Observing Mathematical Learning Experiences in the Primary Years to Examine How Attitudes Towards Mathematics Are Enacted in the Classroom

Abstract

Observing students during mathematical learning experiences provides great insights into their mathematical experiences. Traditionally, observations in the mathematics classroom had a cognitive focus, examining teacher practice or students’ conceptual understanding and engagement in mathematics. There is a dearth of research that uses classroom observations to observe how affective factors, such as attitudes towards mathematics, are enacted during mathematical learning experiences. The purpose of this paper was to document a replicable process to examine primary students’ attitudes towards mathematics from the perspective of how attitudes are enacted during mathematical learning experiences. Drawing on three separate studies conducted with Australian primary students (5–12 year olds), this paper documented an observational framework and schedule that can be used to investigate attitudes towards mathematics. The framework and schedule have been used to conduct 50 observations in all primary year levels, including multi-year classrooms. The aim of the study reported here was to bring to the surface the complex and multifaceted aspects of attitudes towards mathematics as children implement and talk about the mathematics that they learn. As well as contributing to the literature on attitudes, these findings have provided insights into methodological considerations of conducting observations.

1. Introduction

Class observations provide the opportunity to collect live data from ‘naturally occurring’ situations, yielding more authentic data than that collected through inferential means (L. Cohen et al., 2011, p. 456). Further, L. Cohen et al. (2011) posits that observation provides opportunities to record interactions and non-verbal behaviour whereby factors sensitive to the context in which the observation occurs can be understood. It is through these ‘naturally occurring’ situations that aspects of affect can be identified and inferred, such as attitudes towards mathematics, including the emotional dimension (Di Martino, 2004). Researchers have lamented “the need to infer aspects of affect from observable behaviour” and in particular, observations that use “specifically designed instruments or settings” (Attard et al., 2016, p. 89). While observing mathematical learning experiences is attracting recent attention, many studies have focused on teacher instruction or conceptual development. For example, Stites and Brown (2021) observed preschool teachers and their use of pedagogical strategies to develop mathematical conceptual understanding. At the same time, they developed a “diagnostic instrument for observing and analysing children’s mathematical development”. Several researchers have indicated the need to investigate students’ attitudes towards mathematics and how they are enacted during mathematical learning experiences (Attard et al., 2016; Di Martino & Zan, 2010; Grootenboer & Marshman, 2016; Ingram et al., 2020; Quane et al., 2023).

Qualitative research has been used to investigate affective factors in mathematics for decades. However, studies using quantitative methods far exceed studies that employ qualitative research techniques. Researchers have used a range of qualitative research methods, including focus groups (Grootenboer & Marshman, 2016; Grootenboer et al., 2002), creative interview techniques (McDonough, 2002; McDonough & Ferguson, 2014; McDonough & Sullivan, 2014) and children’s drawings (Bachman & Neal, 2018; Picker & Berry, 2001; Quane, 2024; Quane et al., 2023) to investigate a variety of affective factors, including attitudes towards mathematics. However, despite the considerable amount of research, there is still a dearth of research investigating the nature of primary students’ attitudes, particularly in a lesson context (Attard et al., 2016; Ingram et al., 2020; Quane et al., 2019). There is more to learn about children’s mathematical learning experiences, specifically research that is conducted in classroom contexts (Attard et al., 2016; Niss, 2007).

Students need various mathematical experiences to develop conceptual understanding, aspects of working and thinking mathematically, and the growth of positive attitudes towards mathematics (Tucker, 2014; Yelland et al., 2014). Three studies conducted by the author are reported in this paper with the aim of investigating the complex and multifaced aspects of primary students’ attitudes towards mathematics (Quane et al., 2023) during mathematical learning experiences. To achieve this aim, observations of mathematical learning experiences in the primary years were conducted in order to ascertain their attitudes towards mathematics. Before the observations were conducted, individual children’s attitudes were ascertained as a baseline measure using a drawing methodology whereby participants drew themselves doing mathematics, provided a written description of their drawing, and participated in a semi-structured interview (Quane, 2024; Quane et al., 2023). Specially designed instruments were designed to “infer aspects of affect from observable behaviour” as recommended by (Attard et al., 2016, p. 89). The research reported in this paper comes from a multistage research project that investigated the range and nature of primary students’ attitudes towards mathematics, which included research conducted in a non-lesson context (via children’s drawings, written descriptions and interview responses) and lesson contexts (via observations). It is the latter that is reported in terms of the methodological approach employed in all three studies.

This investigation answered the broad question: How are primary students’ attitudes towards mathematics enacted during mathematical learning experiences? It is important to distinguish between the range and nature of attitudes towards mathematics. In this article, a distinction is made to ensure clarity around these two words. The range refers to the scope or extent of primary students’ attitudes towards mathematics, providing a broad view of the issue. The nature is descriptive, providing the basic qualities, structure and essence of individual attributes of primary students’ attitudes towards mathematics—in other words, the nuances or fine-grain view of attitudes (Quane et al., 2023).

1.1. Attitudes Towards Mathematics

Attitude comprises affective, behavioural, emotional and cognitive factors (Grootenboer et al., 2008) as well as sensory, environmental, personal (Yasar, 2016) and external factors (Goodykoontz, 2008; Grootenboer & Marshman, 2016). These factors contribute to an understanding of student participation, performance and engagement in mathematics (Parsons et al., 2014). Students’ attitudes have a profound influence on student achievement and success in mathematics (Hemmings et al., 2011; Kibrislioglu, 2015; Köğce et al., 2009; Stiles et al., 2015; TIMSS, 2016). In addition, schools are becoming more accountable for students’ achievement (OECD, 2012; Parsons et al., 2014). Thus, it is necessary to understand attitudinal factors and how they influence attitudes towards mathematics. Such an understanding would provide valuable insight into understanding student achievement in mathematics.

Extensive research has been conducted to measure students’ attitudes towards mathematics, which has been instrumental in identifying negative attitudes towards mathematics. However, early research has “not focused on understanding the underlying reasons for the problem” (Grootenboer & Marshman, 2016, p. 19). Also, limited research has been conducted into primary students’ attitudes towards mathematics and research into how attitude develops before Year 3 is scant (Ingram et al., 2020). Further, understanding the developmental trajectory of attitudes towards mathematics is required “to deepen our understandings” of how attitudes “are lived in different contexts” (Pepin, 2011, p. 536). The research reported in this paper was conducted across all levels of primary school, adding to the field of research into the construct of attitudes towards mathematics.

1.2. Theoretical Framework

To reiterate, the aim of this study was to investigate the complex and multifaced aspects of primary students’ attitudes towards mathematics during mathematical learning experiences. To capture the complexity of these attitudes requires a robust theoretical framework. The Three-dimensional Model of Attitude (TMA) is a framework that can “classify attitudes towards mathematics, identify typical or recurrent patterns” in students’ attitudes (Di Martino & Zan, 2010, p. 44) and can be used to investigate attitudes towards mathematics in general and towards specific aspects of mathematics such as problem-solving (Di Martino, 2019). However, there are some limitations of using the TMA, as identified by (Di Martino, 2019; Di Martino & Zan, 2010, 2011). The original TMA reduces each dimension to a dichotomy. The Emotional Dimension (ED) is scored as positive/negative, while the Vision of Mathematics (VM) is scored as relational/instrumental and Perceived Competence (PC) as high/low (Di Martino & Zan, 2010). Using a dichotomy for each dimension overlooks the range of attitudes. Further, the TMA does not take into consideration the development of attitude within the child, as reported by Di Martino (2019) when investigating young children’s attitudes towards mathematical problem solving. This may be due to the original use of the TMA to analyse adolescents’ autobiographical essays.

Di Martino (2019) proposed the need for “the development of new methods to collect data” to investigate young students’ attitudes towards mathematics (p. 305). The TMA was extended to cater to the developmental nature of attitude, making it a suitable framework to investigate primary students’ attitudes towards mathematics. Extending the framework provided an opportunity to listen to students’ voices, the importance of which Di Martino and Zan (2010) acknowledged, stating “our study made us convinced of how important it is to let these voices talk” (Di Martino & Zan, 2010).

With this acknowledgement, the TMA was modified to analyse and report the research findings. The TMA was modified to accommodate the age of participants and the research techniques employed. The TMA was modified in two main ways. First, each dimension was divided into two sub-dimensions to capture a more nuanced view of primary students’ attitudes towards mathematics. The sub-dimensions were developed to analyse the generated data and develop thick descriptions that form the basis of rich narratives describing the nature of primary students’ attitudes towards mathematics. Additionally, the sub-dimensions were developed to be able to identify and report the range of possible attitudes and any possible significant contributing factors to primary students’ attitudes towards mathematics. Figure 1 presents the Modified TMA where the original dimensions are shown at the top of each dimension and the sub-dimension are below the original dimensions.

The second modification was to expand the dichotomy to include six attitude classifications for each sub-dimension. The attitude classification labels are not the same for each sub-dimension due to the varying phenomena being described and observed and are shown in

Table 1. Description of the Modified TMA Sub-Dimensions.

Dimension	Sub-Dimension	Description	Attitude Classifications
Emotional Dimension	Emotional Tendency (ET)	Students’ feelings and emotional responses experienced during mathematical activities and tasks	Cannot be Classified
			Extremely Negative
			Negative
			Neutral
			Positive
			Extremely Positive
	Overall Sentiment (OS)	Students’ general reactions and emotional beliefs regarding mathematics, including non-verbal cues (posture, gestures and body language) and verbal cues over a series of mathematical learning experiences	Cannot be Classified
			Extremely Negative
			Negative
			Neutral
			Positive
			Extremely Positive
Vision of Mathematics	Tasks, Topics and Processes (TTP)	Types of mathematical learning experiences and processes students engage with during mathematical learning experiences; the mathematical topics and how students communicate their mathematical understanding and learning	Cannot be Classified
			Minimal Vision of Mathematics
			Low Vision of Mathematics
			Developing Vision of Mathematics
			High Vision of Mathematics
			Exemplary Vision of Mathematics
	Value and Appreciation (VA)	How and what students view as important and acknowledge as worthwhile about mathematics during mathematical learning experiences. That is, what students engage or disengage (actively or passively) with during mathematical learning experiences	Cannot be Classified
			No Value of Mathematics
			Low Value of Mathematics
			Some Value of Mathematics
			High Value of Mathematics
			Very High Value of Mathematics
Perceived Competence	Mathematical Mindset (MM)	Students’ perceptions of themselves related to their ability to do mathematics during mathematical learning experiences	Cannot be Classified
			Fixed Mindset
			Low Growth Mindset
			Mixed Mindset
			Growth Mindset
			High Growth Mindset
	Self-Concept (SC)	Students’ beliefs in their mathematical ability and their expectancy for success during mathematical learning experiences	Cannot be Classified
			Extremely Low Perceived Competence
			Low Perceived Competence
			Neutral Perceived Competence
			High Perceived Competence
			Very High Perceived Competence

. All sub-dimensions included the attitude classification “cannot be classified”, which took into consideration situations where a child does not describe or demonstrate the main theme of the sub-dimension. Additionally, the sub-dimensions were developed to be able to identify and report the range of possible attitudes and any possible significant contributing factors to primary students’ attitudes towards mathematics.

2. Materials and Methods

2.1. Participants

The observational framework and schedule have been employed in three separate studies conducted in South Australian Primary Schools by the author. A total of 13 classes have been observed with participants from the first year of formal schooling (5-year-olds) to Year 6 (11–12-year-olds). The first year of formal schooling in South Australia is known as Reception. Other jurisdictions may know this first year of schooling as Kindergarten, Preparation or Foundation. Observations were conducted pre and post the COVID-19 pandemic. Most observations occurred in multi-year classes, which combined two or more consecutive year levels in one class. The number of individual students observed was only a fraction (18%) of all students (N = 277) who participated in all three studies. The year groups were strategically and purposively selected for all observations to develop a detailed understanding of primary students’ attitudes towards mathematics. Each observation was for a minimum of 50 min, whilst the longest observation was for 100 min.

2.2. Methods

This research was both exploratory and descriptive. The broad lens applied to the research is the exploratory element—to understand the nature and range in both lesson and non-lesson contexts. Further, the research is descriptive in nature. As the name suggests, descriptive research highlights the existence and extent of an issue (de Vaus, 2014). This research aimed to accurately and systematically describe primary students’ attitudes towards mathematics in a lesson context where children were observed during mathematical learning experiences. Geertz’s (1993) notion of ‘thick descriptions’ has been used as children’s attitudes are “not reducible to simplistic interpretation” (L. Cohen et al., 2011, p. 17). Thick descriptions have been used to describe the complexities of the two contexts and the nature of attitudes towards mathematics.

The purpose of the classroom observations was to collect firsthand information about actual child behaviour during mathematical learning experiences, generating additional data about children’s relationships with school mathematics. McLeod (1987), recommended that researchers observe, in realistic settings that focus on overt behaviour, what students say and their physical reactions. The case presented draws on three classroom observations where the researcher was a passive observer (L. Cohen et al., 2011), making every effort not to intrude on the learning experiences. All observations were audio-recorded.

Analytical Approach to Developing an Observational Framework

Developing an observational framework that was suitable to ascertain and describe primary students’ attitudes towards mathematics in a lesson context was a critical component of this research. An observational framework was developed, drawing upon the following four methods of analysis:

(a)

Flanders’ Interaction Analysis Categories (FIAC)

(b)

Social Discourse Analysis

(c)

Pragmatics (Social Communication)

(d)

Non-verbal Communication

Interaction analysis using FIAC has long been used to study children’s attitudes and behaviour (Delamont, 1976). FIAC provides a systematic and structured method that can be used to analyse teacher and student interactions (L. Cohen et al., 2011). Ned Flanders, an American education professor, developed FIAC between 1955 and 1960 to ascertain how interactions influence students’ attitudes (Flanders, 1965). Each interaction is classified in regard to response and initiation for both the teacher and children. Four elements of FIAC have been used to analyse students’ communication: response, initiation, silence and confusion (Kim & Ahn, 2017). These four types of communication are outlined later in

Table 2. Observational Framework.

Framework	Child’s Communication	Description
Interaction Analysis (FIAC) Flanders (1965)	Response	Talk by a student in response to the teacher. The teacher initiates the contact or solicits a students’ statement or structures the situation. Freedom to express personal ideas is limited.
	Initiation	Talk by students that they initiate; expressing their own ideas; initiating a new topic; freedom to develop opinions and a line of thought, like asking thoughtful questions; going beyond the existing structure.
	Silence	Pauses, short periods of silence, no talking
	Confusion	Periods of confusion in which the observer cannot understand the communication.
Pragmatics (social communication) Fraser (1983)	Beliefs	A student expresses beliefs that propositions are true.
	Desires	A student expresses a desire concerning the action specified in the proposition.
	Commitment	A student expresses an intention to undertake a commitment associated with the action specified in the proposition.
	Evaluation	A student expresses a personal evaluation towards some past action.
Sociocultural discourse analysis Mercer (2004)	Disputational talk	Disagreement and individual decision making, short exchanges consisting of assertions and challenges or counter assertions. There are few attempts made by the student to pool resources, to offer constructive criticism or make suggestions.
	Cumulative talk	A student speaks positively but uncritically on what others have said. Characterised by repetitions, confirmations, and elaborations.
	Exploratory talk	Students engage critically but constructively with each other’s ideas. Statements and suggestions are offered for joint consideration that may be challenged or counter-challenged, but the challenges are justified, and an alternate hypothesis is offered. Students all actively participate, and opinions are sought and considered before decisions are jointly made. Compared to disputational and cumulative talk, exploratory talk is where knowledge is made more publicly accountable, and reasoning is visible.
Non-Verbal Communication Andersen (2008) Pease and Pease (2006)	Facial Expressions	The face is the primary site of emotional communication. During social interactions, we examined faces to determine an individual’s emotional state. “Facial expressions are usually spontaneously communicated to others without conscious thought or linguistic representation” (Andersen, 2008, p. 145).
	Posture and Body Language	“Emotions can be reliably identified from postures” (Andersen, 2008, p. 147) while “body language is an outward reflection of a person’s emotional condition” (Pease & Pease, 2006, p. 11).
	Gestures	Students’ hand gestures are used to support learning, or to communicate emotion. Gestures are read in clusters and context while looking for congruence; that is, words match gestures. Gestures can provide valuable insights into what a child may be feeling at a given time (Pease & Pease, 2006, p. 11).

, which brings together the features of all four methods. Interaction analysis, where the focus is on children’s attitudes towards mathematics, has to acknowledge that there are several confounding factors contributing to a child’s attitude. One such factor is the teacher, their attitude towards mathematics and their interaction with the child. Observations were analysed regarding significant teacher interactions with the three children and only reported if the interactions were considered to contribute to a child’s emotions, gestures, reaction or behaviour.

Social Discourse Analysis is used in social settings such as classrooms, and Mercer (2004) describes three archetypical forms of children’s talk during lessons. Disputational talk is characterised “by disagreement and individual decision making” and can be in the form of assertions such as “Yes, it is!” Cumulative talk “is characterised by repetitions, confirmations and elaborations”. Exploratory talk is characterised by suggestions and statements that may challenge, and counterchallenge, as well as opinions that are sought and considered (p. 146). For example, a child may offer another child an alternative solution or strategy to a mathematical task, contribute to class discussions, or make statements based on their observations.

In addition, Fraser (1983) described four types of social communication or speech acts that can express attitudes that occur during a conversation. A child can express beliefs that propositions are true, desires concerning the action specified in the proposition, the intention to commit to the action, and finally, personal evaluations towards the action. For example, children may provide propositions that indicate enjoyment (“this is fun”) or disdain (“this is boring” or “I can’t do this”).

Students communicate both verbally and non-verbally, and it would be neglectful not to examine non-verbal communication forms that children exhibit in a lesson context. Non-verbal communication is considered the “original communication system” allowing another person to be able to “read a person’s attitudes and thoughts” (Pease & Pease, 2006, p. 7). For example, body language is considered an outward display of an emotional condition, with each gesture providing insight into what an individual is feeling. Just as an individual’s body language can convey emotion, so too can their facial expression. “The human face is hugely expressive”, allowing someone else to have insight into a person’s feelings at a particular time (D. Cohen, 2007). Further, the production of facial expressions in children is a crucial emotional component that allows others to understand children’s emotions and social motivation. Facial expressions are more easily recognised when the producer is younger (Grossard et al., 2018). This research examined gestures, postures and facial expressions to better understand the emotional dimension of primary students’ attitudes towards mathematics.

Non-verbal communication used in mathematics is attracting increasing attention in the mathematics education literature. One example is the recent research regarding the use of gestures by children and teachers (Alibali & Nathan, 2012; Krause, 2016). Gestures are evidence of an individual’s thought process (Alibali & Nathan, 2012; Krause, 2016; Yoon et al., 2011). Gestures used in the mathematics classroom are considered important in understanding performance, assessment and instruction (Alibali & Nathan, 2012) and in the learning of mathematical concepts (Alibali & Nathan, 2012; Yoon et al., 2011). However, it is yet to be seen how children’s gestures used in mathematics are connected or related to their attitude towards mathematics.

This research aimed to explore children’s attitudes towards mathematics in a lesson context using TMA. The Emotional Dimension of TMA connects emotions to attitude. Emotions are outwardly displayed through an individual’s facial expression, gestures, and posture. In other words, emotions are evident in non-verbal communication modes. Due to the complex nature of classifying and interpreting the many forms of non-verbal communication used by children, this study focused on interactions, significant events or overt behaviour children exhibit during mathematical learning experiences.

Children’s gestures were observed during focused and selective observations. Some of the possible gestures or postures of children may exhibit during the observations include the following: putting their head in their hands, thumping a fist on the table, giving themselves a pat on the back, giving a high-five to another student or teacher, and stamping their foot or pushing objects away.

Combining the four areas of interactional analysis, pragmatics, socio-cultural discourse analysis and non-verbal communication provides the framework used to analyse the observational data (see Table 2).

The observational framework described above has several limitations. For example, FIAC ignores the physical and social setting of the classroom. Descriptive observations were conducted to overcome this issue. A second limitation regarding FIAC focuses on the collection of instances of overt behaviour rather than small actions (Delamont & Hamilton, 1976). Combining the four techniques aims to reduce the limitations and make for a more robust interpretation. Focused and selective observations were used to provide a systematic approach to observing children during mathematical learning experiences. Several tools (the observational framework, observational protocol, procedure and schedule, and attitude classification rubrics) were specifically developed for this research and were instrumental in conducting the observations. In this study, observations could also be checked for corroboration with data collected from non-lesson contexts. Triangulation of observations, drawings and children’s responses regarding their drawings was used to find evidence to support descriptions and theme development (Creswell, 2012).

2.3. Observational Procedure

The following procedure was employed to ensure consistent observations:

• Children’s drawings, written descriptions and interview responses were analysed to determine the nature of each child’s attitude towards mathematics;
• Three children were selected from each class who had completed research phases 1 and 2 and were classified as having three different attitudes towards mathematics;
• Descriptive observations of the classrooms were conducted;
• Focused systematic observations were conducted;
• Five-minute intervals were used to observe the three children;
• Observations of the children were conducted on a rotational basis;
• Selective observations were conducted when a significant event occurred or overt behaviour was exhibited by one of the three children;
• Focused observations were resumed when the child returned to the act before exhibiting overt behaviour or when the significant event ended.

In order to focus on children’s behaviours relevant to the study, there was a need to adopt a framework to guide data generation and the extensive field and reflective notes that were recorded after each observation. Field notes included nine variables, as suggested by Spradley (2016), namely, space, actors, activities, objects, acts, events, time, goals and feelings. The observation of these nine variables is the enacted element of a child’s attitude towards mathematics. Figure 2 summarises the type and scope of classroom observations using these variables in three steps: descriptive observations, focused observations and selective observations.

An initial broad and descriptive observation of the classroom environments was conducted, as suggested by Spradley (2016). Descriptive observations documented the space, planned activities, actors, objects, goals and times (refer to Figure 2 for more details). They occurred before and after the focused and selective observations and without the presence of children. Consultation between the researcher and classroom teachers was an essential part of the descriptive observations and provided the opportunity to document the variables accurately and to clarify any information. An observational protocol and field notes were used to record observations.

Focused observations were conducted systematically with equal time intervals spent observing the three children.

Table 3. Focused Observation Schedule for Systematic Observations.

Focused Observations
Class	Time Interval	Child	Nature of Attitude
Class 1	0–5 min	1	Neutral attitude
	5–10 min	2	Positive attitude
	10–15 min	3	Neutral attitude
	15–20 min	1	Neutral attitude
	20–25 min	2	Positive attitude
	25–30 min	3	Neutral attitude

shows a sample of the time intervals used for observations. Children’s emotions, facial expressions, gestures, interactions, and talk were documented during focused observations.

The observer switched to selective observations whenever a significant event occurred with one of the identified children, or what L. Cohen et al. (2011) refer to as critical incidents. A critical incident is one whereby a child may behave unexpectedly, offer valuable insights or may reveal an emotional response (L. Cohen et al., 2011). Critical incidents may be in the form of children’s classroom talk. Selective observations occurred when a child displayed overt behaviour, emotion or reaction to an event, interaction, object or situation. Significant events were analysed using a combination of socio-cultural discourse analysis (Mercer, 2004), interaction analysis (Flanders, 1965), and pragmatics (Fraser, 1983).

Observing interactions during mathematical learning experiences can be complex. Using an observational framework provides a structure that can be used to assist in analysing a situation systematically and providing opportunities for intra-observer reliability. An observational protocol and field notes were used to record observations and to help write thick descriptions that would identify the range and describe the nature of primary students’ attitudes towards mathematics. Together, the observational protocol, field notes, and audio or video recordings were a form of triangulation to ensure reliability in data collection. An analytical approach was used to develop an observational framework drawing upon the four methods pertaining to interaction, social discourse and communication, including non-verbal communication, to analyse the mathematical learning experiences.

2.4. Data Collection and Analysis

Observations of mathematical learning experiences were either audio or video recorded and analysed in conjunction with the observational protocol and field notes to ensure consistency in data analysis. The data generated from the observations were used to develop ‘thick descriptions’ and a narrative of primary students’ attitudes towards mathematics (L. Cohen et al., 2011). “A ‘thick description’ describes the complexity of a situation, as classroom events and interactions” (L. Cohen et al., 2011, p. 17). An observational framework was used to develop a series of indicators for each of the six sub-dimensions of MTMA (Quane et al., 2023). After the observations, the rubric was used to classify students’ attitudes for each sub-dimension.

Numerical coding was used to classify the range of primary students’ attitudes towards mathematics in a lesson context using the developed rubrics (Quane et al., 2023). Each sub-dimension was divided into six categories, and each category was allocated a score from zero to five. The six scores for each sub-dimension were then added to give an overall score out of 30, providing a quantitative measure to ascertain the range of children’s attitudes towards mathematics. Six categories were developed to move beyond the dichotomy of attitudes as either ‘negative’ or ‘positive’, recognising that attitudes towards mathematics is a spectrum (Zan & Di Martino, 2007). Additionally, it was noted that for some sub-dimensions, a student’s attitude could not be classified and if there were numerous sub-dimensions that could not be classified, then this student was excluded from analysis. The six categories varied for each dimension as shown in Table 1.

3. Results

The findings are reported in three sections based on the type of observation that was conducted. It is important to note that all observations of mathematical learning experiences used all three types of observations, and while all variables were documented for each observation, only variables that provide direct insight into students’ attitudes have been included. Vignettes from the observations and excerpts from the field notes are included in the reports of the findings.

3.1. Descriptive Observations

Descriptive observations are fundamental to subsequent observations as they not only provide context, but they also situate the observations. Several commonalities were noted in the descriptive observations. Each of the 13 classrooms where the observations occurred varied in space, in particular, their design and layout. However, all classroom spaces utilise flexible seating, compromising different sizes and configurations of tables arranged in small groups and a common floor space for students to gather with a smart whiteboard as a focal point in the room. Another commonality was the activities used in the teaching and learning of mathematics as captured in the following field notes:

B56 was in a Year 3 class where the teacher utilised a range of pedagogical practices, including but not limited to direct instruction, group work, problem-solving, using multiple representations, game-based activities, incorporating digital platforms and body-based learning experiences.

Students (actors) from all year levels were observed.

Table 4. Participant details by year level.

Year	R/1	R/1/2	1/2	2	2/3	3	3/4/5/6	Total
Number of classes	1	2	3	2	3	1	1	13
Number of observations	3	24	9	6	9	3	16	70
Number of participants for selective observations	3	12	9	6	9	3	8	50

provides numerical counts for the number of classes, observations conducted within each class and the number of participants observed categorised by year level.

While this study focused on students’ attitudes towards mathematics, other actors were noted in the descriptive observations. All teachers and student support officers were women and classified themselves as experienced teachers. Observations of learning experiences typically occurred in the morning (n = 40, 57%) or the middle of the day, between recess and lunch (n = 30, 43%), and were either 50 (n = 24, 34%) or 100 (n = 46, 66%) minute learning experiences.

3.2. Focused Observations

Focused observations were informed by phase 1 of the research, which involved students completing a drawing, writing a description of their drawing, and participating in a semi-structured interview. Phase 1 was conducted by withdrawing individual students from the class to collect the three artefacts on a one-to-one basis, referred to as the non-lesson context. The data from phase 1 were analysed to classify students’ attitudes towards mathematics. Students were purposefully selected for focused observations based on their non-lesson context attitude classification. In all observations, the three selected students were observed once before selective observations began.

The following field notes (

Table 5. Focused observations from three students using Spradley’s (2016) nine variables for participant observations.

Variable	B54	B53	B56
Event	B54 is happily working with a male partner to discuss fractions. The boys are playing a matching game using fraction cards with circle images and numerical fractions. Both boys are taking turns, pointing to different shaded sectors and matching the pictorial representation to the numerical representation.	B53 is sitting cross-legged with three other students close by, not talking to others. All four students are working through a set of questions on Mathletics. At 11:08, B53 starts a conversation about a question. Another student points to part of the screen on the iPad, B53 finishes the set of questions and states, “I’m up to 300”. Another student replies, “340! I’m way ahead of you”. Once B53 has completed the set of questions, she turns around and begins to talk to other students. Then, she engages in a conversation with another student about going away on holiday. At 11:12, she returns to the iPad and begins to answer questions about numbers.	At 11:15, teacher calls “hands on top” and the class responds with “that means stop”. Second rotations begin. Students move to return or get the equipment required for the next task. At 11:17, B56 is given the task “Erika is buying 3 poppies. They cost $4.25 each. How much did it cost? How much change did she get from $20”. B56 is sitting with another female student, both girls on their iPads, whispering, playing with slime and involving other students nearby. B56 is eating and conversing with other students at the table. At 11:20, B56 has not started the problem. Another girl has opened her book and stuck the task in her book, stating, “I’m going to start my work”. B56 opens her book to a blank page and talks to another student, continuing to play with the slim. She stands and walks to the other side of the room *
Overt Behaviour	N/A	N/A	Yes.
Artefacts	Fraction cards: pictorial and numerical representations. Fractions are 4/8, 5/12, 2/3 and 3/5	iPads–Mathletics	Task card, workbook, iPad.
Emotions	B54 appears content and happy to be working with a partner and playing the game. Happy to respond to teacher’s questions. Smiling. Appears confident.	Appears happy. Smiling, celebrating achievements. Head against windowsill	Distracted, actively disengaged, content in not completing the work. Appears to be unhappy.
Acts	Teacher checks work and gives feedback to the pair. B54 replies “I’m done!” Teacher asks further questions “What would this one be here be?” Pointing to a card. B54 is quick to respond, providing a clear justification. Teacher asks the boys to represent the fraction and then glue the task sheet in the book.	Observation occurs in the first set of mathematical rotations. Girls sitting by the window in a row each with their own iPad.	Observation begins at the end of the first round of rotations where students were completing a task on the iPads. Task given and not completed.
Activities	Small group work, game-based learning situated in mathematical rotations. The game has a focus on explanations and justifications.	Mathematical rotations: iPad–Mathletics.	Small group work with a word-based question. No manipulatives.

* selective observation begins.

) were taken from an observation in a Year 3 class and are an indicative example of the notes recorded for the selective observations. Times have been included to help sequence the events. Each focused observation was for five minutes. The field notes show the key variables for each of the initial three selective observations for one observed lesson.

From the focused observations, several aspects of the MTMA were noted. Students’ emotional disposition was observed from the body language, facial expressions, gestures, and statements made. The types of Topics, Tasks and Processes students engaged in during mathematical learning experiences influenced their emotional responses and the Perceived Concept. As stand-alone observations, the focused observations provided insights into students’ attitudes towards mathematics. However, when coupled with selective observations, the enactment of individual attitudes and the factors influencing their attitude could be documented.

3.3. Selective Observations

Building on the focused observation, the selective observations (Figure 2) included here are from B56, who began to show overt behaviour during the first observation. B56’s drawing and interview responses has been included here as a reference to what she drew in the non-lesson context (see Figure 3). B56 is sitting at her desk working on her “times tables”. She is asking the teacher, “So, why do I have to do this?” with the teacher responding, “Cus [sic: because] it’s good for your learning”. In her drawing, B56 is completing times tables in her workbook. During the interview, she is feeling sad “when I forget, or I get the answer wrong”, resulting in the child feeling “a bit angry”. These emotions are indicators of the child’s Emotional Dimension. Based on B56’s drawing and interview responses, she was selected for observations during mathematical learning experiences.

Data reported here are from the observational protocol and the field notes are digital recordings of the mathematical learning experiences. The learning experience used group rotations. In one rotation, B56 was given six mathematical word problems to solve, and she was required to write the solutions in her workbook. B56 slowly finds her mathematics book and sits with a group of girls. Peers were also an influential factor in B56’s attitude during the lesson context. B56’s peers provided her with a positive role model and encouragement. However, B56 ignored this modelling. B56 was observed procrastinating in starting the set task and not staying on task even after several efforts from others to redirect her to complete her work. B56 engaged in non-mathematical conversations with the group of girls, playing with slime and eating her lunch. One of the girls in the group has opened her book and tells B56 “I am going to start my work”. B56 follows the lead of her peer and opens her book to a blank page. However, the attempt to focus on her work is short-lived, and B56 returns her attention to playing with the slime. The other girls try to focus on their work. The teacher approaches and directs the girls to complete their work with B56 taking the teacher’s advice and beginning to write a numerical algorithm in her book. Again B56’s attention is diverted, this time to eating some food, stating:

B56: I’ve just realised that I’ve eaten all of my lunch.

Peer: I’ll give you a chicken nugget.

B56: I’ve never had a chicken nugget before.

B56 had several reminders from her peers and teachers during the learning experience. However, she chose only temporarily to engage in the set task, and her reluctance to sustain her engagement was evident. Her peers offered subtle guidance and model engagement. B56’s active disengagement, as determined by her non-mathematical conversations with her peers and opting to engage in other activities, are signs of her Vision of Mathematics, particularly her Mathematical Mindset. Additionally, B56’s disengagement was a result of her preference for how mathematics is taught, displaying higher levels of engagement in game-based activities to written word problems. This preference was not indicated in B56’s drawing; rather, B56 drew herself sitting alone at a desk questioning the teacher as to the relevance of the multiplication facts she was completing in her workbook.

In a second rotation, B56 was working in a small group on the floor under the direct supervision of the teacher. The activity involved using domino-like cards to match a pictorial representation of a fraction to the fraction written as a vulgar simple fraction. B56 is closely working with the same female peer from the previous learning experience. She is lying on her stomach and was observed picking up a card, looking at the other cards to see if they match. B56 appears to tentatively place a card. She asks her peer “Is this right?” The teacher hears B56’s question and asks, “Can you explain why?” B56 provided an explanation to the teacher and the teacher responded, “That’s good”. This is an indication that B56 is confident in answering the teacher’s questions in this situation. The teacher moves to another group of students and the two girls resume matching the cards. By the end of the rotation, B56 and her peer have made a total of seven matches. This second learning experience confirms that B56 prefers tasks that use game-based learning compared to tasks that involve writing and representing mathematics more abstractly. B56’s preference for tasks that involve game-based pedagogies provides insight into what the child values and appreciates. During this second learning experience, B56 appeared to be more engaged in the learning, which impacted on her Mathematical Mindset. For example, after the teacher’s question and feedback, B56 confidently continued participating in the game.

In the third rotation, B56 returned to being a reluctant learner. The task required students to represent a vulgar fraction as a collection (pictorially) and to place this number on a number line. At the commencement of the task, B56 was observed taking her time to locate her mathematics workbook. Once located, the child returns to her desk and delays starting the task further by flipping through the pages of her book several times to find a blank page. With the book open, B56 leaves her desk to find a pencil, even though there were several pencils on her desk. The child’s active disengagement consumes ten minutes of the lesson. B56 returns with a pencil and begins to write in her workbook. She has written the fraction five-tenths and has drawn a rectangle with ten divisions (two columns and five rows). She leaves the rectangle unshaded. Upon completing the rectangle, B56 leaves her desk again and does not return to her work for the remainder of the lesson.

By the third learning experience, a pattern in B56’s attitude is emerging. B56 has a Positive Emotional Tendency when mathematical tasks are in the form of a game and a Negative Emotional Tendency when mathematical tasks are not game-based. That is, B56 enjoys and likes some tasks, whereas she clearly exhibits negative emotions for other tasks, resulting in a Neutral classification. B56’s Emotional Disposition of attitude is closely related to the Topic, Tasks and Processes that she is or is not engaging in. Further, when B56 Values and Appreciates the type of Topic, Tasks and Processes, she displays positive emotions, with the converse holding true. B56’s preference for a particular task also impacts on her Mathematical Mindset and Self-Concept.

Peers, teachers, and pedagogical practices influenced B56’s attitude towards mathematics, impacting how the child either engaged or disengaged in mathematics. The reasons for the child’s engagement are yet to be determined. However, a potential reason could be that B56 finds game-based learning less cognitively demanding than other pedagogical practices. Additionally, the child found using the correct written mathematical conventions difficult.

4. Discussion

The focus of this paper was to address the need to have specifically designed instruments that can be used to observe affective factors during mathematical learning experiences (Ingram et al., 2020). The observational data presented in this paper employed three different observational techniques using descriptive observations, focused observations, followed by selective observations to observe primary-aged students during mathematical learning experiences. Using a triad of observation techniques afforded data to be collected regarding how students’ attitudes towards mathematics are enacted during a mathematical learning experience, which Creswell (2012) states is a distinct advantage of using observations. Together, the three observational techniques provided a structure whereby data could be collected in a systematic manner but also have the flexibility of being student-centred. That is, the observer was switching between the selected participants when a critical incident or overt behaviour was evident. It was these critical incidents that ‘illuminated’ (L. Cohen et al., 2011, p. 464) an aspect of a child’s attitude which in turn, formed the thick descriptions and rich narratives as in the case of B56.

Classroom observations collected firsthand information about actual student behaviour during mathematical learning experiences, generating additional data about children’s relationship with school mathematics. McLeod (1987) recommended that researchers observe in realistic settings that focus on overt behaviour, what students say and their physical reactions. The observational framework (Table 2) and observation procedure provided a replicable, reliable and valid method to collect data in a realistic classroom setting. In doing so, the act of observing mathematical learning experiences has been ‘operationalised’ to ensure that evidence is “consistent, unambiguous and valid” (L. Cohen et al., 2011, p. 474).

Field notes included nine variables, as suggested by Spradley (2016), namely, space, actors, activities, objects, acts, events, time, goals and feelings, which were used to assist in the operationalisation of observations. The observation of these nine variables was instrumental in documenting how children’s attitudes are enacted during mathematical learning experiences. For example, the activities, objects, acts and events were beneficial in documenting students’ Vision of Mathematics, in particular for the sub-dimension Topic, Tasks and Processes.

The feeling variable provided insight into students’ Emotional Dimension. Single selective observation provided rich examples of students’ Emotional Tendencies, whereas collective observations provided evidence of students’ Overall Sentiment towards mathematics. In engaging in activities, the space of the classroom and the other students and teachers or Spradley’s actor variable provided evidence towards the Perceived Competence dimension. In particular, the utterances, behaviour and interactions provided examples of students’ Mathematical Mindset. However, it is important to note the interconnectedness of the MTMA model, whereby a particular incident could provide evidence of multiple dimensions. For example, when B56 appeared happy for more knowledgeable peers to direct and dominate learning experiences she showed an unwillingness to contribute and participate in the learning experience and a lack of perseverance. These indicators highlight a concerning materialisation regarding B56’s Mathematical Mindset and Self Concept, impacting both the Emotional Dimension and B56’s Vision of Mathematics.

In this study, observations corroborated with data collected from non-lesson contexts. Triangulation of observations, drawings and children’s responses regarding their drawings was used to find evidence to support descriptions and theme development (Creswell, 2012). The Observational Framework (Table 2), the schedule for systematic observations (Table 3) and the Observational Protocol to record field notes provided a repeatable procedure that could be employed in future observations. The findings from this study suggest that attitudes towards mathematics can in fact be observed and documented during mathematical learning experiences, thus addressing the dearth of research being conducted into the enactment of attitudes during mathematical learning experiences (Attard et al., 2016). It is argued that this work constitutes an important contribution to future studies on how to observe attitudes in a lesson context.

5. Conclusions

Attard et al. (2016) raised the need for research to include ‘observational data’ in researching aspects of the affective domain. This current study contributed to the field by using ‘observational data’ to ascertain the range and nature of primary students’ attitudes towards mathematics. The ‘observational data’ provided a baseline measure of the nature of these attitudes and how attitudes are enacted and change during mathematical learning experiences. Using ‘observational data’ provided a different perspective on students’ attitudes and established a means for making comparisons between contexts. Significant methodological contributions were made regarding a well-developed classroom observation protocol and used in conjunction with a schedule for systematic observations and observational framework to provide a repeatable procedure that could be employed in future observations. Using qualitative research techniques and the specifically designed instruments provided the means to identify and describe factors that were found to influence primary students’ attitudes towards mathematics (Quane, 2024; Quane, 2021).

It is important to acknowledge that methodological limitations exist in both the design and in terms of how the interviews were conducted. The observational framework did not take into consideration the influence of external entities such as parents and students’ previous experiences and attitudes prior to the observations. Further, the observations focused on overt behaviours that were observable and did not attend to small actions or reactions. It is not that these small actions, reactions or behaviours are considered inconsequential, rather, they are hard to detect in a classroom situation. Therefore, it was difficult to ascertain whether any personal events, such as personal birthdays, illness, or other circumstances, influenced students’ attitudes in either context. Unknown circumstances may have affected what the students were experiencing on the day they completed their drawing, and on the days of the observed mathematical learning experiences.

The second limitation identified was related to the analysis of students’ emotions. During the non-lesson context, students were explicitly asked about the face they had drawn and their feelings towards mathematics and were provided multiple opportunities to discuss their emotions through a range of questions. However, the classroom observations did not provide the same opportunity to ascertain and clarify students’ emotions. Due to the nature of the observations, students’ emotions were inferred using semiotic resources as well as verbal acclamations and social discourse. While the inferences were made on several pieces of evidence to reduce the likelihood of an incorrect conclusion, there is nevertheless the possibility that errors based on inferences were made during the observations.

A third limitation relates to the type of observations. Observations were non-participatory with minimal interaction or communication. Consequently, the affordances available in the non-lesson context were not available during the lesson context—for example, asking clarifying questions and checking children’s responses to ensure a true account of what transpired. Creswell (2012) claims that researchers who do “not actively participate” in the experience may find that their observations “may not be as concrete as if you had participated in the activities” (p. 215). Changing the role of the observer and the nature of observations may provide deeper insights into primary students’ attitudes towards mathematics.

This research was limited to Australian primary students attending South Australian State schools. This limitation identifies a potential threat to the external validity regarding the “lack of representativeness of available and target populations” (L. Cohen et al., 2011, p. 186). Thus, the use of the observation framework and schedule is worthy of further exploration in a larger heterogeneous sample.

Observing primary students’ mathematical learning experiences is a step towards understanding the enactment of attitudes in a lesson context. Repeated and ongoing classroom-based observations are required to thoroughly examine the range of influencing factors on students’ attitudes towards mathematics. In addition, interactions with the students’ observed are recommended. While the framework has been used to observe a particular aspect of the affective domain, further research is needed into whether this framework can be used to observe other affective factors and for cognitive-based research.