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2022. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). Proceedings of the 45th Conference of the

International Group for the Psychology of Mathematics Education, Vol. 2, pp. 59–66). PME.

HOW DO FIRST-GRADE STUDENTS RECOGNIZE PATTERNS?

AN EYE-TRACKING STUDY

Lukas Baumanns1, Demetra Pitta-Pantazi2, Eleni Demosthenous2,

Constantinos Christou2, Achim J. Lilienthal3, Maike Schindler1

1University of Cologne, 2University of Cyprus, 3Örebro University

Recognizing patterns is an important skill in early mathematics learning. Yet only few

studies have investigated how first-grade students recognize patterns. These studies

mainly analyzed students’ expressions and drawings in individual interviews. The

study presented in this paper used eye tracking in order to explore the processes of 22

first-grade students while they were trying to recognize repeating patterns. In our

study, we used numerical and color pattern tasks with three different repeating patterns

(i.e., repeating unit is AB, ABC, or AABB). For each repeating pattern task, students

were asked to say the following object of the given pattern. For these patterns, we

identified four different processes in recognizing repeating patterns. In addition, we

report differences in the observed processes between the patterns used in the tasks.

INTRODUCTION

Mathematics can be described as the science of patterns (Steen, 1988). In early

mathematical learning, patterns play a decisive role in the development of algebraic

thinking (Carraher & Schliemann, 2007; Clemens & Sarama, 2007). Being aware of

patterns provides primary students with a mindset that is useful in the later study of

algebra (Schoenfeld, 2007). In addition, pattern recognition is a central content of

mathematics education in primary school (NCTM, 2000), which makes it a significant

topic for mathematics education.

Yet, pattern recognition poses challenges to many students. For example, Clarke et al.

(2006) found that for repeating patterns (e.g., ABABAB) first graders are successful in

recognizing and extending them in only 31% of the tasks. This calls for support of

young children in their ability to recognize patterns. For being able to foster children’s

pattern-recognition ability, it is crucial to investigate and understand their pattern-

recognition processes (Lüken, 2018; Papic et al., 2011). Empirical studies mostly

explore pattern-recognition processes based on children’s expressions and drawings.

The present pilot study in the context of the Erasmus+ project DIDUNAS investigates

students’ pattern-recognition processes using eye tracking, the recording of eye

movements. Eye tracking has shown to be useful for investigating processes of children

on early primary level before (Schindler et al., 2020; Sprenger & Benz, 2020). Our

study uses eye movement video analysis to explore pattern-recognition processes.

In this pilot study, the aim is to investigate (1) what processes first-graders use in

recognizing patterns and (2) whether there are differences in students’ use of pattern-

recognition processes between different kinds of patterns.

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

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THEORETICAL BACKGROUND

Pattern recognition in early mathematics learning

One goal of early mathematical learning is the development of algebraic thinking

(NCTM, 2000). Algebraic thinking in early grades includes noticing structures (Kieran,

2004). Fostering students in their ability to notice structures and recognize patterns can

therefore contribute to their development of algebraic thinking. (Carraher &

Schliemann, 2007). In the first grades of primary school, repeating patterns can be used

to foster students’ ability to recognize patterns (Clemens & Sarama, 2007). Repeating

patterns refer to patterns that have one unit (e.g., AB) that repeats multiple times

(ABABAB). Patterns differ from each other if the repeating unit is of different length

or the elements of a repeating unit are arranged differently. For example, pattern

ABABAB is different from pattern ABCABCABC, or pattern AABCAABC is

different from ABACABAC.

For pattern recognition, it is of particular importance to recognize the recurring unit of

a pattern (Papic et al., 2011). The repeating unit can be represented, for example,

symbolically (e.g., AB), numerically (e.g., 2 5), or by colors (e.g., ● ●). The present

study uses numerical and color patterns. Common pattern tasks for preschoolers and

first graders consist of identifying and extending given repeating patterns (Rittle-

Johnson et al., 2015). For example, students are given a color pattern (e.g., ● ● ● ● ●

● ●) and asked to say what color the next dot should be. In this paper, student

approaches to extend repeating patterns, which are anticipated to entail identification

of the repeating unit, are referred to as pattern-recognition processes.

Research on pattern recognition at the beginning of primary school

Some studies investigate the abilities of preschoolers or first graders in the context of

repeating or growing patterns. Clarke et al. (2006) found that 76% of the students at

the beginning of first grade can copy a repeating pattern, but only 31% can extend it.

Rittle-Johnson et al. (2015) found similar results in a study with 64 four-year-old

preschoolers. Further studies have used children’s expressions and drawings in

individual interviews to investigate processes of preschoolers and first graders in

recognizing patterns (Lüken, 2018, Papic et al., 2011). For example, Lüken (2018)

found that three- to five-year-old children use a process of comparison to compare the

beginning of the repeating pattern with the part that has to be extended. Lüken also

found that the repeating unit was identified and used by the students. Papic et al. (2011)

identified similar pattern-recognition processes.

In this study, we were interested in pattern-recognition processes of first-grade

students. In contrast to previous studies on this topic, we use eye tracking videos (not

student utterances or drawings) to explore student pattern-recognition processes, since

eye tracking has proven itself valuable to identify student processes in mathematics

(e.g., Schindler et al., 2019, 2020). We ask the following research questions.

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

PME 45 – 2022 1 - 3

1. What processes do first-grade students use in recognizing repeating patterns?

2. Are there differences in students’ use of pattern-recognition processes

between different kinds of repeating patterns?

METHODS

Participants, procedure, and tasks

The study was conducted with 22 first-grade students (age: M = 6.80 years; SD = 0.24

years) from a primary school in Cyprus. Fourteen (~63.6%) of the students had Greek

as their mother tongue, the others Arabic (n=7, ~31.8%) and Bulgarian (n=1, ~4.5%).

In addition to the eye-tracking study (see below), we conducted the standardized

mathematics test ZAREKI-K for assessing students’ mathematical performance level

at the transition from kindergarten to primary school (von Aster et al., 2009). We used

the adapted version by Walter (2020). The test indicates that twelve of the 22 students

(~54.5%) are not at risk for developing math difficulties, but ten are at risk. Thus, the

sample has a good spectrum in terms of performance levels.

In the main part of the study, the students worked individually on eight pattern tasks

on a computer screen (see Figure 1). Each task consisted of at least three repetitions of

a unit followed by a white blob. The students were asked to name the number or color

of the object that was hidden behind the white blob (e.g., 1, yellow). Before the first

numerical and the first color pattern task, the students worked on a sample task, to

ensure that the students understood the task correctly. The following three repeating

units were used in the pattern tasks: (1) AB (four tasks), (2) ABC (two tasks), and (3)

AABB (two tasks). The students answered by saying aloud the number or color they

thought was behind the white blob. The students did not receive feedback and incorrect

answers were not corrected. Four tasks had numerical repeating units in form of digits

(e.g., 4 1), four in the form of colored dots (e.g., ● ●).

Figure 1: Numerical and color repeating pattern tasks used in the study.

Eye tracking

Students’ eye movements were recorded with the screen-based eye tracker Tobii Pro

X3-120 (infrared, binocular), with a sampling rate of 120 Hz. The tasks were presented

on a 24” monitor. The students’ heads were about 60–65 cm away from the monitor.

The eye-tracking data showed an average accuracy of 1.37°, which corresponded to an

error of about 1.44–1.55 cm on the screen at a head distance of 60–65 cm. The centers

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

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of the digits and dots in our tasks were on average 4.16 cm apart from one another on

the display on the monitor (3.2–5.5 cm), which means that the eye-tracking accuracy

was sufficient to reliably determine what element the students looked at.

Analysis of eye-tracking data

Raw gaze-overlaid videos provided by Tobii Pro Lab software were used to analyze

students’ pattern recognition processes. In addition, notes were taken during the data

collection describing student actions (e.g., when students pointed to the monitor).

Although pattern-recognition processes have already been elaborated in research, an

inductive approach was chosen for this study. We performed a qualitative content

analysis through a data-driven inductive category development (Mayring, 2000),

similar to Schindler et al. (2019, 2020), in four stages: Stage one: A randomly selected

half of the gaze-overlaid videos were viewed and for each video, the gazes were

described. Stage two: Similar descriptions of the gaze-overlaid videos were subsumed

into one category, while categories in this study refer to pattern-recognition processes.

A first general description of the respective pattern-recognition process was

formulated. Stage three: The second half of the gaze-overlaid videos were viewed and

coded using the pattern-recognition processes elaborated in stage two. During this

coding, existing descriptions of pattern-recognition processes were revised and

specified, and new processes were added when existing processes did not seem suitable

for describing the gaze-overlaid videos. Stage four: Finally, with the complete category

system, the first half of the gaze-overlaid videos were re-coded to check the fit of the

revised process descriptions and to assess the emergence.

All gaze-overlaid videos were coded by the first author. 22.7% of the videos were

analyzed independently by the last author. The interrater agreement was calculated

using Cohen’s Kappa (Cohen, 1960). With κ = 0.87, the inter-rater agreement is almost

perfect (Landis & Koch, 1977).

Statistical analysis

To determine differences between the pattern-recognition processes and the different

patterns, a two-tailed Fisher–Freeman–Halton exact test for r×c contingency tables

was performed (Freeman & Halton, 1951) using SPSS 28. This test is an extension of

chi-square test and is especially suited for small sample sizes for which the chi-square

approximation does not hold (Fagerland et al. 2017). For this analysis, we have

grouped the different patterns according to their structure (see Figure 1). For example,

the pattern tasks with repeating unit 4 1 and ● ● in Figure 1 fall into group AB.

RESULTS

Pattern-recognizing processes

In the following, we describe the pattern-recognition processes found through the

analysis of gaze-overlaid videos of the first-grade students. We use gaze plots to

visualize the processes for this paper, even though the analysis was based on the videos.

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

PME 45 – 2022 1 - 5

Figure 2 shows idealized gazeplots of these processes as illustrations. These idealized

gazeplots are not actual gazeplots of children, but idealized illustrations for the

identified processes.

(1) Identifying one repeating unit of the pattern

The gazes go to one repeating unit (sometimes multiple times)—mostly the repeating

unit before the white blob. The gazes partially also touch one dot/number before the

repeating unit.

(2) Identifying one repeating unit and validating/applying it

(a) Identifying and validation: The gazes go to the repeating unit before the blob

(sometimes multiple times) and then go to another repeating unit in the pattern

(sometimes multiple times). Afterwards, the pattern is extended by continuing the

repeating unit before the blob.

(b) Identifying and application: The gazes go over a repeating unit in the beginning or

the middle of the pattern (sometimes multiple times). Afterwards, the pattern is

extended by continuing the repeating unit before the blob.

(3) Looking at each element

The gazes go to each dot/number of the pattern individually usually from left to right

(sometimes multiple times). Up to three dots/numbers are skipped from the beginning.

(4) Unsystematic jumping over the pattern

The gazes jump fast over the pattern. Often the blob is not looked at. There is no

systematic process recognizable.

Figure 2: Idealized gazeplots of the pattern-recognition processes with numbers

indicating the order in which the students looked at each dot.

Differences in pattern-recognition processes between different patterns

To determine differences between the identified pattern-recognition processes (see

Figure 2) and the different kinds of patterns (see Figure 1), Fisher–Freeman–Halton

exact test for r×c contingency tables was performed (see Table 1). The test revealed

that the pattern-recognition processes used by the students differed significantly

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

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between the three kinds of patterns (p = 0.031). Cramér’s V = .20 indicates a moderate

relationship between the kind of pattern and pattern-recognition processes.

Pattern-recognition process

(1)

(2)

(3)

(4)

Total

Repeating

unit

AB

35

24

23

0

82

ABC

12

22

5

1

40

AABB

15

16

7

2

40

62

62

35

3

162

Table 1: Observed number of pattern-recognition processes for the different patterns.

Figure 3 shows the distribution of the pattern-recognition processes over the three

kinds of pattern tasks based on the absolute values in Table 1. Figure 3 illustrates that

for patterns of kind AB, students used process (1) identifying one repeating unit, the

most with 42.68%. Process (2) identifying one repeating unit and validating it, and (3)

looking at each element, were equally distributed in the pattern tasks with the repeating

unit AB. Process (4) unsystematic jumping over the pattern, in contrast, did not occur.

In the pattern tasks with repeating unit ABC, process (2) was identified most often with

55%. The other processes occurred less often. Processes (3) and (4) together were

identified only half as often as process (1). For patterns of kind AABB, processes (1)

and (2) were identified with almost equal frequency, 37.5% and 40%, respectively.

Processes (3) and (4) were identified slightly more often than in patterns of kind ABC.

Category (4) appeared exclusively together with wrong answers.

Figure 3: Distribution of pattern-recognition processes over the kinds of patterns.

DISCUSSION

This pilot study aimed to investigate what processes first-grade students use in

recognizing repeating patterns and whether there are differences in students’ use of

pattern-recognition processes between different kinds of repeating patterns. The study

was conducted with 22 first-grade students and eight repeating pattern tasks in which

students were to name the number or color of the next object of a given pattern. The

results of our study show that the first-grade students used four different pattern-

recognition processes in numerical and color repeating pattern tasks: (1) Identifying

Baumanns, Pitta-Pantazi, Demosthenous, Christou, Lilienthal, & Schindler

PME 45 – 2022 1 - 7

one repeating unit of the pattern, (2) identifying one repeating unit and

validating/applying it, (3) looking at each element, and (4) unsystematic jumping over

the pattern. Our results connect to some of the pattern-recognition processes identified

in previous studies (Lüken, 2018; Papic et al., 2011) and extend them. Furthermore,

we found that these processes were used differently between different kinds of patterns

(see Figure 3).

These results should be interpreted considering the following limitations: With 22

students, a relatively small sample was available. It cannot be discounted that with a

larger sample, additional pattern-recognition processes could be identified. Also, in

future studies, more than eight pattern tasks should and will be conducted. In particular,

other patterns than those used in this pilot study (i.e., AB, ABC, AABB) need to be

investigated (e.g., AAB, AABC, ABAC).

The results of this study hinted at the value of using eye tracking to explore students’

pattern-recognition processes. With regard to the overall purpose of supporting

children in pattern recognition processes, the study has shown that eye tracking can

inform about student strategies and that these insights can help to support students

adaptively and individually in developing further their pattern-recognition processes.

In line with this, one aim of the DIDUNAS project is to develop teacher materials that

serve to support students, for example, in pattern recognition. The results of this pilot

study provide initial insights for the development of such teacher materials.

Acknowledgment

This project has received funding by the Erasmus+ grant program of the European

Union under grant agreement No 2020-1-DE03-KA201-077597.

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