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DOI: 10.30738/union.v11i3.16006 © Author (s), 2023. Open Access
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Mathematics teachers’ conceptions about infinity: A preliminary study at the secondary and high school level
Irving Aarón Díaz-Espinoza*, José Antonio Juárez-López
Faculty of Physical and Mathematical Sciences, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
* Corresponding Author. Email: [email protected]
Received: 30 September 2023; Revised: 18 October 2023; Accepted: 19 October 2023
Abstract: The literature indicates that students consistently exhibit limited conceptions of infinity in various contexts. Furthermore, it is more resistant in geometric than arithmetic contexts. On the other hand, there is a smaller amount of literature that explores teachers' conceptions and how these conceptions may be similar to those of their students. Therefore, this work presents the results obtained from applying a five-item instrument with secondary and high school mathematics teachers who are in public or private service in the state of Puebla, Mexico. This study had a dual purpose: to show that teachers also possess limited conceptions of infinity, as students do, thereby expanding the existing literature on this topic; and secondly, to emphasize that the results shown here will enable subsequent work to design an intervention for a profound conceptual change regarding infinity among teachers.
Keywords: Conceptions; High school level; Infinite; Teachers
How to cite: Díaz-Espinoza, I. A., & Juárez-López, J. A. (2023). Mathematics teachers’ conceptions about infinity: A preliminary study at the secondary and high school level. Union: Jurnal Ilmiah Pendidikan Matematika, 11(3), 426-435. https://doi.org/10.30738/union.v11i3.16006
INTRODUCTION
In research involving students and teachers in training, there is evidence of incomplete understandings in expressions such as 0.999 … = 1 (Díaz-Espinoza et al., 2023; Dubinsky et al., 2005b; Juter, 2019; Kattou et al., 2010; Schwarzenberger & Tall, 1978; Vinner & Kidron, 1985;
Wistedt & Martinsson, 1996; Yopp et al., 2011). Recently, in Krátká et al. (2021) proposed four conceptions of infinity: natural, potential, omega-epsilon position, and actual. Similarly, in Cihlář et al. (2015), they suggested the omega-epsilon position of infinity as a previous phase of the actual infinity. Thus, constructing a complete conception of infinity involves progressing through each of the stages from natural infinity to actual infinity, which is not easy to achieve since it is not a linear journey, but rather, a continuous oscillation between the stages.
For example, Montes et al. (2014) mentioned that teachers must have a good practical knowledge of the stages through which students must go to achieve an understanding of infinity. As Manfreda Kolar & Čadež (2012) state, “only on the basis of sound and broad knowledge can a teacher acquire the confidence required to face new challenges and to meet the needs of inquisitive students” (p. 399).
It is essential to design instructions that enable us to transcend the preceding stages of infinity and perceive it as an actual infinity. If teachers convey errors to their students, these mistakes can become a source of difficulties for them in understanding this concept (Kattou et al., 2010; Schwarzenberger & Tall, 1978). Simultaneously, as Date-Huxtable et al. (2018) maintain “[it is] strongly influenced by tacit models and teachers’ and/or students’ conflicting
beliefs” (p. 546), making their learning even more challenging. This is underscored by the investigation of Cihlář et al. (2015), revealing that only two students — 10% of all respondents
— possess a conception of actual infinity.
Similarly, in a study conducted with in-service teachers, it was found that approximately 72% of teachers conceive of infinity as an endless process — potential infinity — while only 28% define infinity as an object — actual infinity — (Kattou et al., 2010, p. 1775). In Díaz- Espinoza et al. (2023) it is concluded that the interviewed professor lacks a clear understanding of the concept of infinity. Infinity as a process is limited by the need for real meaning, and the misconception persists that an infinitely decreasing sum cannot result in a finite number, and that an infinitely periodic decimal cannot represent a finite number. These various misconceptions of infinity presented in Díaz-Espinoza et al. (2023) along with the typology outlined in Krátká et al. (2021) are the driving factors behind this research.
Therefore, the objective of this work is to identify and classify the conceptions that in- service secondary and high school mathematics teachers have about infinity according to the four stages: natural, potential, omega-epsilon position, and actual proposed by Krátká et al.
(2021). Although, there are research articles that have investigated the conceptions of infinity presented by teachers, examining both potential and actual infinity (Date-Huxtable et al., 2018;
Díaz-Espinoza et al., 2023; Kattou et al., 2010; Manfreda Kolar & Čadež, 2012; Montes et al., 2014; Tsamir, 1999; Yopp et al., 2011). This research aims to expand the literature by exploring the new categories that suggested by Krátká et al. (2021). These categories were reported in conjunction with students, and the objective is to contrast whether teachers’ conceptions exhibit similarities with those of the students.
As mentioned by Krátká et al. (2021):
Most existing research does not work with an infinity concept that would fit the omega-epsilon position. Therefore, students whose ideas did not correspond to the conception of natural infinity or actual infinity are often categorized into the conception of potential infinity. (p. 19)
CONCEPTUAL FRAMEWORK
In the research by Krátká et al. (2021), four conceptions of infinity that students can present are suggested: natural infinity, potential infinity, omega-epsilon position, and actual infinity. In this research, the aim is to expand the study of infinity at each of the stages, now with in- service high school teachers. On the other hand, Cihlář et al. (2015) describe natural infinity as a first phase connected with real objects, sets, points and lines seem infinite to individuals as long as they extend within their “horizons”. Thus, a first approach to infinity occurs when numbers that are very large, for example, the number of grains of sand on Earth or numbers that are finitely large, are thought of as infinite 10101010 (Dubinsky et al., 2005a).
The transition from natural infinity to potential infinity is described in Kratka (2013):
The difference between natural infinity and potential infinity is clearly manifested, for example, in the problem of the existence of the intersection of divergent lines whose images on the paper do not intersect. While a student with the concept of natural infinity rejects the existence of an intersection, a student with the concept of potential infinity lengthens the images of straight lines and confirms the existence of an intersection. (p. 98)
On the other hand, the omega-epsilon position of infinity:
[It] is a transitional developmental phase between potential and actual infinity and that it is created predominantly by means of primary intuition, now when the individual is forced to change his potential approach to infinity to the actual one by applying a new context. (Cihlář et al., 2015, p. 70)
Lastly, according to Krátká et al. (2021), in the actual conception of infinity, all horizons are already destroyed. According to these authors, “the concept of infinity is inevitably closely linked to the concept of the horizon as ‘a line’ separating the illuminated (visible) part of an observed (or known) object from the unilluminated (or unknown, respectively)” (Krátká et al., 2021, p. 4). These ways of categorizing and exemplifying each of the stages through which the subject goes to conceive infinity will serve to classify the conceptions of the teachers in this study.
METHOD
As mentioned earlier, a study was conducted with the purpose of contrasting the results obtained from this application with student results reported in the literature. The hypothesis expected to be tested is that the instrument is equally applicable to teachers and reveals the same conceptions as students. The instrument for the study consists of five items and is the same as the one shown in Krátká et al. (2021, pp. 9–10) with certain modifications, allowing teachers to justify their answers in an open manner. Each one with the intention of exploring the concept of infinity in two contexts: arithmetic and geometric, and in two types of views:
‘infinitely large’ and ‘infinitely close’. This questionnaire is chosen because, according to the authors, it enables the identification of the four conceptions of infinity in each item in both arithmetic and geometric contexts, as well as in an ‘infinitely large’ perspective — according to Manfreda Kolar and Čadež (2012) — and an ‘infinitely close’ perspective — according to Manfreda Kolar & Čadež (2012) — (Krátká et al., 2021, pp. 12–13). Teachers were provided with the printed instrument and given 50 minutes to complete it. Participants signed to acknowledge and voluntarily agreed to participate in this research.
Participants
For the implementation, eight in-service teachers were considered — five at the secondary level and three at the high school level — and one teacher in training at the secondary level, all from different educational institutions with public or private support in the State of Puebla, Mexico. All teachers were selected from a group studying mathematics didactics in a master’s degree program in educational mathematics at the time of the application. Only one teacher did not respond to item 4. For classification and subsequent analysis, the acronyms P1-BL, P2- BL, P3-BL, P4-SL, P5-SL, P6-SL, P7-SL, P8-SL, and P9-IT were used. These acronyms represent, respectively, Baccalaureate Level (BL), Secondary Level (SL), and In Training (IT).
RESULTS AND DISCUSSION Results
In Table 1, the absolute frequencies of the conceptions of infinity that the teachers presented in each of the contexts and views of the instrument are shown. For item 1, which constitutes an arithmetic context in an ‘infinitely large’ view, the majority — six teachers — have a conception between potential infinity and omega-epsilon position. Item 3 is analogous, considering the same view, but there are contradictions in the teachers’ responses to these two items.
Table 1. Results of the Conception of infinity present in in-service teachers
Items Context View
Conception Natural Potential Omega-
Epsilon
Position Actual
2 and 5 Geometric ‘infinitely large’ 5 3 1 0
‘infinitely close’ 7 2 0 0
1, 3 and 4 Arithmetic ‘infinitely large’ 1 3 3 2
‘infinitely close’ 0 5 1 2
For instance, a professor, when asked about the largest number he satisfies 𝑥 − 1 > 2, mentions that it would be infinite as a representation of a large number (see Figure 1), but in item 3, when asked about the largest real number, indicates that it does not exist (see Figure 2).
With ∞, we can refer to any number greater than 3
Figure 1. P6-SL’s response to item 1c) — What is the largest number that satisfies the equation 𝑥 − 1 >
2?
It doesn’t exist. There is always the possibility of writing a number greater than the previous one; that’s where ∞ makes sense
Figure 1. P6-SL’s response to item 3) — What is the largest real number?
This indicates that in the preceding inequality, the teacher attempts to provide a number as a solution, and since it is always increasing, it demonstrates a conception of potential infinity.
However, later, the teacher states that said number does not exist; therefore, there is an attempt to transition towards actual infinity.
The interval that satisfies the inequality is [3, ∞]. But considering that the set of real numbers is infinite, it is not possible to determine the largest number that satisfies the inequality. The ∞ does not represent a number
Figure 2. P9-IT’s response to item 1c) — What is the largest number that satisfies the equation 𝑥 − 1 >
2?
The set of real numbers, ℝ, is an infinite set; therefore, it is not possible to define what the largest real number is
Figure 3. P9-IT’s response to item 3) — What is the largest real number?
Between two real numbers, there are infinite real numbers; therefore, it is not possible to define what the smallest real number greater than zero is.
Figure 4. P9-IT’s response to item 4) — What is the smallest real number larger than zero?
Considering that the line segment 𝑝 is infinite, then the lengths of 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅
will also be infinite. But considering that the line segment 𝐴𝑌̅̅̅̅ starts before the line segment 𝐵𝑍̅̅̅̅, its length will be smaller. There are infinities larger than others!
Figure 5. P9-IT’s response to item 2e) — What can you say about the lengths of the 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ lines?
Another teacher demonstrated a conception of actual infinity in items 1, 3, and 4 (see Figure 2, Figure 3 and Figure 4), which correspond to an arithmetic context. However, in items 2 and 5, in a geometric context, the teacher presented conceptions of potential infinity (see Figure 5 and Figure 6). This is possibly due to the limitations of the illustrations provided to understand the differences between a line segment and a straight line, or other factors, as mentioned by Krátká et al. (2021):
In a geometric context, such misunderstandings may occur, for example, when the teacher says that a segment has infinitely many points. This means, among other things, that the number of points on the segment does not depend on its length. For a student making use of the natural infinity conception, however, two segments of two different lengths have a different number of points, although in both cases they are infinite. (p. 16)
Even in item 5, said teacher presents contradictions in two sections: 5d), by stating that the area cannot be known because it is infinitely small, but in 5f), he compares said area with another and even determines that it is larger (see Figure 6). Therefore, it could be assumed that the professor presents a conception of natural infinity in an attempt to overcome potential infinity.
It is not possible to determine the triangle ∆𝐴𝐵𝑌 with the smallest possible area, considering that there are infinite points between points 𝐵 and 𝐶.
The area of ∆𝐴𝐵𝑌 is greater than the area of ∆𝑆𝐵𝑌 since the length of its base is double that of the ∆𝑆𝐵𝑌 triangle
Figure 6. P9-IT’s responses to items 5d) — Find the point 𝑌 on the 𝐵𝐶̅̅̅̅ side so that the triangle ∆𝐴𝐵𝑌 will have the smallest possible area — and 5f) — What can you say about the areas of the triangles
∆𝐴𝐵𝑌 and ∆𝑆𝐵𝑌? — respectively.
Thus, only one teacher evidenced a conception of omega-epsilon position in a geometric context, particularly in item 2. In his created image, he places the points at the end of the straight line, indicating a conception of infinite potential. It can be said that his conception is still at the limit of his horizon, but surpassing potential infinity since he considers points at infinity when explaining his answer in the next section (see Figure 7).
Given that ∞ can be different, meaning that there can be infinities larger than others, we cannot conclude if they are greater, smaller, or equal
Figure 7. P3-BL’s responses to items 2) — Construct segment 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ with point Y and Z, which lies given straight line p with the segment 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ as long as possible — and 2e) — What can you say
about the lengths of the 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ lines? — respectively.
He even argues that straight lines cannot be compared because they are infinite. It is worth mentioning that said teacher commented that after handing in the test, he realized that he had identified a contradiction in that item because his drawing did not reflect what he wrote.
Therefore, he considered himself in an omega-epsilon position on the way to an actual infinity.
Contrarily, another professor drew the points at infinity without placing the points at the end of the given figure (see Figure 8). However, when arguing his answer, although he acknowledges that the straight line is infinitely extended, he still compares the size of the resulting lines as if they were infinities of different sizes, attempting to analogize the cardinality of infinite sets (see Figure 5). For this reason, it is considered that he presented a conception of infinite potential.
Figure 8. P9-IT’s response to item 2) — Construct segment 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ with point Y and Z, which lies given straight line p with the segment 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ as long as possible.
Discussion
In general, in the geometric context, the teachers drew the points at the ends of the figure and showed limitations in the concept of a straight line. Although some of them indicated that the points are at infinity, they drew them at the ends, which implies an obstacle in their mental image of a straight line. On the other hand, in the arithmetic context, the teachers included infinity as a representation of a number that never ends — potential infinity —. They accepted that they cannot write a larger number in situations of the ‘infinitely large’ view, but when it comes to a view ‘infinitely close’, there are limitations in accepting that there is no last number closer to zero. This is in accordance with what was reported in research with students and teachers (Date-Huxtable et al., 2018; Eisenmann, 2008; Juter, 2019; Kattou et al., 2010; Krátká et al., 2021; Manfreda Kolar & Čadež, 2012; Wistedt & Martinsson, 1996; Yopp et al., 2011).
The results of this study present similarities to other investigations, for example, in Kattou et al. (2010), more than 70% of the teachers interviewed have a conception of potential infinity, and a little less than 30% have a conception of actual infinity. In this study, teachers showed a tendency towards a natural and potential conception in both geometric and arithmetic contexts. Contrary to the fact that there is a greater conception of omega-epsilon position and actual infinity in arithmetic than geometric situations. In fact, in geometric situations, there were no teachers who presented conceptions of actual infinity. This is worrying since, according to Krátká et al. (2021), it should disappear with the age of the students, and conversely, it should promote the appearance of the actual infinity as they advance in their training.
In research with students by Manfreda Kolar and Čadež (2012), it was reported that students perform better with ‘infinitely large’ tasks than with ‘infinitely close’ ones. In this study, in general, regardless of whether they are in geometric or arithmetic contexts, the items related of ‘infinitely large’ or ‘infinitely close’ are approximately similar for the surveyed teachers. However, in Kratka (2013), it is mentioned that sometimes finite quantities exceed the student's horizons and therefore are considered infinite. This was the case with a professor who was asked about the largest real number, mentioning that it is impossible for him to write it, as he could not even give it a name (see Figure 9). Thus, in this case, the teacher presents a conception of natural infinity.
I don't know it; I could mark 99999 … until I finish the sheet and wouldn't reach it, because I also wouldn't know its name. ℝ almost ∞
Figure 9. P4-SL’s response to item 3) — What is the largest real number?
On the other hand, as mentioned before, in Krátká et al. (2021), it is stated that the conception of natural infinity disappears with the age of the students, however, the study observed a greater tendency toward said conception in geometric rather than arithmetic contexts.
Some teachers expressed that if the straight line were extended beyond the figure shown, there is no endpoint, but if they stayed with the figure shown to them, it would have an endpoint at the ends. Here we can sense confusion between a line segment and a straight line or an epistemological obstacle to the mental image they have of a straight line (see Figure 5 and Figure 10).
It could be A if the limit of line p is the figure, but if line 𝑝 continues its path, we could say that points 𝐴𝑌̅̅̅̅ and 𝐵𝑍̅̅̅̅ would be endless. Although in my opinion, based on the figures, 𝐴𝑌̅̅̅̅ < 𝐵𝑍̅̅̅̅ since, even though the point closest to 𝑝 is 𝐴𝑌̅̅̅̅, it is a longer line than 𝐵𝑍̅̅̅̅
Figure 10. P2-BL’s response to item 2e) — What can you say about the lengths of the AY̅̅̅̅ and BZ̅̅̅̅ lines?
While there are limitations in this study, such as the fact of a small sample of teachers; this research expands the study of the stages of infinity with in-service teachers, and future work could generalize the findings with a larger sample. Additionally, another future objective is the development of activities in geometric contexts that allow teachers to overcome misconceptions of infinity. Measuring the impact of these activities on teachers to overcome misconceptions would be an interesting subject for another research project.
CONCLUSION
Teachers have limitations in expressing their arguments in each of the situations. For example, teacher P3-BL, who at the end of the test, commented to the instrument’s applicator that he made a mistake in item 2 because the drawing has a “cheat” since it is a line segment and not a straight line. The teacher knows the difference between a line segment and a straight line; however, this may suggest that the mental image he has of a straight line conflicts with the image he can represent on paper. Thus, the need for real meaning can limit the conception of infinity.
As has been previously revealed, the majority of teachers present natural and potential conceptions of infinity in both contexts, although in a greater proportion in geometric than in arithmetic contexts. Likewise, there are contradictions in the arguments of some teachers. For
example, teacher P9-IT presents conceptions superior to potential in arithmetic contexts, but this is not the case in geometric contexts. In fact, no professor presented conceptions of actual infinity in geometric contexts, which is concerning because, according to Krátká et al. (2021), conceptions of natural infinity disappear as subjects advance in their academic training.
Therefore, teachers need new situations where infinity emerges so that they can construct it as actual infinity.
As stated at the beginning of this research, the purpose was to identify and categorize the conceptions that in-service teachers have about infinity and compare them with those reported in the literature with students. The evidence from this study leads to verifying that the conceptions presented by the teachers here and the results shown in Krátká et al. (2021) are similar. Thus, the hypothesis that teachers exhibit misconceptions about infinity in each of the stages, similar to students, has been confirmed. However, future work must contemplate geometric situations that allow the natural or potential conception of infinity to be overcome and for the teacher to build an actual conception to a greater extent.
Declarations
Author Contribution : IAD-E: Writing - Original Draft, Formal analysis, Data Curation, Investigation, Funding acquisition, Methodology, Software; JAJ-L:
Conceptualization, Writing - Review & Editing, Project administration, Resources, Supervision, Validation.
Funding Statement : This research was funded with support from the National Council for Humanities, Sciences, and Technologies (CONAHCYT, for its acronym in Spanish) under scholarship number CVU 569097.
Conflict of Interest : The author declares no conflict of interest.
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