the development of this understanding should be found in the elementary and high school mathematics curricula.

Literature and Framework

In their related work, the authors have used the explanatory framework of the theory of conceptual change (Biza et al. 2005) to study the learning processes of both high school and first-year university students, as they engage with material presented in textbooks and other teaching materials. This theory is appropriate for a longitudinal study, as it illuminates the temporal dynamics of the development of misconceptions in one’s learning. Kajander and Lovric (2009) write: “Based on a certain amount of information (learned from a book, heard in a class, etc.), a learner uses her/his own ideas and current understanding to create an initial explanatory framework. The presuppositions that inhabit this framework, also known as naive beliefs or alternative conceptions, will form student’s current beliefs about the material. Faced with having to absorb the material that is in some way incompatible with the prior knowledge, the student will try to assimilate new information into their existing framework.” (p. 2).

This existing framework, usually referred to as the synthetic model, contains learner’s misconceptions, i.e., their beliefs, understandings, and compromises made in an attempt to connect prior knowledge with incompatible new information. We will demonstrate that in the cases studied in this paper (division by zero, limits and infinity), that prior knowledge is so vague and inconsistent that almost any new piece of information (for instance, as heard in a university calculus lecture) is incompatible with learner’s existing synthetic model.

Modifying synthetic models, which are known to be robust and resistant to change (Biza et al. 2005), requires considerable time and effort, both from a teacher and a learner. Tall (1991) states that mathematics “by its very nature, includes concepts which are subtly at variance with naïve experience. Such ideas require an immense personal reconstruction to build the cognitive apparatus to handle them effectively” (p. 252). A teacher has an important role to identify students’ synthetic models and to apply adequate strategies to facilitate this “immense personal reconstruction” that is required. Kajander and Lovric (2009) suggest possible strategies to accomplish that when students learn about the concept of a line tangent to the graph of a function. Further suggestions involving learning and understanding infinity in several contexts in an introductory calculus course can be found in Burazin and Lovric (2018).

Challenges to creating narratives about division by zero, asymptotes, and limits (and many other mathematical objects and concepts) are in part due to the language that we use. It is well documented that sometimes subtle, but often obvious, differences between colloquial language and the language of mathematics contribute to students’ difficulties in understanding (Kim et al. 2005). Kajander and Lovric (2009) show that colloquial, “reader-friendly” language leads to the development of misconceptions in textbook presentations of the concept of a tangent line. The Ontario grades 11 and 12 curriculum document (Ontario Ministry of Education 2007) does not require a clear conceptualization of infinity (i.e., there is no suggestion to introduce, and discuss infinity in a precise and well-defined context), nor does it suggest a vocabulary that is to be used in articulating reasoning about infinity.

Methodology

The three Ontario school mathematics curriculum documents, cited in the Introduction, formed the data sources for our study. First, we created a list of keywords (key terms) determined to be related to infinity and limits (see Table 1).

Table 1 List of keywords (key terms)
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Our intent was to locate the earliest occurrences of these terms, and then follow the development of the related concept and ideas over the subsequent grades, as well as attending to the context and meaning (as, for instance, infinity has multiple meanings). All text from the Grade 1–8 curriculum document, including the Glossary (a list of definitions provided at the end, mainly for teachers), was searched for the keywords listed in Table 1. In cases where the exact matches to the keywords in Table 1 could not be found, parts of the phrases, or even individual words, were searched (for example, “zero”), to ensure that we had not missed a different phrasing of the concepts we were looking for. This process was then repeated with the two remaining curriculum documents (noting that the grades 11 and 12 curriculum document does not have a glossary).

In parallel, we noted places in the curriculum where it was felt that the concepts of limits and infinity would first be needed. With respect to rational numbers, for example, we assumed the curriculum would suggest some way—whether formally or informally—of discussing zero occurring in the denominator of a fraction. Similarly, the circle area formula development (see Kajander & Lovric, 2017), relies on a different approach to the limit, namely a geometric one, so we looked for opportunities where such limits could be introduced. Our goal was to track the early development and formalization of such ideas in places where the relevant concepts would be needed in order to develop conceptual understanding. Lastly, we looked for topics that we felt would be strengthened by exploration of infinity-related ideas, even if not essential, as recommendations for curriculum enhancement.

As it turned out, our initial plan to track the development of the ideas around division by zero and the concepts related to infinity was eye-opening, to say the least.

Results

Given that division with whole numbers is first mentioned in grade 3, it was expected that fairly soon at, or after, grade 3, the issue of division by zero might emerge. The curriculum states that students are to use “... a variety of strategies to investigate multiplication and division” (Ministry of Education 2005a, p. 43), and in grade 4 (p. 73), the concept of multiplication and division as inverse operations is mentioned in the Expectations. However, even though multiplication by zero is present, no mention of the associated inverse operation—division by zero—was found in the Expectations. A more systematic word search of related terms, such as “undefined,” “not defined,” “division by zero,” “dividing by zero,” and “domain and range,” as per Table 1, yielded no results. Even simply searching Expectations using the word “zero” produced no hints as to what was to happen when zero was a divisor. For example in Grade 4, the curriculum (Ministry of Education 2005a, p. 73) states “determine, through investigation, the inverse relationship between multiplication and division (e.g., since 4 x 5 = 20, then 20 ÷ 5 = 4; since 35 ÷ 5 = 7, then 7 x 5 = 35),” with no mention of zero.

Further lack of addressing division by zero in the Expectations was identified in the context of fractions and rational numbers. The possibility of a zero denominator is simply not mentioned. Continuing the search process as described earlier yielded similar omissions. The results are summarized in the starting rows in Table 2.

Table 2 Instances (appearances) of words, phrases, and terms in Ontario curriculum Expectations grades 1 to 12, related to division by zero and infinity. A hyphen [-] indicates no mention at all, and a check mark [✔] indicates an appearance of the word or the phrase
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Searching forward in the documents, Expectations in the grades 9 academic, 10 academic, 11 university, and all three grade 12 university level mathematics courses were examined in a similar manner, with results appearing in Table 2.

The glossaries of terms, provided in the back of the grades 1 to 8, and grades 9–10 curriculum documents, were also checked for the identified keywords, yielding the results in Table 3.

Table 3 Instances of key terms (key words) in curriculum Glossaries. A hyphen indicates no mention at all, a check mark indicates an appearance of the word or the phrase
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In some instances, in the high school curriculum documents, occurrences of the key terms (key words) were identified in other mathematical contexts, noted in Table 4.

Table 4 Occurrences of key terms (key words) which are not related to division by zero or infinity (in the sense of limits). MCR3U is the grade 11 Functions course, and MDM4U is grade 12 Mathematics of Data Management
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Discussion

The definition of “rational number” in the Glossary (grades 1 to 8 Curriculum) mentions explicitly the requirement that the denominator cannot be zero. However, the rationale as to why this idea is needed is not at all addressed in the actual curriculum Expectations. We note that although Expectations (p. 111) require representation, comparing, and ordering of positive and negative fractions, there is no mention of fractions whose numerator is zero. The Grades 9 and 10 Curriculum Glossary includes a similar definition of a rational number: “a number that can be expressed as the quotient of two integers where the divisor is not 0.” But, yet again, considering the case of division by zero is not at all suggested by Expectations within the particular course descriptions. (The grades 11 and 12 curriculum document does not contain a glossary.)

Given the lack of treatment of division by zero in defining either whole number division or rational numbers in the Expectations, it was impossible to follow the development of the ideas related to these concepts, as planned. While the definition of a rational number in the elementary curriculum Glossary does contain the information that the denominator cannot be zero, it is not in the Expectations as something students would learn.

The slope of a line is introduced and discussed starting in grade 9; even though Expectations state to “identify, through investigation, properties of the slopes of the lines [...] positive or negative rate of change, steepness” (p. 34) there is no suggestion to investigate what happens, say, when steepness increases (i.e., what can be said about the slope of a vertical line). We note that the slope of zero (horizontal line) is not discussed either. A grade 10 curriculum expectation states “to identify special cases x=a and y=b,” (p. 56) but without suggesting that the slopes of these lines be examined. Vertical lines are not mentioned in the Glossaries of either of the two documents in which they are provided. Slopes are revisited in grades 11 and 12, mostly in the context of examining rates of change; again, however, the slope of a vertical line is not mentioned.

Although the definition of the derivative (grade 12 MVC4U) involves dividing by smaller and smaller numbers (p. 102), no discussion is suggested to offer an explanation that although both the numerator and the denominator in the average rate of change approach zero (so division by zero is involved), we can (and do) make sense of the limit, in many cases. As part of investigation of properties of the derivatives, it is suggested that students look at the slopes of the tangents to the square root function f(x) = x ^ (1/2) (p.103). There is no mention, nor hint, about the fact that the slopes of tangents approach infinity as x approaches 0.

It seems to us that an effort was made to avoid facing an issue that is right in front of us.

The word “infinity” is not mentioned, not even once, in grades 1 to 8 and 9 and 10 curricula. In grades 11 and 12, it is mentioned three times, in reference to an infinite set: “an infinite number of polynomial functions satisfy” (MHV4U, p. 92); “an infinite number of graphs is possible” (MCV4U, p. 105); and “infinite number of possible outcomes” (MDM4U, p. 116).

Several learning objectives in the grades 11 and 12 curriculum include working with asymptotes, for instance: “[determine] by using graphing technology and by reasoning that there are vertical asymptotes” (MHV4U, p.92); and “determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes” (MCR3U, p.48). In the latter, the curriculum suggests narratives for answers “e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases”, etc.; yet, there is no mention of the language that should be used to describe the behavior of a function near its vertical asymptote. Noting that “infinity” in the context of the size of a set (actual infinity) and “infinity” in the context of asymptotes (potential infinity) are completely different notions, we see that there is absolutely no curricular treatment of “approaching infinity” as required for analyzing, for instance, vertical asymptotes.

Recommendations and Conclusions

The analysis of curriculum documents (related to division by zero, limit processes, asymptotes, and infinity) reveals an obvious conclusion—our students struggle with these concepts in university because they have never been adequately introduced and discussed, not in elementary school, not in high school, and not in first-year university.

To follow, we provide examples, one from each of the elementary and high school level curricula, to illustrate how the treatment of the existing topics could be deepened to include these critical concepts. The first example evolves from the fundamental operation of division.

As mentioned, division is introduced in grade 3, with references to concrete models such as “equal groups” and “repeated subtraction” (p. 56). These models in fact are the partitive and measurement models of division, both of which are important for students to understand (see Kajander and Boland, 2014). In the partitive model, the divisor refers to the number of groups, while in the measurement model, the divisor is the size of each group. As part of this exploration, students could also be encouraged to think about whether it makes sense to think about “zero groups” or “groups of zero.” It is not necessary to fully resolve these issues in grade 3, but it would be useful to question whether dividing by zero makes sense in this case. Such questions would plant the seed that a zero divisor might be something different—and might not even “make sense.”

Later in the curriculum, students are asked to explore the meaning of dividing by decimal numbers less than one. For example, in grade 7 (p. 100), division by a decimal such as 0.2 is suggested; here, students could be specifically encouraged to explore what happens when the decimal divisor gets smaller and smaller. Technology would make this a game-like activity; students could be asked “can you divide a number by something to make the answer greater than 100? Greater than 1000?” Such an exploration would crucially set the stage for various concepts to be developed. Not only would this allow a richer understanding of what happens when the denominator gets closer to zero, (and the idea that, with appropriate accuracy, you can “keep going” and get “closer and closer”), but such ideas also set up fraction division concepts, which are introduced formally in grade 8.

Understanding fraction division relies crucially on an understanding of the measurement model of division. To begin, students might be asked how they could determine, for example, 2 ÷ \( \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \), using a model and reasoning. If division by \( \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \) is understood as counting the number of one-eighths in the dividend (here, in 2), then the algorithm of multiplying 2 by the denominator (8) makes sense; this is understood as “there are 8 one-eighths in each whole, and for two wholes, there must be 16 parts of size \( \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \).” (For a full explanation of the rest of the modeling process for division of fractions, see Kajander and Boland, 2014). In this case, as the denominator of the unit fraction in the divisor gets larger (e.g., moving from \( \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \) to 1/10 to 1/20 to 1/100 or 1/1000), the fraction quantity gets smaller. It has already been determined that 2 ÷ (1/1000) can be calculated with 2 × 1000. Extending the reasoning should allow students to explore and conjecture that when dividing a quantity by a smaller and smaller fraction, the quotient will be getting bigger and bigger. This open-ended exploration can provide infinite enjoyment for some students. Investigating such questions is both possible and important in grade 7 and 8. Not only would this help students better understand division of fractions (as well as division by numbers closer and closer to zero), it sets up the stage for subsequent learning to follow in high school and university.

In discussing infinity, it is necessary to distinguish actual infinity (e.g., when we say that there are “infinitely many integers”) from potential infinity (e.g., when we discuss asymptotes). Lovric (2012) writes: “Perhaps the most common misconception comes from the teaching practice by which infinity is rarely (in either high school or in university calculus) clearly defined and discussed in a precise, well-defined context” (p. 140). In other words, we should not shy away from infinity, but rather approach it by providing a clear conceptualization as a background for students’ explorations and learning.

Introducing and discussing infinity provides rich opportunities for learning. For instance, a grade 6 Expectation (Ontario Ministry of Education 2005a) suggests a study of patterns of numbers, and asks students to articulate what happens as the pattern continues: “Determine a term, given its term number, by extending growing and shrinking patterns” (p.23). By studying an increasing pattern generated by an arithmetic sequence (potential infinity), students could be asked to determine “how big” the terms become. As suggested already, answering questions such as “Can you find a term larger than 100”? Larger than 1000? Larger than 10,000? students are engaging with the dynamic nature of a process that “goes to infinity.”

Another opportunity to discuss infinity is presented by the slope of a line, and later, by the general notion of the average rate of change of a function. Students could be asked to draw lines with increasing slopes, and asked to hypothesize whether the slope of a vertical line could be a number. If the slope is 100, does in represent a vertical line? What if the slope were 1000, or 50,000, is it a vertical line then? The conclusion, namely that no real number can identify the slope of a vertical line, is quite valuable. Now, we can clarify the meaning of the phrase “the slope does not exist,” found in “gray zones” in both high school and university. It simply means that the slope is not a real number.

Students familiar with functions (grades 11 and 12) could be asked to investigate the behavior of a function whose denominator can be zero, such as 1/x or tan(x). Again, emphasizing the potential infinity aspect of “infinity,” they could be asked to articulate what happens near a vertical asymptote. (How) can we make the values of 1/x larger than a thousand? Larger than a million? Larger than a billion? Being exposed to such questions from their early years (as suggested in this paper), and thus having developed a “feeling for large numbers,” means that answering these questions, now in the context of functions, should not be a big challenge.

To contrast, in describing sets, such as the domain and the range of a function, we could say that they are “infinite,” but emphasize the actual infinity nature of “infinity,” by clarifying that the reference is to the fact that they are not bounded. (Of course, a bit of work is needed to define and understand what a bounded interval is.) For instance, the range of the function f(x) = x^2 is an unbounded interval [0, infinity).

The examples above are meant to suggest how the concepts in the existing Ontario curricula could be strengthened to include an understanding of infinity and division by zero at the earliest stages. At the moment, teachers are left to their own devices if students ask questions such as “but what if you divide by zero?” Asking several elementary teacher colleagues how they handled such questions resulted in the consistent response along the lines of “we just tell them you can’t do it, or that it doesn’t exist”. These would be important concepts for preservice teachers to study also (Ball 1990; Crespo and Nicol 2006). We can only hope that future iterations of the curriculum allow the natural development of these ideas with reasoning and understanding, from the earliest stages of the introduction of division. Such a progression could be highly beneficial in the senior years of high school and early university, where consistent gaps and misconceptions around these concepts have persisted for well over a decade since the new curriculum was introduced.