action research on mathematics

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Action research projects.

Using Cooperative Learning In A Sixth Grade Math Classroom , Teena Andersen

Algebra in the Fifth Grade Mathematics Program , Kathy Bohac

Real Life Problem Solving in Eighth Grade Mathematics , Michael Bomar

Holding Students Accountable , Jeremy Fries

Writing In Math Class? Written Communication in the Mathematics Classroom , Stephanie Fuehrer

The Role of Manipulatives in the Eighth Grade Mathematics Classroom , Michaela Ann Goracke

Reasonable or Not? A Study of the Use of Teacher Questioning to Promote Reasonable Mathematical Answers from Sixth Grade Students , Marlene Grayer

Improving Achievement and Attitude Through Cooperative Learning in Math Class , Scott Johnsen

Oral Communication and Presentations in Mathematics , Brian Johnson

Meaningful Independent Practice in Mathematics , Michelle Looky

Making Better Problem Solvers through Oral and Written Communication , Sheila McCartney

Student Understanding and Achievement When Focusing on Peer-led Reviews , Ryon Nilson

Students Writing Original Word Problems , Marcia Ostmeyer

Cooperative Grouping Working on Mathematics Homework , Maggie Pickering

Making Sense of Word Problems , Edie Ronhovde

Oral and Written Communication in Classroom Mathematics , Lindsey Sample

Written Communication in a Sixth-Grade Mathematics Classroom , Mary Schneider

The Use of Vocabulary in an Eighth Grade Mathematics Classroom: Improving Usage of Mathematics Vocabulary in Oral and Written Communication , Amy Solomon

Enhancing Problem Solving Through Math Clubs , Jessica Haley Thompson

Communication: A Vital Skill of Mathematics , Lexi Wichelt

Mathematical Communication through Written and Oral Expression , Brandee Wilson

Oral Presentation: Exploring Oral Presentations of Homework Problems as a Means of Assessing Homework

Building Confidence in Low Achievers through Building Mathematics Vocabulary , Val Adams

An Uphill Battle: Incorporating cooperative learning using a largely individualized curriculum , Anna Anderson

Using Descriptive Feedback In a Sixth Grade Mathematics Classroom , Vicki J. Barry

Does Decoding Increase Word Problem Solving Skills? , JaLena J. Clement

Using Non-Traditional Activities to Enhance Mathematical Connections , Sandy Dean

Producing More Problem Solving by Emphasizing Vocabulary , Jill Edgren

Reading as a Learning Strategy for Mathematics , Monte Else

Perceptions of Math Homework: Exploring the Connections between Written Explanations and Oral Presentations and the Influence on Students’ Understanding of Math Homework , Kyla Hall

Homework Presentations: Are They Worth the Time? , Kacy Heiser

Reduce Late Assignments through Classroom Presentations , Cole Hilker

Mathematical Communication, Conceptual Understanding, and Students' Attitudes Toward Mathematics , Kimberly Hirschfeld-Cotton

Enhancing Thinking Skills: Will Daily Problem Solving Activities Help? , Julie Hoaglund

Can homework become more meaningful with the inclusion of oral presentations? , Emy Jones

Confidence in Communication: Can My Whole Class Achieve This? , Emily Lashley

Exploring the Influence of Vocabulary Instruction on Students’ Understanding of Mathematical Concepts , Micki McConnell

Using Relearning Groups to Help All Students Understand Learning Objectives Before Tests , Katie Pease

Cooperative Learning in Relation to Problem Solving in the Mathematics Classroom , Shelley Poore

How Student Self-Assessment Influences Mastery Of Objectives , Jeremy John Renfro

RAP (Reasoning and Proof) Journals: I Am Here , Bryce Schwanke

Homework: Is There More To It Than Answers? , Shelly Sehnert

Written Solutions of Mathematical Word Problems , Marcia J. Smith

Rubric Assessment of Mathematical Processes in Homework , Aubrey Weitzenkamp

Calculators in a Middle School Mathematics Classroom: Helpful or Harmful? , Leah Wilcox

Pre-Reading Mathematics Empowers Students , Stacey Aldag

The Importance of Teaching Students How to Read to Comprehend Mathematical Language , Tricia Buchanan

Cooperative Learning as an Effective Way to Interact , Gary Eisenhauer

Generating Interest in Mathematics Using Discussion in the Middle School Classroom , Jessica Fricke

“Let’s Review.” A Look at the Effects of Re-teaching Basic Mathematic Skills , Thomas J. Harrington

The Importance of Vocabulary Instruction in Everyday Mathematics , Chad Larson

Understanding the Mathematical Language , Carmen Melliger

Writing for Understanding in Math Class , Linda Moore

Improving Student Engagement and Verbal Behavior Through Cooperative Learning , Daniel Schaben

Improving Students’ Story Problem Solving Abilities , Josh Severin

Calculators in the Classroom: Help or Hindrance? , Christina L. Sheets

Do Students Progress if They Self-Assess? A Study in Small-Group Work , Cindy Steinkruger

Why Are We Writing? This is Math Class! , Shana Streeks

Effects of Self-Assessment on Math Homework , Diane Swartzlander

The Effects Improving Student Discourse Has on Learning Mathematics , Lindsey Thompson

Increasing Teacher Involvement with Other Teachers Through Reflective Interaction , Tina Thompson

Increasing Conceptual Learning through Student Participation , Janet Timoney

Improving the Effectiveness of Independent Practice with Corrective Feedback , Greg Vanderbeek

Using Math Vocabulary Building to Increase Problem Solving Abilities in a 5th Grade Classroom , Julane Amen

Departmentalization in the 5th Grade Classroom: Re-thinking the Elementary School Model , Delise Andrews

Cooperative Learning Groups in the Eighth Grade Math Classroom , Dean J. Davis

Daily Problem-Solving Warm-Ups: Harboring Mathematical Thinking In The Middle School Classroom , Diana French

Student Transition to College , Doug Glasshoff

The Effects of Teaching Problem Solving Strategies to Low Achieving Students , Kristin Johnson and Anne Schmidt

The Effects of Self-Assessment on Student Learning , Darla Rae Kelberlau-Berks

Writing in a Mathematics Classroom: A Form of Communication and Reflection , Stacie Lefler

Math in the George Middle School , Tiffany D. Lothrop

Bad Medicine: Homework or Headache? Responsibility and Accountability for Middle Level Mathematics Students , Shawn Mousel

Self-Directed Learning in the Middle School Classroom , Jim Pfeiffer

How to Better Prepare for Assessment and Create a More Technologically Advanced Classroom , Kyle Lannin Poore

Cooperative Learning Groups in the Middle School Mathematics Classroom , Sandra S. Snyder

Motivating Middle School Mathematics Students , Vicki Sorensen

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Action Research in Mathematics: Providing Metacognitive Support (as a Heutagogical Technique) to Grade 3 Students

By Zoriana Myburgh

Self-determination, as one of the 21st-century skills, prepares students for our constantly changing world. Can metacognition help students become self-determined and is it worth starting to introduce metacognitive elements at an elementary school? Based on the current literature from teachers’ and administrators’ points of view, metacognition positively influences students’ performance and wellbeing. Therefore, there is a need to continue researching the effects of metacognition at a young age but from students’ perspectives.

The purpose of this research is to enhance grade 3 students’ metacognitive abilities to help them manage their learning. The data was collected using the students’ personal thoughts and emotions during semi-structured interviews and the researcher’s observation notes in order to summarise whether or not metacognitive interventions were helping students become self-determined. A lack of qualitative action research in existing literature highlighted the need for this research design.

It has been found that metacognition as a heutagogical technique can be used to improve students’ self-determination in an elementary school. Students mostly showed improvement in the Commitment category: goal-setting, self-questioning and self-monitoring. Less than half of the students showed the development of metacognitive skills in the Capacity category. Most of them already had some skills before the experiment started (e.g., strategies for solving problems, improvement strategies, and asking for help when needed). The Value category represented how students developed their metacognitive experience and supported the finding of the previous categories. Considering the significance of metacognitive development, teachers should constantly strengthen the metacognitive abilities of their students.

University of Southampton, 2021

FACULTY OF SOCIAL SCIENCES

ACTION RESEARCH IN MATHEMATICS: PROVIDING METACOGNITIVE SUPPORT (AS A HEUTAGOGICAL TECHNIQUE) TO GRADE 3 STUDENTS

by Zoriana Myburgh

A dissertation submitted in partial fulfilment of the degree of

<MSc Education (online) PT>

by taught course.

You can view this dissertation in its entirety in PDF form here.

1. INTRODUCTION

Background of the study.

Due to constant and rapid changes in our society, the importance of lifelong learning is also increasing. People need to constantly evaluate their ideas and experience. This ceaseless “transformation of information, the creation, construction and renewal of knowledge, is at the heart of reflexivity” (Dyke, 2009, p. 295). Traditional learning models might not be satisfactory for current learners nowadays who look for greater independence and integration (McLoughlin and Lee, 2008). Taking into account COVID-19 pandemic restrictions all over the world and the availability of online learning resources, it’s essential for students to learn how to manage their learning. Furthermore, some of the jobs that we have now might be replaced by the time our primary students will finish school. In that case, it is necessary to teach 21st-century skills, namely critical thinking, self-determination, and socialization, that will help students adapt to these new requirements (Irgatoglu and Pakkan, 2020).

Being an elementary school teacher, I often noticed that when students are given a word problem to solve, they will say, “I do not know, teacher” or “Teacher, what to do here?”, or will patiently wait until the teacher scaffolds it, or someone else finds the answer. Can coaching students on metacognition as one of the heutagogical techniques solve this issue? The main goal of developing students’ metacognition is to make them autonomous and self-determined learners (Thomas, 2003). Self-determined learners set their own goals and learning paths, monitor timing, and reflect on their final outcomes.

Based on Self-determination Theory, students have three primary needs : autonomy, competence, and relatedness (Deci and Ryan, 2000). Autonomy means the freedom of choice and the wish to control the learning process. Competence refers to having the confidence and capabilities to complete a task. Relatedness is about working with others and feeling connected (Marshik, Ashton and Algina, 2017).

Definition of Terms

This paper will be based on three main terms: heutagogy, self-determination, and metacognition.

First, it is important to differentiate between self-determination, self-regulation, and self-efficacy as these three terms will appear in this paper.

• Self-determination refers to individual motivation that is influenced by internal impetuses (Ryan and Deci, 2000).
• Self-regulation is defined as self-management, the ability to control the thoughts and emotions that direct human’s behaviour (Panadero, 2017).
• Self-efficacy is one of the tools of self-regulation and means the beliefs people have in achieving a task successfully (Bandura, 1977).

Self-determined learning is also known as heutagogy , a concept introduced by Hase and Kenyon (2000) and refers to student-centred learning “where the individual student’s interests and motivations create a focus area for new learning” (Jones et al ., 2019, p. 1172) and the teacher acts more like a mentor in a classroom. One of the main heutagogical principles is self-reflection (Blaschke, 2012) which is known as metacognition. Metacognition is “thinking about thinking”, “cognition about cognition” (Pritchard, 2013, p. 27) or “our ability to know what we know and what we don’t know” (Costa and Kallick, 2008, p. 24).

Metacognition is not only about planning and knowledge activation but also the intentional monitoring of students’ cognitive processes, reflection, time management, and self-evaluation (Bol et al ., 2016) and determining new ways to proceed and learning from the experience (Edwards and Costa, 2012).

Metacognition is one of the 16 Habits of Mind (HoMs) developed by Costa and Kallick (2000) that are a collection of behaviours that can help students tackle different problems they encounter at school or in other settings. Students need more than just academic knowledge and skills in order to succeed. HoMs focus not only on how much students know but also on what they do when they do not know an answer (Costa and Kallick, 2008). Teachers can help them practice these behaviours by modelling them, immersing these habits in the school curriculum and culture, and constantly checking their growth (Edwards and Costa, 2012).

Problem Statement

Considering that learning is a lifelong process, it is logical to start teaching self-determination skills as soon as possible (Palmer & Wehmeyer, 2003). Moreover, it is recommended by some researchers to start teaching self-determination at a young age as it will become more difficult at later stages (Danneker & Bottge, 2009). Stein (2018, p. 4) underlines that “effective teaching and learning [should] take place early on so that students can be successful in secondary school and beyond”. For instance, elementary school students can be taught how to set goals, make decisions, assess, and reflect on one’s own work. Knox (2017) emphasises that students who developed metacognitive skills can organise, examine, and assess their thoughts. However, there is not much qualitative research that investigates the impact of metacognitive support on elementary students, particularly in the context of mathematics. Most of the studies focus on quantitative data based on students’ performance. Previous literature was more focused on the teachers’ perspectives disregarding the views of students.

Self-reflection tasks are often treated as some extra time-consuming work for students, especially during mathematics classes because students do not see any value in them (Kiles, Vishenchuk and Hohmeier, 2020). Metacognitive skills that lead to self-determination are not instinctive and might be challenging (Bouldin, 2017). Considering that self-determination plays an important role in students’ well-being (Martinek and Kipman, 2016), it is important to dedicate this study to research the effects of metacognition on learners’ self-determination at a young age from students’ vision.

Fifteen grade 3 students (9-10 years old, 8 girls and 7 boys) took part in this experiment at an elementary school in Cambodia. Data from the participants were collected as semi-structured interviews with students recorded via Google Meets and the researcher’s observation notes taken during math classes. During the research period, all schools in Cambodia were operating online because of COVID-19 pandemic restrictions. The experiment lasted 9 weeks from April 29 to June 30, 2021.

Purpose of the Study and Research Questions

The purpose of this study is (a) to understand the necessity of a heutagogical framework from the perspectives of students during maths classes in an elementary school in Cambodia, and (b) to find out whether metacognition can help students become self-determined at a young age. Action research cycles will be used to examine self-determined learning in the context of mathematics at an elementary school. Action research can be conducted by teachers in their classrooms with the aim of refining pedagogy and student learning (Nasrollahi, 2015). Coding student perspectives and researcher’s notes will help understand the phenomenon of self-determination and the use of metacognitive elements, namely goal-setting, self-assessment, self-questioning, self-monitoring, responding to and reflecting on feedback, note-taking, problem-solving strategies, and improvement strategies, etc.

The aim of the study was supported by the following research questions:

Central Question: How can students’ needs be met using the heutagogical framework while teaching maths in an elementary school in Cambodia? Subquestion 1: What tools and strategies should teachers use to implement the heutagogical framework? Subquestion 2: How can metacognition as a heutagogical technique be used to improve students’ self-determination

By answering these research questions and gaining insight from students, we can better understand the conceptions and misconceptions about the heutagogical framework in an elementary school. We will see whether metacognitive elements in a lesson plan encourage self-determination based on students’ responses and the researcher’s observation notes. With this undertaking, elementary teachers might be inspired to design better lesson plans.

Organization of the Study

The structure of the study is as follows. The literature review provides the context for current research, explains the importance of metacognition as a heutagogical technique in the learning process from teachers’ perspectives. It was considered whether metacognition can be taught and what techniques work best based on the current empirical data. The methodology chapter explains the rationale for research design and the methods chosen as the most appropriate for this study. Plus, it gives an insight into how data was collected and analysed. The findings chapter presents the themes developed during the analysis of the collected data. The research outcomes are compared with the existing literature in the discussion chapter. Based on the research findings, implications for further research are suggested and conclusions are made.

2. LITERATURE REVIEW

Metacognition will be studied through constructivism and individualism leading to postmodernism. Literature was reviewed to examine the importance of metacognition as one of the heutagogical techniques and a Habit of Mind in the learning process. It was taken into account whether metacognition can be taught and what approaches on the current empirical evidence work best to enhance self-determination. Finally, some quantitative and qualitative tools to measure metacognition were analysed. The search was not limited to specific dates although preference was given to the last two decades.

2.1 Metacognition through the lens of constructivism, individualism, and postmodernism

From a constructivist viewpoint, metacognition is “the result of mental construction” (Pritchard, 2013, p. 18). According to constructivist theory, learning is a metacognitive process (Wray and Lewis, 1997). Reuer states that constructivism happens when students connect their experiences and ideas (Reuer, 2017). Constructivists claim that we learn better if we constantly build our understanding (Pritchard, 2013). Since metacognition is also about self-assessing and self-monitoring, it complements constructivism perfectly. Different people learn things differently. We, as teachers, might not know which approach will be the best for a specific topic for different students but we can encourage them to experiment with various methods and decide metacognitively how to approach and solve a mathematical problem. Pritchard (2013) also claims that if students are asked to share their approaches or evaluate their classmates’ ideas in a constructive way in a safe and supportive environment, it will eventually lead them to develop their processes of thinking and help them solve problems.

It’s important to understand the teacher’s role in this domain. Constructivism is often criticised because students are left “to teach themselves” (Hubbard, 2012, p. 160). According to heutagogy, teachers act more like coaches (Mohammad et al ., 2019) and facilitators (Akyildiz, 2019). Akyildiz interviewed 40 educators from Turkey who implemented heutagogical frameworks within their classes and summarised that 30% of teachers reported that they had to reflect more if they wanted to progress with heutagogical education and 40% of them claimed that they lost power in the classroom. However, Horrigan (2018, p. 57) argues that “empowering students is not the same as a teacher losing power”. Can metacognition help students and teachers because both need to adapt to the 21st-century requirements?

Furthermore, metacognition is connected to the values of individualism, encouraging self-directedness and heutagogical principles in general. Working at one’s own pace and reflecting on one’s learning are paramount skills for students in order to transform into independent learners (Sart, 2014).

Seeing each child as an individual with their own personality, mental and physical capabilities is an example of individualism (Fevre, Guimarães and Zhao, 2020). According to constructivists, knowledge is built by an individual through the explanation and combination of different ideas (Hubbard, 2012). Thus, both the constructivist methods and the individualism of heutagogy lead to postmodernism because postmodernists believe that the goal of education is “teaching critical thinking, production of knowledge, development of individual and social identity, self-creation” (Hossieni and Khalili, 2011, p. 1307). This aligns with 21st-century skills discussed in the previous chapter. Following this philosophical approach, teachers as mentors guide students to come across new things and reflect on them on their own (Hossieni and Khalili, 2011), therefore, students alter from passive recipients of knowledge to active constructivists (Reuer, 2017).

2.2 Metacognition through self-determination, self-regulation, and self-efficacy

Self-determination and even heutagogy are not new terms in education but their importance in elementary school has emerged within recent years. Some findings suggest that nurturing metacognitive beliefs in kindergarten children will increase their behavioural self-regulation (Compagnoni, Sieber and Job, 2020). Therefore, teachers must “maximize the learning classroom climate for self-regulated learning” (Panadero, 2017, p. 23). Using action research cycles (Plan – Act – Observe – Reflect – Revise – Plan), the behaviour of students engaging in self-directed learning was investigated in Ireland (Newman and Farren, 2018). They claimed that reflections and critical self-analysis motivated students to become more self-directed. However, the authors believed that the terms “self-directed” and “self-determined” are interchangeable. This definitional vulnerability questions the validity of this study. Self-directedness is based on single-loop learning (correcting the mistakes without reflections) and self-determination is founded on double-loop learning (questioning beliefs and assumptions) (Peeters and Robinson, 2015).

Self-determination techniques are often analysed alongside metacognitive strategies because according to heutagogy, metacognition is a major characteristic of how people naturally learn (Blaschke, 2012). Moreover, “the autonomy and flexibility of heutagogical models are managed well when incorporated into a reflective practice” (Newman and Farren, 2018, p. 6). If learners reflect on their results through the problem-solving process and take their actions and beliefs into account, new learning situations can be adapted to various learning styles (Blaschke, 2012). Blaschke is talking about the double-loop learning model, the key principle in the heutagogical framework. It was examined in Indonesia among 48 middle-school participants who were split into two groups (Nur et al ., 2019). Using questionnaires, pre-tests and post-tests students’ progress was measured. It was summarised that this constant reflection using a double-loop learning model positively affected students’ results. However, the analysis is lacking in rigour. First, it is not mentioned whether multiple observers were invited to this study and the duration of the experiment is not clear. Secondly, it is difficult to judge the validity of the data without the examples of pre-tests and post-tests.

Some researchers even consider metacognition as a general term combining other definitions, such as self-regulated learning, thinking skills, etc. (Perry, Lundie and Golder, 2019). Winne and Hadwin (2008) explored self-regulated learning and its effects on motivation from a metacognitive perspective and described it as a cycle: define a task, set goals, perform the task and metacognitively modify if needed. According to their study, self-regulated learners can use metacognitive strategies to monitor their goals and progress in completing them. Analysing this model, Greene and Azevedo (2007) noticed that metacognitive monitoring is a significant part of self-regulated learning. However, these authors together with Panadero (2017) in his meta-analysis, argued that it was unclear how the last phase worked in Winne and Hadwin’s cycled model, specifically they believed that metacognition should occur throughout each phase and not as an end product.

Students who are confident in their abilities, try to achieve their goals, look for academic help and self-reflect will show better performance at school (Martinek and Kipman, 2016). Martinek and Kipman (2016) argue that if supporting students’ autonomy increases students’ motivation, then self-determined learning will also improve students’ self-efficacy (self-confidence in succeeding in the task). Self-efficacy is one of the variables that influences self-regulated learning (Panadero, 2017). According to the Self-determination Theory, there are three basic needs: autonomy, competence, and relatedness. Therefore, if students have a choice and understand the purpose and relevance of the tasks, students’ intrinsic motivation will be increased (Ryan and Deci, 2000). Martinek and Kipman (2016) predicted that self-determination, academic self-regulation and self-efficacy (𝑆𝐸³) will have positive consequences on students’ subjective well-being ( W ). 131 students from an Austrian primary school completed questionnaires regarding this study called 𝑆𝐸³𝑊. It was concluded that 𝑆𝐸³ decreased noticeably from grade 1 to grade 2, but the general W did not change. Martinek and Kipman (2016, p. 130) assumed that the teachers did not provide enough autonomy to their students because they did not “trust in the learning abilities” of their students or felt “a loss of control” in actual students’ work. However, some researchers believe that using questionnaires in elementary school is not appropriate because of the students’ age and observer’s reports are essential in future studies (Morison et al ., 2000). Others think that teachers do not have enough training on self-determined learning (Panadero, 2017).

Teachers’ behaviour as an example of self-determination is truly crucial. If students do not feel these supportive interactions, their self-determination might not be ignited. 2,587 students in the USA took part in a survey that examines how the social aspects of schools foster students’ self-determination (Adams and Khojasteh, 2018). Having applied quantitative methods, the authors concluded that the school climate does have consequences. At the same time, a similar experiment in the USA was conducted with 106 university students (Hayward et al ., 2018). The authors used a mixed-methods research approach to examine how inner directedness (in particular, metacognition) will affect students’ motivation. For each experimental section instructors recruited three to five volunteers called Student Pedagogical Teams. Their role was to collect feedback from other students and to present it to their instructors. The instructors followed this repeating cycle: teach a class, receive feedback, modify the classroom activities, receive students’ feedback again, reflect on the results and plan the next task. Metacognition through self-assessment, reflections, and self-regulated learning promoted teachers’ self-directed professional development (Hayward et al ., 2018).

2.3 Metacognition as a Habit of Mind

Metacognition is one of the Habits of Mind (HoM), “the characteristics of what intelligent people do when they are confronted with problems” (Costa and Kallick, 2008, p. 15). The authors of 16 HoMs, Costa and Kallick, believe that the main purpose of all HoMs is to create self-determined, lifelong learners. Therefore, students need to self-monitor and self-regulate to achieve this goal. This HoM is subjectively the heart of all Habits of Mind and a core of heutagogy. If an individual can self-regulate their thinking processes, they can choose other appropriate habits to succeed in the task. Based on the empirical evidence presented by Muscott, students who were trained in and demonstrated how to think about their thinking did better in assessments (Muscott, 2018).

There are two main components of metacognition: metacognitive knowledge (MK) and metacognitive experience (ME) (Flavell, 1979). MK is the strategy students use to solve problems and ME is about contact with the environment. ME could inspire both cognitive and metacognitive strategies necessary to solve mathematics word problems. “Cognitive strategies are invoked to make cognitive progress, metacognitive strategies to monitor it” (Flavell, 1979, p. 909). Efklides separated metacognitive skills (MS) from MK and ME. She defines MS as “procedural knowledge” (Efklides, 2006, p. 5). Students must get used to “linking and constructing meaning from their experiences” (Costa and Kallick, 2008). Costa and Kallick add two more components: inner awareness and the strategy of recovery. They explain it with an example of reading an unfamiliar story. When a person reads a text and suddenly loses the meaning, he/she will reread it and get back the connection to understand what is happening.

Costa and Kallick (2008, p. 63, fig. 4.1) developed five dimensions within which students can grow regarding Habits of Mind: meaning, capacity, alertness, commitment, and value.

By exploring meaning the authors mean that students understand the meaning of HoMs. Increasing alertness is about applying HoMs without being asked. Extending values is understanding why some HoMs are more applicable in specific situations so if the students extend the value of metacognition in mathematics, it might turn into a pattern of behaviour eventually. Expanding capacities is about developing various techniques when solving problems or making decisions. If students develop metacognitive strategies, they might be able to apply other HoMs more effectively. Building commitment is constant development in the use of HoMs when students progressively become self-determined. This dimension is closely related to heutagogy as students learn to self-manage, self-monitor, self-reflect, and set higher expectations for themselves (Costa and Kallick, 2008).

Wagaba, Treagust and Chandrasegaran (2016) were also trying to conceptualise the dimensions that characterise metacognitively-oriented learning environments: (1) metacognitive demands, (2) student-student discourse, (3) student-teacher discourse, (4) student voice, and (5) teacher encouragement and support. If we are talking about metacognitive demands, it is recommended to model metacognition and not assume that the students already know organisational techniques or how to set goals, collaborate, etc. but explicitly teach these metacognitive strategies. The student-student dimension refers to whether students think interdependently and discuss how they learn with their peers. Student-teacher interaction means whether students discuss how they learn with their teacher. By “student voice” the author means students’ contribution to lesson planning. That is especially important in heutagogy. Thomas (2003) argues if students have control over their learning tasks, it will be easier for them to meet their learning goals. Lastly, teacher encouragement and support might not influence students’ metacognitive abilities directly, but it might be the first step in developing those skills. As Gallucci (2006, p. 19) mentioned, “the heart of teaching is providing students with the tools to make them more effective learners” and not just teaching the content.

2.4 Can metacognition be taught?

Some researchers argue that metacognition can be taught, and it might improve academic achievement (Muscott, 2018). The effects of metacognitive training on students’ achievement in maths were tested recently in the USA (Bol et al ., 2016). 116 randomly selected participants who were split into two groups (control and experiment) took part in this three-week research. The authors concluded that this training in self-regulated learning (specifically, such skills as goal-setting, self-monitoring and self-reflection) improved not only students’ maths results but also their time management skills. However, the validity is questionable since this questionnaire is a self-reported measure and some students might have overestimated their use of self-regulatory strategies (Young and Ley, 2005). Their paper states that questions about self-regulation could be difficult for poorly self-regulated students.

A year earlier a similar experimental design was employed in Italy among 135 elementary school students (Cornoldi et al ., 2015). By using regression analysis, the authors concluded that training in metacognition transferred positively onto students’ abilities to solve maths problems. Mathematical word problems are “complex cognitive tasks” (Cornoldi et al ., 2015, p. 434) and apart from mathematical abilities, children also need reading comprehension, metacognitive abilities, and motivation. In this experiment, not only were activities focused on metacognition, but the students reflected on their beliefs about mathematics. Referring to this study, Davey (2016) also emphasised the importance of metacognitive training programmes. Having conducted a case study, she claimed that the teachers’ actions and words are crucial in developing students’ metacognitive abilities.

Mutambuki et al . (2020) combined metacognition and active learning and by applying a quasiexperimental design investigated the influence of both on the first-year undergraduate students in chemistry courses. By active learning, the authors mean frequent questions, discussions, problemsolving tasks, etc. Such metacognitive prompts as planning, reflecting, developing self-awareness, and making adjustments were mentioned in this paper. The authors concluded that active learning implemented with metacognitive instruction impacted students’ results in General Chemistry, “particularly on cognitively demanding chemistry concepts” (Mutambuki et al ., 2020, p. 1832). Having 240 students in the control group and 270 students in the treatment group increased the validity of the findings. However, the two groups had different instructors with different teaching styles which could have influenced students’ learning in a course.

Belenky and Nokes (2009) took into account not only students’ performance but also engagement while using metacognitive prompts. Based on the questionnaires, low-achievers “showed better learning and transfer when getting metacognitive prompts” (e.g., How does the solution relate to what you did?) and high-achievers “showed better learning and transfer when getting the problemfocused prompts” (e.g., What is your goal now?) (Belenky and Nokes, 2009, p. 102). The study proves that manipulatives alone do not necessarily make students think and reason deeply, it is also the way they are engaged with them. Plus, students’ engagement does not depend on whether concrete or abstract materials are used.

As it was mentioned in the previous chapter, writing reflections can be met with resistance among learners. Though reflective journals are commonly used among researchers (O’Loughlin and Griffith, 2020; Ramadhanti et al ., 2020), there are other audio or video tools to record self-reflections and at the same time engage students, for example, Flipgrid (Flipgrid, Inc., 2021), an online video forum platform. Flipgrid was recently implemented as a pilot study in the USA (Kiles, Vishenchuk and Hohmeier, 2020). In 2019 about 30-35.7% of students wrote reflections in journals and in 2020 87-93.4% participated in Flipgrid forums. 96% of students preferred to record short videos instead of writing journals. The authors believed that the casual nature of Flipgrid might have motivated students to share deeper reflections. It was concluded that this online platform has the potential as “a self-reflection tool used in combination with other pedagogical techniques to facilitate learning” but the depth of reflections should be explored further (Kiles, Vishenchuk and Hohmeier, 2020, p. 4).

As for students, changes for teachers might be difficult too. To promote metacognition during math classes, a teacher should model it first (Knox, 2017). That is why it is important to have professional development about this topic. Many teachers do not see the connection between mathematics, reading, and metacognition (Tok, 2013). Tok believes that it’s essential to grasp reading and mathematical skills in order to succeed. Both Tok (2013) and Knox (2017) argue that classroom instructions usually pay more attention to content rather than analysing the problem-solving process.

Noteworthy research was conducted in the USA for a similar duration (4 weeks) (Siegle and McCoach, 2007). However, the sample in this experiment was even larger (872 grade 5 students) and the aim was to investigate whether teachers who are trained in self-efficacy will have a better effect on students’ performance. It was also organised with a control group and a treatment group, but the teachers had professional development regarding metacognitive support in the latter. As a result, the students from this group were more confident and it had a positive effect on their self-efficacy.

Therefore, factors that support metacognitive development are (1) attention to learning goals (Buzza and Dol, 2015), (2) constant self-assessment, (3) reflection using different tools, (4) applying one strategy at a time (Davey, 2016). Meanwhile, (a) predetermined curricula, (b) predefined answers to open-ended questions, and (c) ineffective classroom management (Greene, Costa & Dellinger, 2011) can decrease metacognitive development and educators should be aware of that.

2.5 How to measure metacognition?

There are different quantitative tools to measure metacognition: the Metacognitive Awareness Questionnaire (Schraw and Dennison, 1994), the Metacognitive Awareness Inventory for Children (Jr. MAI) B Form (Sperling et al ., 2002), the Motivated Strategies for Learning Questionnaire, (Valencia-Vallejo, López-Vargas and Sanabria-Rodríguez, 2019).

Wagaba, Treagust and Chandrasegaran (2016, p. 5379) argue that sometimes it’s hard to check how students are progressing metacognitively because most tests assess only cognitive abilities. Similar to the current study, action research was conducted but quantitative data analysis methods (Metacognitive Support Pre- and Post-questionnaires) were applied. Seventy-nine students took part in this study, but the number varied throughout the research cycles. Frequently, students reported that they didn’t discuss with their peers how they learned science because they either did not have an opportunity to do so or, as the authors stated, most of them were low achievers. The researchers concluded that some results could have been “misleading because many of the scales had generally high pre- and post- mean scores in the three cycles, therefore there was not much room to move up on the Likert scale” (Wagaba, Treagust and Chandrasegaran, 2016, p. 5392). They also believed that the 20-weeks cycle is too long to enhance the necessary changes and it is better to have the same topic and the same students in all three cycles. It is also recommended to find a better instrument to test metacognition in the classroom.

One of the tools to measure metacognition is called the Metacognitive Awareness Inventory for Children (Jr. MAI) B Form (Sperling et al ., 2002). This questionnaire consists of 18 items (5-point Likert-type scale). It was adopted while studying metacognitive abilities among 150 gifted and 150 non-gifted students (Ogurlu and Saricam, 2016). Researchers claimed that gifted students possessed greater metacognition which should be considered while creating lesson plans. However, they mentioned social-desirability bias as a common vulnerability in questionnaires. Valencia-Vallejo et al . (2019, p. 15) also underlined that “the subjectivity of students’ answers is a limiting factor of self-reporting questionnaires”. Thus, further qualitative research is needed to observe metacognitive awareness among young gifted and non-gifted students, and it proves the importance of the current qualitative study.

Another fascinating research was conducted in Malaysia among 378 students to show the correlation between metacognitive abilities and achievement in mathematical problem solving (Zakaria, Yazid and Ahmad, 2009). A Metacognitive Awareness Questionnaire, modified from the one created by Schraw and Dennison (1994), was used in this research to prove that the higher students’ metacognitive abilities, the higher their results in a Mathematical Problem-Solving test. However, Muscott (2018) questions the authenticity of the problems used in this test. Employing quantitative methods, he concluded that HoMs, particularly HoM 5 Metacognition, have positive effects on students’ performance outcomes but an assessment tool to measure HoM competencies is still required.

Qualitative measuring tools use the coding of responses. The codes will depend on student answers or the researcher’s interpretation of metacognition. Responses might be scored as high, medium, or low levels (Strauss and Corbin, 1998). Stanton et al . (2015) used students’ answers to make deductions about their levels of metacognition and developed a coding system as Sufficient/Provides Evidence or Insufficient/Provides No Evidence. 245 undergraduate students in the USA took part in this experiment. Researchers coded all answers individually. Then, they discussed all findings together and came to a consensus for the final coding. Following this model allowed the researchers to get some valid data. It was summarised that metacognitive skills had an effect on learners’ performance and what is more important would assist them to become more self-regulated students in the future.

These findings support the significance of metacognitive training as a heutagogical technique among teachers and students to enhance cognitive abilities and mathematical reasoning. There is some strong empirical evidence that metacognition has positive effects on students’ performance, specifically in mathematics. However, there is still not enough evidence to prove its importance in elementary school, especially from the students’ point of view, which validates the purpose of this qualitative action research. In the next chapter, the research design and methods to solve this problem will be discussed.

3. RESEARCH METHODOLOGY

This study aims to explore the perceptions of students on the heutagogical framework while teaching maths at an elementary school in Cambodia. The study focuses on some metacognitive elements that can assist students to become more self-determined.

Central Question: How can students’ needs be met using the heutagogical framework while teaching maths in an elementary school in Cambodia? Subquestion 1: What tools and strategies should teachers use to implement the heutagogical framework? Subquestion 2: How can metacognition as a heutagogical technique be used to improve students’ self-determination?

This chapter aims to explain the research design and methods that were selected as appropriate to research the issue related to the research questions.

3.1 Research design

A lack of qualitative action research in existing literature highlighted the need for this investigation into self-determination through metacognition at a young age. Compared to traditional experimental studies, action research is an amalgam of theory and practice (research and action) that “focuses on specific situations and localized solutions” (Stringer, 2007, p. 1). It is a compass leading teacher in the right direction and helping them see changes in their practice (Simon and Wilder, 2018). Action research is “grounded in a qualitative research paradigm whose purpose is to gain greater clarity and understanding of a question” (Stringer, 2007, p. 19).

Action research was promoted in the mid-1940s with the purpose to solve some practical problems in everyday life. The goal was to distinguish the problem, try to change the situation, and check the results (Coolican, 2014). Since the main purpose of action research is to improve the practice of education by studying issues or problems (Creswell, 2012), it was decided to choose this research design to align with the purpose of the study. Self-determination is like “mini action research”. It also has a cyclical model: setting goals, attempting to achieve them, self-assessing and making adjustments (Zimmerman, 2002). Metacognition itself has a cyclical model too: planning, thinking about it and making changes if needed, reflecting, and making new plans based on the results (Costa and Kallick, 2008). Thus, it was the other reason for choosing action research as a research design for this investigation.

The main attribute of almost all action research models is the cycles, specifically that each cycle is based on the conclusions of the previous cycle (Edwards and Willis, 2014) which “guides teacher preparation and instruction” (Stringer, Christensen and Baldwin, 2010, p. 1). Sometimes they are named spirals or helices (Punch and Oancea, 2014). Edwards and Willis’ (2014) model starts with reflection: Reflect – Plan – Act – Observe, while Stringer’s (2010) model commences with the observation cycle: Look – Think – Act. Even though it seems straightforward, it is not as simple as it looks. There are no arrows between them so it is not a linear process and the researchers can go back to any cycle when changes or adjustments are needed. The “Act” stage also includes reflection and evaluation. Reflection is the key characteristic of each cycle, and the results indicate whether changes should happen or an additional cycle or “mini-experiment in practice” is needed (Wagaba, Treagust and Chandrasegaran, 2016, p. 5378). Since its purpose is not just understanding the problem but finding the solution, it is a suitable form of research for this investigation. Moreover, Nasrollahi (2015, p. 18667) noticed that Stringer’s model does not only involve teachers but also students “as action researchers collaborating in the action research process”. It is a standard model that has been similar to hundreds of other models created in the last eighty years.

The present action research will include these research tools : interviews with students and observations using the researcher journal. The research questions that require qualitative data to investigate students’ views justified the qualitative approach of this study. Interviews (qualitative methods) show the complexity of the data provided by participants (Creswell, 2012). At the same time, some quantitative methods, such as questionnaires, can be “inappropriate because of the child’s age” (Morison et al ., 2000, p. 113). Qualitative methods and students’ involvement can provide more light on heutagogy in elementary schools and explore any misconceptions.

3.2 Setting and participants

Metacognition is one of the sixteen Habits of Mind (Costa and Kallick, 2008) adopted by the participating school in Cambodia. Thus, it provided an ideal setting to apply action research on metacognitive support in mathematics. Secondly, being a subject coordinator allowed the researcher to take action and adapt the math curriculum using metacognition as a heutagogical tool. Stringer (2007) underlines that conducting action research helps in curriculum construction and evaluation. Finally, as a participant-researcher, the researcher had a chance to work closely with the participants and gather data during math classes.

Previous research in self-determination in elementary school is mostly from the teachers’ (Stein, 2018) or school administrators’ points of view (Akyildiz, 2019) which indicates the need for the present study. Earlier, children were treated as “dependent” on others to guide them on what to do (Elden, 2013, p. 78). Later, Arnold and Triki (2017) argued that children might be participants in experimental research, however, the chance exists that students would only say what they think researchers want to hear instead of their honest statements. It is understandable that writing a high-quality study with children may be a noteworthy challenge (Ponizovsky-Bergelson et al ., 2019), yet open-ended questions and encouragement can elicit some valuable data. Hoover (2018) recommends using short semi-structured individual interviews while working with young children. Semi-structured interviews “have in-built flexibility to adapt to particular respondents” (Punch and Oancea, 2014, p. 184) which is also preferable for interviews with children. Punch also suggests having natural settings and making sure that the language is age appropriate. All these recommendations were taken into account during this study. Interview questions can be found in Appendix III. Each interview with each student was not longer than 15 minutes, with three interviews during the experiment (the beginning, middle and end of the unit). Interviews were conducted via Google Meet, the platform used at school during online learning.

23 grade three students (9-10 years old) and their parents were informed about this research. 15 of them returned the signed consent forms, 8 parents did not reply to the email sent, so their children were not involved in research. In the end, 7 boys and 8 girls took part in the study.

All the students were Cambodian. Their native language is Khmer (the official language of Cambodia). The interviews were conducted in English as this is the language they use to study at this International School. This language barrier is why math in English might be particularly challenging and metacognitive strategies might be beneficial to them. All interviews were transcribed verbatim by the researcher.

Consent forms were sent on April 28, 2021, as soon as permission was granted by ERGO. Parents were called by the academic assistant from the school who informed them about an email sent. As soon as some consent forms signed by parents were returned, consent forms were sent to their children. Observations began as soon as both parents and students signed the consent forms. The first interviews were held on the same or following days as soon as the consent forms were returned:

29/04 : S2, S3, S13, S14 03/05 : S17, S21, S22 04/05 : S16, S23 05/05 : S5, S7, S8, S12, S19 07/05 : S20

The second round of interviews was held on June 17-21. The third round was recorded on June 14- 15, 2021.

3.3 Data collection

The experiment lasted 9 weeks starting on April 29 until June 30, 2021. Three cycles of Stringer’s model were incorporated into three phases of instruction (Stringer, 2007, p. 9, fig. 1.1).

PHASE 1: PLANNING

1. LOOK (3 days)

Consent forms are sent to parents and students ( Appendices I and II ). The first semi-structured interviews with some students are recorded. Field notes are taken every day.

This first stage allows the researcher to collect data about participants’ perspectives and to define the problem.

2. THINK (1 week)

Data are organised, the answers are coded and analysed, and observations continue during this time. Metacognition Diaries (MD) and the final assignment that incorporated metacognition elements are created based on the results of the interviews and on the theory of the process of metacognition which consists of three dimensions – commitment, value, and capacity (Costa and Kallick, 2008). Even though the authors presented five dimensions in their book, and they are discussed in the previous chapter, the core of this study constituted only three dimensions. First, HoMs are included in the school curriculum since kindergarten, it is assumed that by grade 3 most of the students should know the meaning of 16 HoMs, thus, Expanding Meaning as a dimension was not covered by this research. Secondly, considering the age of students and the time constraints it would have been overwhelming for students to reflect properly on all five dimensions at the same time. Since the focus of this paper was self-directedness through metacognition, Increasing Alertness as a dimension was not included in this study.

3. ACT (1 week)

MDs are shared with the students in Google Classroom (Appendices VI, VII, VIII). The final assignment is explained to students (Appendix IV). Before being given to participants, MD and the final assessment were examined by our school curriculum coordinators who supported that both assignments can be used to monitor students’ metacognitive growth.

The Flipgrid platform is introduced to students. Students have a choice to either complete MD in Google Classroom or answer these questions by recording videos in Flipgrid (video diaries). To set the tone of reflection, at the end of each math lesson students have five to ten minutes of “silent thinking time” where they have a chance to reflect and answer the questions in the diaries or record the videos. Observations continue during this time. Organisational skills are taught by using checklists.

This part of implementing practical solutions distinguishes action research from other types of research. Stringer (2007, p. 142) calls it “the sharp end of the stick”. This is the part where the action happens.

PHASE 2: INSTRUCTION

1. LOOK (1 week)

The second semi-structured interviews are recorded to check what modifications should be made. Observations continue during this time. MDs and Flipgrid Videos are checked by the researcher. Most of the student answers in the diaries are short and not specific. Multiple students do not stay online until the end of the class and do not complete the MDs or do not record the videos.

Answers are coded and categories are added or modified. Observations continue during this time. Reasons for students leaving online classes early could be family circumstances, bad Internet connection, no interest in the topic, other distractions. Student answers might be improved by providing more scaffolding and teaching how to set goals, how to monitor learning by using checklists and rubrics, how to take notes, etc. The benefits of reflections should be discussed with the students.

3. ACT (2 weeks)

This cycle takes longer because it is necessary to spend more time on interventions. All math lesson plans are modified and include metacognitive elements (Appendix V), namely goal-setting, selfassessment, peer feedback, HoM Discussion, assessing using a rubric, exit tickets, note-taking, post-assessment. Not all lessons include all these elements at once because of time constraints. It is decided to add metacognitive elements in the beginning and middle of the lessons so that all students have a chance to reflect before they leave the class. Observations continue during this time.

PHASE 3: EVALUATION

The third semi-structured interviews are recorded on June 14-15 and observations continue during this week. All lesson plans continue to have metacognitive elements and students continue to write MDs or record video diaries on Flipgrid.

Final interviews are analysed. Student MDs, videos and observation notes are reviewed, final codes and categories are added, the strengths and weaknesses of the experiment are identified. All core categories, abstract concepts and specific indicators were organised in a spreadsheet.

3. ACT (3 days)

General and brief results were discussed with colleagues during the subject meeting at the end of the school year. These continuous cycles of looking, thinking, and acting allowed the researcher to identify the necessity of metacognition as a heutagogical technique in fostering self-determination in young students.

3.4 Data analysis

Metacognition is not only about planning and knowledge activation but also the intentional monitoring of students’ cognitive processes, reflection, time management, and self-evaluation. That is why the participants frequently had chances to set goals in using HoM 5 Metacognition, self-reflect on whether they achieved those goals, monitor their progress in solving math problems, reflect on their progress using the rubric, apply different strategies in solving word problems and reflect on them. These topics were observed by the researcher and discussed during each of the three interviews. The analysis began after the data was collected from the initial interviews, starting from April 29, 2021, as soon as some parents and students returned the signed consent forms. From that point onwards, data collection (from the following interviews and field notes) and analysis occurred simultaneously. This approach has become a regular practice in qualitative research (Charmaz and Belgrave, 2015).

In order to understand the context, Stringer (2007, p. 100) suggests researchers should start with interviews and then move to other sources of information in the next cycles of action research. Observations written in the research journal were held during all three cycles of action research. The focus was on behaviours related to self-determination, particularly the metacognitive skills related to goal-setting, self-reflection, reflecting and monitoring the progress in solving math word problems, and the emotional aspect after solving problems (feeling of difficulty, feeling of confidence, etc.). As soon as the student showed some evidence of applying metacognition, it was recorded as a memo or a sentence. Thus, the protocols were kept for each student who signed the consent form. Observations were overt since both students and parents were asked to give their consent before the beginning of the study. The primary data were derived from interviews but observations further clarified or extended understanding of the issue being investigated. Punch (2014) underlines that combining interviews and observations is a good method that can lead to high-quality data.

Students’ interview data, as well as observation notes, were coded and analysed using a grounded theory approach. Creswell (2012) believes that it helps the analysts remain close to the data during the whole process. Codes and themes were developed, and connections were identified between different themes in order to generate conclusions. Based on the systematic design of the grounded theory, there are three types of codes represented diagrammatically by Punch and Oancea (2014, p. 238, fig. 10.4):

a) substantive (or open coding) is done at the beginning of the analysis to generate more abstract concepts, b) theoretical (or axial coding) to see how data is interconnected, and c) core (or selective coding) concentrates on core categories around which the theory is designed.

Charmaz (2006) emphasizes that it’s not a linear process and a researcher can go back to the initial data and make new codes at any moment of action research. Moreover, the grounded theory implies that the experiment doesn’t commence with an already formulated theory but rather “allows the theory to emerge from the data” (Strauss and Corbin, 1998, p. 12). Analysts are encouraged to (a) remain open to various opportunities, (b) produce a few options, (c) investigate different opportunities before selecting one, (d) use cycles to go back to the experiment and get a new vision, (e) believe in the study, (f) circumvent shortcuts, (g) enjoy the research (Ezell, 2017, p. 72).

Information was collected, analysed, and compared until data saturation was achieved.

3.5 Ethics and risk assessment procedures

The approval to conduct this research was confirmed through the Ethics and Research Governance Online system of the University of Southampton. In order to start the study, it was obligatory to obtain confirmation. After that, the potential participants and their guardians were invited to take part in the study via email.

First, the Board of Directors of the participating school were informed about the study. Since the children are 9-10 years old, the consent forms were first signed by students’ guardians and after that, if the parents approved, they were sent to students. Both the guardians and the participants agreed that the interviews will be recorded. Confidentiality was guaranteed to all participants. The participants had the right to withdraw from the interviews at any moment before June 30. Personal information was anonymised during the transcription process and coded as Student 2, Student 3 etc. Since February 20, all schools in Cambodia moved to online learning, thus, both interviews and observations took place via Google Meet. Only the researcher had access to the audio-recorded data which were held on a University of Southampton file storage space and were destroyed after the transcriptions were complete. The transcribed interviews and the observation notes were stored on a password-protected University of Southampton file storage space too.

3.6 Validating findings

To avoid an incorrect interpretation of data, multiple methods of qualitative data collection should be used (Oliver-Hoyo and Allen, 2006). The accuracy of the findings was validated through the triangulation analysis:

– interview with students, – observation notes, – literature review.

Triangulation of data ensured the chosen key themes, providing an insight into self-determination from the students’ views. Punch (2014) also suggests getting feedback from responders. After the interviews were transcribed verbatim, the participants were asked to check them. Triangulation and member checking are the most common validation techniques (Creswell, 2012).

In the next chapter, the results of qualitative action research will be presented.

4. FINDINGS

4.1 overview of the chapter.

Using an action research framework, it was investigated how engaging in reflection can change grade 3 students’ behaviour, making them more confident in their abilities to solve mathematical tasks and motivating them to be more active and consistently remain self-determined. Data was collected over a period of 9 weeks from April 29th to June 30th, 2021 during a lockdown (all classes were online). The data analysis stage was conducted concurrently with data collection during the three cycles of action research.

This chapter presents the data collected during the interviews with the participants and the observation notes collected by the researcher while teaching mathematics. The participants were 15 grade 3 students (9-10 years old) in one of the private schools in Cambodia. The responses and the field notes were grouped into categories and analysed to answer the research questions.

Core categories, abstract concepts, and specific indicators (Punch and Oancea, 2014) were selected. Having analysed all interviews, it was clear that open coding and deductive analysis did not answer the research questions because they focused more on the categories selected rather than how students developed their metacognitive knowledge and skills throughout the unit. Therefore, during the axial and selective coding, inductive analysis was conducted, new codes were created based on the previous analysis.

The following core categories were decided on during the analysis and were based on the Habit of Mind Dimensions of Growth by Costa and Kallick (2008):

A. Commitment (ability to self-assess, self-direct and self-monitor in their development of HoM 5 Metacognition within the unit) B. Value (ability to recognise the benefits and advantages of engaging in the HoM 5 Metacognition) C. Capacity (ability to develop skills, strategies, and techniques through which they engage in the HoM 5 Metacognition within the unit)

Each core category covered three to five abstract concepts. The specific indicators are the same for each abstract concept in each category and coloured accordingly:

• Indicator 1: not attempting to do (red).
• Indicator 2: attempting to do (yellow).
• Indicator 3: able to do successfully (green).

After reading each student’s response, it was categorised to a core category and then assigned to an abstract concept. After that, it was colour-coded based on the indicators above (Appendix IX).

4.2 Core category: Commitment

In general, most of the students showed the development of metacognitive knowledge. Four abstract concepts were defined during the analysis based on students’ responses:

1) Setting goals 2) Self-questioning 3) Self-monitoring 4) Responding to feedback

Table 1: Core Category Commitment – Overall Picture during 3 Phases

* Major improvement – moved from red to green indicator. * Minor improvement – moved from red to yellow indicator or from yellow to green indicator. * Stay in red – did not show any progress. * Stay in yellow/green – already had good self-determined skills at the beginning of the research.

More examples and notes can be found in Appendices X – XIII that complement Table 1.

4.2.1 Setting goals

It is interesting to find out how all students understood and could define the importance of setting goals during the first interviews but most of them did not set goals in mathematics or did it only when teachers asked them.

After completing the tasks with the metacognitive elements, some students shared their views on how goals helped them:

“don’t give up and try to solve a problem” (S17-P3), “know what to do and you will not lounge, so you’ll be better” (S5-P3), “learn and stay happy and, like, don’t get bored of learning” (S13-P3).

However, according to the field notes Students 5 and 17 were not persistent in achieving their goals during the unit. Other students believed that it was not necessary to set goals because they did not have time to do them (S20), or they were not important in maths because “you don’t have to calculate” when you write goals (S23-P3). Student 7 did not set any goals because she forgot about them. Based on the observation notes, she also missed some classes or did not complete most of the metacognitive activities before the third interview was held.

4.2.2 Self-questioning

The analysis of the interviews highlighted the importance of self-questioning. Some students were consistent throughout the unit and shared some questions they asked themselves: “is this answer correct” or “have I filled the checklist” or “do I have to start over again?” (S8-P1).

Others claimed that they did not ask themselves questions (S5-P1). However, during the last interview, they mentioned the importance of self-questioning “because if you don’t ask yourself, you might don’t know what to do” (S5-P3) or referred to past knowledge “because you can ask yourself…or just get some ideas from the past” (S20-P3). While during the first interview Student 3 said she would rather wait for the teacher to ask questions, later she noticed “my PT [Performance Task] is not that good so I change my PT, and so I ask question, ‘how good is my PT?’ “ .

Another theme that emerged from the interviews is the lack of persistence: “I ask, ‘what do this task do?’ and sometime even I don’t understand… but sometime I guess” (S22-P3), or “I ask myself ‘Really?’ and go back and sit one more” (S7-P3) or the students simply replied they did not ask any questions (S2). Some participants claimed that they preferred to ask other people because “I don’t know about myself” (S12-P3).

4.2.3 Self-monitoring

Out of all the concepts, this was the only one where all students showed improvement by the end of the unit which is an important sign of self-directed learning. Starting from not using the diary “because I forgot about it” (S14-P1) to using it to find the unfinished assignments “When I go to checklist they will have a link to go in the work for math… it help us know what work that we still haven’t finish” (S14-P2). Field notes show that Student 14 participated more in the middle of the unit. According to the data collected from the interviews, Students 2, 3 and 16 already had good self-monitoring knowledge since Phase 2.

It was thought-provoking to listen to what students think about using rubrics ( Appendices IV and V ) and self-assessment in maths. Most of the participants highlighted that it was useful to have rubrics in the Metacognition Diaries:

“so Teacher can know which task they don’t really understand” (S22-P3), “students write goals and look at rubrics to see what their grades and grade theirself and do important stuff on it” (S3-P3),

On the other hand, some students pointed out such issues as dishonesty, inability to use the rubric without knowing the correct answer, and overconfidence. Student 21 (P2) mentioned that some students might not be honest when they grade their own work: “they want a perfect score and then when they’re bad they just put four and they’re saying that, ‘I’m good, I’m good’ “ . Student 19 (P3) pointed out that it is difficult to use the rubric when you do not know whether the answer is correct or not: “So, when kids do it, no one, he or she in the PT cannot predict if they’re correct or not” . And during Phase 2, he mentioned that he used the rubric when the teacher projected it but not on his own initiative “when you just post the assignment with the rubric under it, it’s very hard for you to make me watch rubric” . Student 20 (P2) also underlined that “it’s kinda helpful if you’re not good, but if you’re good already, you always grade yourself four, I think it’s not really that useful” .

4.2.4 Responding to feedback

This abstract concept was mostly based on the observation notes because students were not asked about feedback during Phases 1 and 3. The planning for the interviews was not done properly. The list of selected questions differed during three Phases. Students were asked specifically about feedback during Phase 2. After the analysis was done, it became evident that they should have been the same questions in all three Phases.

What some students said in the interviews did not match their behaviour during the observations. I believe it happened because they probably wanted to say what they thought the teacher would want to hear or they wanted to present themselves in a positive light (social desirability bias). For example, Students 17, 21 and 22 said that they often checked the feedback in Google Classroom but based on the observation notes, they did not reply to them. Having read the questions that the researcher asked about feedback, some of them could have been reworded or asked indirectly (how a third party would behave) so that students do not feel embarrassed. It proved to be effective while interviewing Student 23:

“Researcher: What advice would you give to students who just finished grade two and are moving to grade three? … Student 23: … I recommend them to use, study more math, use more link and make sure that do more work than me, ‘cause I never do my work. Researcher: Why not? Student 23: You don’t remember at the last interview I said. I’m lazy, but now I do. Researcher: I remember you said so. Student 23: But now I do it.” (P3)

Most students understood that they had to check teacher’s comments and correct their mistakes “If I get feedback I try to make it better, for example, the math what is perimeter, I always do it then you always feedback me, that time I had [a perimeter of] more than 24 and then now I have [a perimeter] over 40” (S20-P2). Some of them checked and replied to comments frequently: “I check on the private comments and I finish some of your private comment and then after, later I will do the next comment and then after that I turn in the work” (S8-P2). Observation notes confirm this data too.

Other students explained why they did not respond to feedback. They either missed the notification “sometime I didn’t see my email to me” (S12-P2) or they were overwhelmed “I have a lot of email and then it comes a lot of email now” (S16-P2), or they still do not understand the feedback “sometime I just don’t understand the question” (S13-P2), or they forgot about it “sometime I forgot” (S14-P2). All these reasons are understandable for grade 3 students who moved to online learning a few months ago. When the students were studying onsite, real-time feedback was provided every day during classes. For example, when students completed a task, the teacher would return their notebooks and they had a chance to ask questions for clarification in person.

Even though some students understood that they could have asked for feedback “Maybe I can ask for feedbacks or I just, when I’m offline or I don’t have anything to do, I’ll just try a little more” (S7-P1), they could not identify why they had not responded to it: “sometimes I just miss it” (S7-P2).

4.3 Core category: Value

In general, most of the students showed the development of metacognitive experience. Three abstract concepts were defined:

1) Making connections to real life and the future 2) Giving advice to other students 3) Emotional aspect after solving problems

Due to the subjective nature of this core category, it was not quantified as the other two.

4.3.1 Making connections to real life and the future

Almost all students could have connected the unit to real-life or to the future starting from the first interview, so these questions were not asked in further interviews:

“I know how to measure, like, when I don’t have a standard unit, I can use the non-standard unit” (S2-P1); “we learn about litres… so, as a scientist, we have to put portion… to invent something” (S8-P1); “because my parents own a business… they just [bought something] from China and we measure the stuff” (S13-P1).

Students who did not make clear connections between reality and the unit did not participate actively in class. They mentioned some general math connections, e.g., counting money (S5), multiplying something (S7) or calculating the cost of the units (S22) which were not relevant to the current unit.

4.3.2 Giving advice to other students

The purpose of this concept was to see if students can apply self-directedness to external situations.

When students were asked to recommend something to children who are moving to grade 3, these themes emerged:

• Use HoMs when solving difficult problems “Persisting and Apply Past Knowledge to New Situations and Listening and Communicating with Clarity and Precision” (S8-P3);
• Watch the recorded lessons and Youtube videos “I will tell them what to watch in the YouTube to help them a success in grade three” (S16-P3);
• Read more books (S14);
• Do extra research “search more about shapes and math because when you go to grade three, now you will learn about the fractions and rhombus, new shapes” (S20-P3);
• Ask questions and no copying (S21);
• Review difficult topics before studying in grade 3 (S22).

4.3.3 Emotional aspect after solving problems

This was the only abstract concept that was not colour-coded based on the specific indicators as they were not applicable here. The purpose of this concept was to observe whether emotions can have some effect on metacognition and self-directedness in general. Students were asked about their feelings directly during the interviews and some of them were also inferred from their answers or observations.

This category explained the answers to other categories. For example, a lot of Student’s 2 answers were simply “No” or “I don’t know”. Thus, in some cases, this lack of persistence could have contributed to less self-questioning and less note-taking.

Students 5, 8 and 20 did not get sad when they made mistakes (P1). They said it was a chance for them to improve more. However, as observation notes show, Student 5 lost his motivation during the unit and did not complete most of the graded tasks. Reasons might be different: family circumstances, no ability to become independent and control his learning or even that it was the last unit of the year, and he was simply tired. Meanwhile, Students 8 and 20 were developing their metacognitive knowledge and skills. Thus, while these reasons helped some students, they hindered others.

During Phase 3, Student 7 was worried that she did not complete most of the tasks, but she also did not ask for help during the unit and missed a lot of online classes. Same as Student 14 who could not identify the difficult topics and thus, could not make an improvement plan. It is possible that lockdown impeded teachers’ ability to reach out more to students in need. In order to help students who struggle, the teacher could have initiated some interventions to learn more about the students’ circumstances.

Student 12 could identify the difficult parts of the unit and was very persistent to learn these topics. Her answers for goal-setting, strategies and self-monitoring concepts, and the observation notes combined into a complete picture to show improvement in self-directedness, similar to Student 13 who also showed enthusiasm to study more on her own. These are the students who were doing well before lockdown and continued doing well during online learning.

4.4 Core category: Capacity

In general, some students showed the development of metacognitive skills. Five abstract concepts were defined:

1) Connecting metacognition to math 2) Applying different strategies when solving problems 3) Asking for help when needed 4) Taking notes 5) Improvement strategies

Table 2: Core Category Capacity – Overall Picture during 3 Phases

* Major improvement – moved from red to green indicator; * Minor improvement – moved from red to yellow indicator or from yellow to green indicator; * Stay in red – did not show any progress; * Stay in yellow/green – already had good self-determined skills at the beginning of the research.

More examples and notes can be found in Appendices XIV – XVIII that complement Table 2.

4.4.1 Connecting metacognition to math

About half of the students were able to connect metacognition to math at the end of the unit. Students could recognise the purpose of using Metacognition Diaries in class:

• to understand the meaning of the HoM (S21-P3),
• to help set goals and self-assess (S17-P2),
• to talk about feelings, improvements, and plans (S16-P2),
• to make it more challenging (S13-P3).

Student 12 connected it to the lesson when she solved word problems about time “because I really struggling about time so I need to think if I go backwards” (P3). Student 8 made a connection to the lesson about perimeter “so that we can express our opinion of our commitment over doing those tasks.” (P2). Student 23 recognised her internal doubts and uncertainty and understood that they need to be eliminated:

“I think about my thinking because if we don’t think about our thinking, for example, I think to do my work, but my mind, my half-mind say don’t know. So, it’s still no. Researcher: So, what do you do then? Student 23: I make that mind go together to get along.” (P3)

Some students still did not see a clear connection to math and said they only used it when completing Metacognition Diaries (S2-P3), when the teacher asked to do the tasks (S19-P2), or when recording videos in Flipgrid “I need to think what I need to say” (S20-P2), “it really help not to always write, we need to talk to people, to not be shy, to show your feeling and your answer, so, I think it’s good” (S13-P2). The students who completed only 1 Metacognition Diary or did not write at all (based on the observation notes) did not see the importance of thinking about their thinking:

“I didn’t think of that” (S14-P3), “I don’t know how to answer this question” (S5-P3), “I do not understand the HoM yet. It’s kind of difficult.” (S3-P3).

4.4.2 Applying different strategies when solving problems

The following strategies were mentioned during the interviews:

• asking the teacher or listening how it was explained to other students (S3);
• asking other family members (S21)
• Striving for Accuracy (S5);
• Persisting (S16);
• Thinking Flexibly (S17, 22);
• Applying Past Knowledge to New Situations (S20).
• completing the shortest tasks first: “Because I was like, ‘What, I still have more task?’, so I have to go do the small task so when I did the small task I will do the long task later” (S8-P2);
• finding clues and drawing a model: “First, I need to find clues. And the second, I try to do a model but drawing a model is very easier” (S12-P1);
• using a calculator (S13).

Five students showed some improvement during the unit. The rest of the students stayed on the same level as they were in the beginning and could not identify any strategies (S2, 7, 14) or said they would just guess the answers “sometimes following your gut you get it right” (S20-P).

I think this concept was not developed well during the unit. The prepared lesson plans did not include any specific strategies to help the students because of time constraints.

4.4.3 Asking for help when needed

Most of the students stated that if they did not understand the problem, they would ask their teacher (S3), family members (S21) or friends (S5).

Even though some students said during Phase 1 that they did not ask anyone “I do not need someone to help when it’s math” (S2), later they mentioned they would ask their relatives if they did not understand. Student 14 remarked that he was scared to ask the teacher “because I ask too much” (P1) and observation notes indicate that he rarely asked for clarification during the unit. This lack of confidence could have hindered his metacognitive skills.

Some students noticed that before solving some problems on their own, they would ask someone to help (S7, 8). Maybe studying at home during the pandemic is the reason for it. Meanwhile, Student 13 mentioned that she would try to solve the problem by herself first and then she would ask someone if needed “or when they’re not home, I would try to research a bit” (P2). Student 20 (P3) also mentioned that he would research by himself before asking anyone. During Phase 1 and 2, Student 22 understood that she needed to ask the teacher more often “but sometime I didn’t ask about, I just do it” . During Phase 3, she stated that she asked the teacher when she needed help which is also mentioned once in the observation notes.

4.4.4 Taking notes

Most students understood the purpose of taking notes: “we take a note and then when we go back to try finishing [a problem], we can copy instead of wasting our time on thinking too long” (S7-P1). However, based on the observation notes, this student did not follow her own advice. During Phase 3, she stated that she only noted on her whiteboard which tasks she had not finished yet. When students were asked to give a piece of advice to their younger peers, Student 13 (P3) suggested watching some videos on YouTube because “mostly people just watch it for fun, not education … so when they have free time they can watch it and take a lot of notes from the video and what do they understand from the video” . She also mentioned that she did not take any notes during Phase 1 and used to forget many things but during Phase 2 she took some notes in order to prepare for the quiz. It was interesting to see how one student understood that he needed to work on controlling his emotions during classes and mentioned that he took notes to remember it (S19).

Students shared how and when they took notes during this unit:

• Whiteboards “Because if you do it like that, the answer is correct” (S2-P1). However, later he said, “I don’t know why I need to take notes” (P3). Similar answers were provided by Student 3.
• Sticky notes “when you show example of one of the EQ [Essential Question], I have to go on the notes and write what you said, so when I go back to my EQ, I know what I, I can learn off that example” (S8-P3).
• Notebook “get notebooks that I write lessons inside” (S12-P1).
• Google Docs or Slides “when we did the clock hand I get a paper and then I write about the short hand and long hand and for this end this unit, for the polygon, I also write it in a Docs” (S16-P3). Meanwhile, during Phase 1 she remarked that she usually forgot to take notes.

Other students mentioned that they took notes when the teacher told them to do it but did not show the initiative themselves “maybe I’m not sure about this, maybe it’s wrong and why did I take note about it if it wrong” (S12-P3). Based on the interview and observation data, some students (14 and 17) did not show any improvement in this concept.

4.4.5 Improvement strategies

Generally, students could identify some specific improvement strategies, such as:

• watching videos on Youtube or the recorded lessons (S12),
• playing online games connected to the topic (S14),
• applying HoM’s Listening with Understanding and Empathy (S17), Creating, Imagining and Innovating (S19) and Applying Past Knowledge to New Situations (S20),
• asking more questions and participating during classes (S19).

Some students shared the view that they would like to have Metacognition Diaries when they study in grade 4 so they could set goals for improvement (S12). This student also said she would do the same extra activities as she was doing in grade 3. Even though she felt they did not help much, she could not identify how to change the situation. A quiet environment when studying online was also mentioned in one of the interviews (S13). In the previous interviews, this student also said she could have asked her family to prepare some extra tasks for her to practice or set a goal to improve. Student 20 emphasised that it was important to know what topics and PT we would study in advance so that he could prepare better.

Changing Behaviour

Another theme that emerged from the interviews was changing behaviour. One of the problems was joining online classes on time. Student 23 started to ask her grandmother to wake her up so she would know how to do the assignments. Student 3 wanted to stop doing the tasks without thinking. To change the situation, she would “[ask] in Hangouts and then just try to do it, one by one and be careful” (P3). Meanwhile, Student 2 could not provide explanations for how to change his behaviour.

Other students either could not define whether they needed to change anything (S5 and 7) or did not specify any improvement strategies apart from “practice about the lesson that I don’t understand” (S22-P3). Student 22 also did not feel she was getting better because she said she did not understand maths. Observation notes show that she missed a lot of classes or did not do the prepared tasks. Finally, some students underlined that they wanted to study at school because they did not have enough materials at home (S20).

These findings are thought-provoking and will be discussed further in relation to academic literature in Chapter 5.

5. DISCUSSION AND CONCLUSION

5.1 overview of key findings.

The research question of this study: How can students’ needs be met using the heutagogical framework while teaching maths in an elementary school in Cambodia? By students’ needs we mean self-determined learning needs. Following the Self-determination Theory (Ryan and Deci, 2000) and the findings from the current research, it is possible that students’ basic needs – autonomy , competence , and relatedness – can be met using the heutagogical framework while teaching maths in an elementary school. Having a choice in choosing their goals, self-monitoring, taking notes and connecting metacognition to maths, eight students were able to control their learning process. Competence was practised by applying different strategies when solving problems and reflecting on improvement strategies. Relatedness was achieved by responding to feedback, self-questioning and asking for help when needed. The data were obtained from 45 interviews (3 phases – 15 interviews each) with students and constant researcher’s observations during online classes for 9 weeks. The data presented in this action research have shown that some metacognitive strategies are necessary at elementary school from students’ perspectives.

Students’ answers were grouped into three categories: Commitment, Capacity and Value – three dimensions of Habits of Mind developed by Costa and Kallick (2008). The majority of students showed improvement in the Commitment category: they often set goals for improvement, questioned themselves and monitored their learning. The results showed that most of the students found them useful, especially when studying at home. Less than half of the students showed the development of metacognitive skills in the Capacity category. Most of them already had some skills before the experiment started (e.g., some strategies for solving problems, improvement strategies, and asking for help when needed). The Value category represented how students developed their metacognitive experience and supported the finding of the previous categories.

5.2 Research outcomes in the framework of existing literature

The findings of this research have several themes that have been discussed previously in the literature review.

Research subquestion 1: What tools and strategies should teachers use to implement the heutagogical framework?

It was found that the following tools and strategies were used to implement the heutagogical framework:

• goal-setting in Google Classroom (at least once a week),
• using checklists, rubrics and self-assessments to self-monitor,
• note-taking,
• discussions at the end of the class,
• recording video diaries on Flipgrid to share strategies implemented during the day and/or completing Metacognition Diaries.

Goal-setting

According to the findings, most students started setting goals regularly during the unit and found them helpful in mathematics. The data reported in this study support the results of Erwin et al . (2016) who claimed that self-determination in adults and children is different, but that some elements can be practised in elementary schools, namely goal-setting. They developed a model with four steps: assess, select, try it, and reflect. The focus was on goals and reflection strategies which promoted good results and consistent communication between parents and teachers. Goal-setting can eventually help students become independent. The learning goal orientation might be a starting point in training self-determination (Compagnoni, Sieber and Job, 2020).

Students who showed high metacognitive knowledge and skills since the first interview noticed that rubrics were not as useful for them as they used to score high and did not feel any improvement. Other students also highlighted that some students might not be honest when they self-assessed and in that case, it was not beneficial. These disadvantages were previously discussed by Jamrus and Razali (2019) who believed that these constraints could be overcome if proper instructions and observations were conducted by a teacher. Moreover, rubrics should challenge students’ abilities, but rubrics’ language should be age-appropriate (Costa and Kallick, 2000).

Another self-monitoring tool that the students used was the checklist. Organising the thought processes and refining the thinking skills is essential for lifelong learners (Knox, 2017). Students found the checklist convenient while studying at home because they could quickly find the tasks and self-monitor. Nidus and Sadder (2016) emphasise the importance of teaching this art of noticing at school. They believe that when students learn how to use the checklist, they can focus on specific tasks, set goals for improvement, thus ask for more detailed feedback and evaluate their progress. The findings of this study about feedback support the results of a quantitative study of Molin et al . (2020) who found that there are positive effects of feedback from teachers or peers on both students’ metacognitive skills and motivation. According to them, responding to feedback is a key element of self-determination. Even though the findings for this specific abstract concept are not complete as the students were not asked about feedback during all three phases, it is still important to take their responses into account, specifically the ones connected to the lockdown.

Metacognitive Diaries (MD)

MD – reflective journals – as a tool of students’ self-assessment positively influenced students’ metacognitive experiences. Knox (2017) recommends journal writing and writing through the problem-solving process during math classes to define students’ mental processes while gaining knowledge. Following the action research framework in this study, students’ reflections were reviewed, and notes were made about their progress. It was also interesting to observe how some reflections reported in the Findings chapter moved from superficial to in-depth in Phase 3. By in-depth reflections, Costa and Kallick (2008, p. 235) mean “making specific reference to the learning event, providing examples and elaboration, making connections to other learning, and discussing modifications based on insights from this experience” . While most of the students from our study reported that they would like to have MD in further grades because “we can tell about the goals that we want to achieve and we can tell how we feel to the teacher and the score that we give to ourself if we give ourself a low score, the teacher will know that we don’t really understand it” (S16-P3), three students did not want to have MDs when they move to grade 4 stating that – “I can think of by my own and then study” (S2-P3). Recent research with elementary students supports the previous findings with university students (O’Loughlin and Griffith, 2020) and illustrate that not only students’ metacognitive skills were affected but also teachers had evidence of how students progressed during the unit. A similar experiment with reflective journals was recently designed in Indonesia (Ramadhanti et al ., 2020). However, the questions in that study were grouped around such aspects of metacognition as awareness, evaluation, and regulation. Fifty students who were involved in Ramadhanti et al. (2020) study went through these processes and were able to become independent learners.

Video Diaries (Flipgrid)

Flipgrid (Flipgrid, Inc., 2021) was used as an alternative online video-response tool to self-reflect and facilitate discussions. Students were already familiar with the concept of journaling. This app was used as an alternative to MDs to increase students’ engagement. Students had a choice of how to answer metacognition questions: writing MD, recording a video on Flipgrid or both. Most of the students reported that they enjoyed using the app because it was something new and they could show their feelings, but student engagement has not dramatically increased. The findings of this study differ from those reported by Stoszkowski, Hodgkinson and Collins (2021) which indicate that participants provided more frequent and more critical answers when using Flipgrid. Although, it might happen because older students took part in that experiment, or the differing amount of scaffolding provided by teachers in various studies. Educators should understand that this platform is not a “magic bullet” to increase participation (Kiles, Vishenchuk and Hohmeier, 2020, p.1).

Research subquestion 2: How can metacognition as a heutagogical technique be used to improve students’ self-determination?

It was found that metacognition as a heutagogical technique might be used to improve students’ self-determination.

The findings of this study are in line with previous literature (Gourgey, 1998; Costa and Kallick, 2008). It proved that students who did not have a habit of thinking metacognitively might not show a lot of enthusiasm in the beginning, especially if they had been passive learners for some time. Panadero (2017) believes that interventions have different effects on students because of their educational level. Moreover, the learning environment played a significant role in developing metacognitive capabilities in mathematics, especially in the current lockdown situation. On the other hand, some students were able to recognise other HoMs during the interviews or even spontaneously during classes, according to the observation notes. The creators of HoMs believe that it is one of the strategies that students can use to build deep reflections in order to become lifelong learners (Costa and Kallick, 2008).

During our maths lessons, some students were able to choose what strategies worked well in achieving their goals or they could change their learning approach if needed. This ability to monitor and regulate is the nature of metacognition (Wagaba, Treagust and Chandrasegaran, 2016). Students reported such strategies as applying other HoMs, finding clues and drawing a model, completing the shortest tasks first, using a calculator, asking the teacher or family members. Nevertheless, when looking at the strategies reported by the students during the interviews, five students stayed in the same yellow or green indicator during three phases and same number of students showed major or minor improvements. Therefore, it is difficult to say whether they were the results of applying past knowledge or strategies offered by previous teachers, peers, family members or they were the strategies taught in this unit. Corresponding findings have also been reported by the studies of Davey (2016). Investigating metacognitive development in early years children, she underlined that even younger students can talk about strategies they implemented when facing a problem to solve. The only thing that is clear is that three students still stayed in a red indicator by the end of Phase 3. Perhaps, more focus should be provided on this area when creating lesson plans.

Different studies argue that if students are taught metacognitive strategies at school, they will perform better and even their general wellbeing will improve (Perry, Lundie and Golder, 2019). This study was not focused on students’ performance, but their emotions were taken into account in the Value Category. The findings support the studies of Gabriel, Buckley and Barthakur (2020) who concluded that motivational and emotional factors affect students’ abilities to self-regulate their learning. Mathematics anxiety is an obstacle to learn maths and might impede students’ engagement and the metacognitive processes in general. It was particularly observed during online classes when students could simply leave the meeting when they wanted or do not join at all. The reports in the findings chapter prove that the students who did that felt worried or stressed by the end of the unit because they did not know how to solve the tasks and since they did not have developed metacognitive skills, they could not self-regulate their learning.

5.3 Limitations of research

The researcher acknowledges several constraints of the study. First, it was difficult to set the tone of reflection while teaching online. Most of the metacognitive tasks were done at the end of the lesson and by that time half of the students left the meeting (bad Internet connection, family circumstances, distractions etc.). The results could have been different if teaching on campus. On the other hand, online settings can test self-directedness even better (Cano-Hila and Argemí-Baldich, 2021) so it probably shows that some students were not ready yet to monitor their learning.

Secondly, it would have been better to have this pilot in the middle of the year. At the end of the year, a lot of graded tasks had to be done and there was not enough time to do as many reflections as was planned. Costa and Kallick (2000) recommend that students should reread their journals from time to time to compare their thoughts and make an action plan. During this study, the students were asked to review their Metacognition Diaries but a lot of them did not do it because of the time limits. Adding to this, the scope of this research project was too broad for the timeframe in which it was conducted.

Finally, the concept of semi-structured interviews was partly misunderstood by the researcher. Some of the questions differed during the three phases, thus the progression might not be completely visible. Moreover, some of the questions could have been reworded to avoid different biases.

Overall, while the results of this study appear promising, they should be treated with caution due to the above limitations.

5.4 Implications for further research

The recommendations for other researchers have been made to provide an impetus to continue research on metacognition from students’ perspectives. The data collected could greatly support teachers and curriculum coordinators in finding solutions to overcome the issue of low self-determination in elementary schools. Stringer recommends being careful while creating the questions so that the interviewers do not integrate their ideas into the interviewees’ answers (Stringer, 2007, p. 65). Having analysed the data, it was noticed that some students’ responses contradicted what was observed in the class. Meanwhile, when they were asked indirectly (e.g., “What advice would you give to students who are moving to grade 3?”) they could make connections to themselves and their learning styles. Kaminska and Foulsham (2013) believe that this social desirability bias is caused because of students’ embarrassment and uneasiness if their answers do not match with teachers’ expectations. It is recommended to reword some questions if this research is about to be repeated. For example, instead of asking “Do you think the rubric that we used in class was helpful or just a waste of time?” it is better to ask “Would you recommend having rubrics like this in grade 3? Why or why not?”.

For teachers, it is crucial to inspire students, at every age, to find how they learn and what benefits them individually (Pritchard, 2013). That is why it is important to have professional development in teaching metacognition at this early age. Plus, further research can be carried out to demonstrate whether different teaching styles have an effect on how students develop metacognitive knowledge and skills.

Regarding future research, there is one more perspective worth investigating, that of the parents. First, parents have a big influence on their children’s achievements (Jezierski and Wall, 2019). Secondly, it is paramount to know whether the students apply these metacognitive strategies or other heutagogical techniques at home and whether their behaviour has changed because of it, especially during the lockdown.

5.5 Practical recommendations

Even though some educators believe that developing metacognitive skills is often difficult and time-consuming (Thomas, 2003), it is an important goal of education. There are a lot of techniques and strategies that teachers might implement to train students’ metacognitive abilities. Students should be given opportunities to learn how to set goals, assess their progress and take ownership of their learning.

However, teachers should not expect that students already know how to monitor their progress, plan and self-evaluate as all these skills should be explicitly taught and modelled (Perry, Lundie and Golder, 2019). We cannot expect elementary students to already have metacognitive knowledge and skills (Wagaba, Treagust and Chandrasegaran, 2016). One of the goal-setting strategies that worked during this study was to have students self-assess first and based on their results ask them to write what they can improve and how. It was important to model that our goals should be specific, achievable, and timely. Furthermore, it was repeated every time before the activity started that these goals are for them and not for the teacher so that students synthesize the importance of goal-setting.

Conducting discussions turned out to be one of the tools that promoted metacognition in the classroom. Following Costa and Kallick’s (2008) advice, some thought-provoking questions were designed for the MDs (Appendices VI-VIII). During these moments, students learned how their peers applied some strategies and grew in this Habit of Mind. If the students are studying onsite, it might be better to conduct discussions at the end of mathematics classes but if lessons are online, it is recommended to either ask the students to complete the journal in their free time or choose a moment when most of the students are present.

It is crucial to demonstrate that solving problems is not only about finding the correct answer but about the process. Moreover, reflective writing might support students in determining their strengths and weaknesses in the topic. Costa and Kallick (2008) also suggest students should reread these journals from time to time, comparing what they have learned in the past and now. Based on the observation notes, teachers should encourage students to complete the diaries regularly to habitualize them. Perhaps teachers may complete the diaries as well as an example.

It is also important not to expect immediate results. Even though this study lasted 9 weeks, it was not enough to coach on metacognition. Students should have enough time to understand their learning processes and to develop a habit of reflecting on their learning and experiences.

5.6 Conclusion

This section is an overall conclusion of the current research. Previous studies from teachers’ and administrators’ perspectives summarised that implementing metacognitive techniques will positively influence students’ performance (Stein, 2018). Nevertheless, the lack of empirical studies from students’ points of view is still a concern in education.

The purpose of this study was to enable grade 3 students to reflect on their learning practices and habits by using metacognitive methods and to see whether it can help them become self-determined learners. Self-determined learners are students who can self-assess, self-direct and self-monitor their learning (Commitment Category), can recognise the benefits and advantages of engaging in metacognitive processes (Value Category) and develop skills, strategies, and techniques through which they engage with their peers (Capacity Category).

Metacognition is the foundation of lifelong learning. Action research cycles with constant reflections helped the researcher as an educator to learn what is best for her students and how to adapt to meet the 21-st century requirements. Based on the observation notes and interview data, most grade 3 students were able to find more motivation for learning mathematics, therefore became more engaged in the learning process. Hence, it might be useful to stimulate deep reflections at an early age. By increasing metacognition, students could find different strategies, apply them, and choose the most effective ones based on the situation. That’s why it is a heutagogical technique. However, some students still could not regulate their learning even after the metacognitive interventions were implemented.

Less than half of the students could connect metacognition to mathematics by the end of phase 3 and three students were able to do it before the experiment started. Four students either stayed in the red indicator or dropped from yellow or green. However, it is important to note that these students missed more than a third of online mathematics classes or did not do most of the metacognitive activities assigned during the unit.

The pandemic lockdown affected children’s learning. Even though there are a lot of physical and emotional limitations connected to the pandemic (Cano-Hila and Argemí-Baldich, 2021), it might be considered a perfect setting to practice metacognitive skills when students can set goals, monitor their learning using different checklists and rubrics, reflect on the feedback provided in Google Classroom and think about improving strategies while writing diaries. Hopefully, these positive aspects of online learning concerning metacognition should remain even after lockdown.

However, recent research in education showed that in fact, the lockdown increased the gap between high and low achievers: stronger students had more ability to concentrate on the tasks while weaker students were less able to focus (Spitzer, 2021). This study supports this statement because when we look at the findings, it is noticeable that the students with yellow or green specific indicators in Phase 1, continued to show progress in other Phases. But if they started in a red indicator and did not join classes or did not do the metacognitive tasks, they stayed in the same colour code. If students are studying at school, they are approximately equal in terms of metacognitive support provided by a teacher. Meanwhile, when studying at home, there are different family circumstances that can increase or decrease their metacognitive skills (e.g., eliminating distractions, helping with monitoring their progress, doing work with parents, etc.) (Cano-Hila and Argemí-Baldich, 2021). Another term used recently in research is “Zoom fatigue” when students are getting tired from overusing virtual platforms (Wiederhold, 2020). A few children reported during the interviews that they were overwhelmed with the number of emails and feedback on different platforms that sometimes they missed and did not reply to the teacher.

The study has revealed that such tools as Metacognition Diaries and video diaries on Flipgrid were effective from students’ perspectives to regulate their learning. This indicates that these tools might be useful in implementing the heutagogical framework. Based on students’ responses, rubrics were useful during mathematics classes by helping students improve their work or informing the teacher about their progress. However, such issues as dishonesty, overconfidence, and the inability to use the rubric without knowing the correct answer were also highlighted and should be taken into account by educators. Checklists, on the other hand, turned out to be very helpful self-monitoring tools during online classes.

Another summary related to the results of the study shows that although some students stated that they wrote goals, responded to the teacher’s feedback, took notes, it disagreed with the researcher’s observation notes. Even though MDs were not graded, the final assessment about HoM 5 Metacognition was included in students’ grade books. It could have influenced some students’ answers. Some students could have hidden what they did not know, and it is not the purpose of reflections (Ramadhanti et al ., 2020). That is why it is better to have it not graded in further studies.

I would like to conclude this paper with the quote of Mark Van Doren, “The art of teaching is the art of assisting discovery” (BrainyMedia Inc, no date). As researchers and as practitioners this should be our aim and metacognition as a heutagogical technique might assist in it. What if teachers were more concerned about students’ abilities after graduation (e.g., problem-solving, decision-making, being a lifelong learner) rather than focusing only on the acquisition and end-of-year exams? Finally, the ultimate goal of education is to develop lifelong learners and, I would add, metacognitive and self-determined lifelong learners.

For references and appendices please refer to the full dissertation in PDF form.

Appendix I: Consent Form (Parents) Appendix II: Consent Form (Students) Appendix III: Interview Questions Appendix IV: Final Assessment Appendix V: Lesson Plan with Metacognitive Elements Appendix VI: Metacognition Diary (MD) – Value Appendix VII: Metacognition Diary (MD) – Capacity Appendix VIII: Metacognition Diary (MD) – Commitment Appendix IX: Coding Legend Appendix X: Qualitative Analysis of Abstract Concepts – Setting goals Appendix XI: Qualitative Analysis of Abstract Concepts – Self-questioning Appendix XII: Qualitative Analysis of Abstract Concepts – Self-monitoring Appendix XIII: Qualitative Analysis of Abstract Concepts – Responding to feedback Appendix XIV: Qualitative Analysis of Abstract Concepts – Connecting metacognition to math Appendix XV: Qualitative Analysis of Abstract Concepts – Applying different strategies when solving problems Appendix XVI: Qualitative Analysis of Abstract Concepts – Asking for help when needed Appendix XVII: Qualitative Analysis of Abstract Concepts – Taking notes Appendix XVIII: Qualitative Analysis of Abstract Concepts – Improvement strategies

LIST OF TABLES Table 1: Core Category Commitment – Overall Picture during 3 Phases Table 2: Core Category Capacity – Overall Picture during 3 Phases

LIST OF ABBREVIATIONS AND ACRONYMS EQ = Essential Question GC = Google Classroom HoM = Habit of Mind MD = Metacognition Diary MK = Metacognitive Knowledge ME = Metacognitive Experience MS = Metacognitive Skills PT = Performance Task P = Phase S = Student S2-P1 = Student 2 – Phase 1 SLO = Schoolwide Learner Outcomes

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Transforming mathematics classroom practice through participatory action research

• Open access
• Published: 16 January 2020
• Volume 24 , pages 155–177, ( 2021 )

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• Pete Wright   ORCID: orcid.org/0000-0002-6926-4237 1  

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This paper explores the potential of participatory action research to bring about significant changes in practice in a context in which more conventional approaches to research have had limited impact. It focuses on secondary mathematics classrooms where teaching approaches characterised by memorising and practising mathematical procedures, with little understanding of their application, purpose or underlying concepts, remain commonplace in many countries around the world and have proved highly resistant to change. The paper highlights the damage caused by such practices in terms of the alienation of large numbers of students and the inequitable outcomes they are associated with, including the strong correlation that persists between students’ socio-economic status and mathematical attainment. It reports on the case of one participatory action research project, involving the author and five teacher researchers, that demonstrated how a vision of mathematics education, involving a genuinely engaging and empowering curriculum, can be translated into classroom practice. The paper considers the extent to which this research project was conducted in a collaborative, systematic and rigorous way. It reflects on the research processes that facilitated critical reflection, enabled the teacher researchers to overcome constraints and generated trustworthy findings. The insights gained from this analysis are used to argue that a participatory action research methodology, which resonates with a critical mathematics pedagogy, has the potential to challenge prevailing discourses in mathematics education and hence lead to genuine transformations in classroom practice.

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Introduction

Mathematics education in England has received a great deal of attention from policy-makers over the past 35 years, with numerous government-commissioned reviews of practice (e.g. Cockcroft 1982 ; Smith 2004 ; Vorderman 2011 ). However, despite large-scale government-initiated curriculum reforms and professional development programmes, including the introduction of functional skills (QCA 2007 ) and the National Strategy (DFEE 2001 ), and seemingly endless reforms of the National Curriculum (Oates 2010 ), school mathematics appears to have changed very little (Morgan 2010 ). Conventional teaching approaches persist, particularly in secondary schools (years 7–11), regardless of calls from the mathematics education community for a more engaging curriculum with greater emphasis on reasoning skills and the development of conceptual understanding (ACME 2011 ; NCETM 2008 ; OFSTED 2012 ).

In this paper, I argue that change is desperately needed in the way that mathematics is commonly taught to avoid successive generations of learners becoming alienated from the subject. I also draw attention to how school mathematics in its present form produces high levels of inequity, with a persistent and strong association between learners’ attainment and socio-economic status. I consider why conventional approaches to mathematics education research have failed to close the gaps in mathematical attainment between different socio-economic groups or bring about significant changes in classroom practice. I highlight the tendency of research to ignore the sociocultural aspects of mathematics education and the constraints and challenges faced by teachers in the mathematics classroom. Teachers are too often relegated to passive consumers of research, encouraged to implement its findings without understanding how these were generated or participating in research themselves.

Brydon-Miller and Maguire ( 2009 ) argue that participatory action research (PAR) involves collaborating with marginalised and oppressed communities and addressing the underlying causes of inequality. And yet PAR, which involves collaboration between teachers (many of whom routinely work with students from marginalised communities) and academics, and taking account of the sociopolitical forces that contribute towards maintaining the status quo, remains under-exploited as a research methodology in the field of mathematics education. Most teachers and learners of mathematics that I have come across appear to be unaware of the exploitation and social inequities that exist, and are perpetuated by school mathematics. I explore in the later sections of this paper the potential of PAR for exposing and challenging these inequities. I draw on my experiences of conducting the Teaching Mathematics for Social Justice (TMSJ) research project, a collaboration between myself and five teacher researchers, which adopted a critical model of PAR (Skovsmose and Borba 2004 ).

The findings of the TMSJ project highlight how the participatory and collaborative nature of the research design facilitated teachers’ critical reflection on existing practice. It enabled teacher researchers to overcome constraints they faced in transforming their own classroom practice and develop teaching approaches which enhanced the engagement and agency of students (Wright 2016 , 2017 ). By considering the characteristics of, and the research processes employed in, the TMSJ project, I argue that PAR has the potential to bring about transformations in classroom practice in a context in which more conventional approaches to research have had limited impact and in which change is desperately needed.

Disengagement and inequity in school mathematics

I draw attention below to high levels of disengagement amongst learners, and inequitable outcomes in terms of attainment and participation, which serve as a strong justification for why change is desperately needed in mathematics classrooms. The majority of students continue to experience an uninspiring mathematics curriculum in which learning is limited to memorising and practising mathematical procedures, with little understanding of their application, purpose or underlying concepts (Foster 2013 , OFSTED 2012 ), resulting in the quiet disaffection of a large proportion of students (Nardi and Steward 2003 ). The aspirations for mathematics education articulated in the latest National Curriculum, i.e. to provide “a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject” (DFE 2013 , p. 2), seem far from being realised. Williams and Choudhury ( 2016 ) report how transmissionist pedagogies (characterised by teacher-centred approaches) were found to accelerate a general decline in students’ dispositions towards studying mathematics over the course of their schooling and were a significant factor in students opting out of further study. Negative experiences of mathematics are not limited to students in England. Skovsmose ( 2011 ) highlights the dominance worldwide of an exercise paradigm in which teachers demonstrate a mathematical procedure to students who then complete a series of almost identical closed questions. As a result, large numbers of children and adults exhibit anxiety towards, and alienation from mathematics, which is commonly perceived by the general public as “dull, irrelevant, useless and often harmful” (Grootenboer 2013 , p. 324).

Prominent mathematics education researchers (e.g. Boaler 2009 ; Skemp 1972 ; Swan 2006 ) have called repeatedly for the adoption of alternative pedagogical approaches that promote problem-solving, discussion and collaborative learning. Most tutors that I have come across in initial teacher education (ITE) over the past 8 years (since becoming a teacher educator), in common with those who helped me qualify as a teacher in 1987, routinely encourage their student teachers to engage with such pedagogies and related research findings. Yet these practices still remain far too uncommon in mathematics classrooms.

Several researchers have drawn attention to the persistent and strong correlation between students’ socio-economic backgrounds and their participation and attainment in school mathematics, which has existed in many Western countries for over 40 years (Boaler et al. 2011 ; Jorgensen 2016 ; Noyes 2009 ). Whilst differences in achievement between other groups (e.g. boys and girls) have narrowed over the same period, social class remains the most decisive factor in determining success in school mathematics (Ernest 2016 ; Jorgensen 2016 ), which in turn regulates access to higher-status university courses and better-paid employment (Black et al. 2009 ). Thus, mathematics education “still serves as a powerful fractional distillation device that separates off different sectors of the population for different rewards” (Ernest 2016 , p. 119). Jorgensen et al. ( 2014 ) highlight how some social and cultural resources are assigned greater value and recognition by schools. These resources include the language and behaviours that tend to be acquired by children from middle-class families as a result of their upbringing, which enable them to make better use of the opportunities provided by schools from the moment they arrive (Noyes 2008 ). As a consequence, they are more likely to have positive learning experiences, identify themselves as mathematics learners and choose to study the subject beyond compulsory level (Williams and Choudhury 2016 ). In contrast, working-class children are more likely to have negative experiences of learning, see themselves as failures and disengage from mathematics (Jorgensen 2016 ).

So what can teachers concerned with equity and social justice do within such a structurally inequitable school system? Jorgensen ( 2016 ) argues that such teachers should strive to help working-class students adopt the middle-class values and behaviours recognised by the school, for example by demonstrating the relevance and application of mathematics to solving real-life problems in order to enhance their motivation and mathematical engagement. She acknowledges that, in doing so, care must be taken to avoid providing an impoverished and purely functional curriculum that fails to provide opportunities to appreciate the beauty of mathematics or the work of real mathematicians. She highlights examples of good practice where schools have raised the mathematical attainment and confidence of disadvantaged students, whilst at the same time providing a rich and stimulating learning environment. One such example is the Railside Project in a deprived area in California (USA) that successfully adopted a collaborative, problem-solving approach to teaching mathematics, resulting in higher attainment and more positive attitudes towards mathematics amongst students than in comparable schools (Boaler 2008 ).

Williams and Choudhury ( 2016 ) argue that critical mathematics pedagogy should confront transmissionist pedagogies, focussed primarily on preparing students for tests, and promote those that have greater use value , e.g. in modelling and solving real-life problems. However, Norton ( 2017 ) highlights the danger of focusing exclusively on knowledge that is immediately relevant to everyday life in an attempt to make the curriculum more authentic and engaging. This might increase students’ intrinsic motivation in the short term but prove counterproductive in the longer term by denying them access to powerful knowledge . Straehler-Pohl and Gellert ( 2013 ) describe mathematics classes in a low-streaming school in a deprived area of Berlin, in which students were routinely provided with closed tasks restricted to repeating techniques. Whilst the tasks were mainly abstract and decontextualised, they did not involve legitimising the use of techniques, thus restricting disadvantaged students’ access to powerful knowledge. Barrett ( 2017 ) stresses the vital role teachers must play in ensuring that “the relation between academic and non-academic discourse is articulated such that the possibility for the development of new meanings and understandings is realized among students” (p. 8).

Critical mathematics educators, such as Skovsmose ( 2011 ), argue that school mathematics should go beyond providing learners with the reasoning and skills they need to solve problems encountered in real life. It must also empower students to become active citizens, to lead satisfying and fulfilling lives and to contribute towards tackling some of the pressing issues currently facing society, such as growing inequality, human rights abuses and environmental sustainability (Cotton 2013 ). A truly empowering curriculum should enable learners to use mathematics to investigate and challenge injustices they experience in their own lives and wider society (Gutstein 2006 ), and involve opening up the mathematical modelling of real-life situations to scrutiny and revealing their often-invisible ethical dimensions (Yasukawa et al. 2016 ).

The problems highlighted above justify why change is desperately needed in school mathematics. I turn my attention now to consider why, despite consistent calls amongst the mathematics education research community for a more engaging and equitable school mathematics curriculum, there has been little change in practice in mathematics classrooms.

The implications for mathematics education research

There is an abundance of published research focusing on equity and social justice in mathematics education, including several edited books and special issues of journals devoted to these issues (e.g. D’Ambrosio et al. 2013 ; Gates and Jorgensen 2009 ; Strutchens et al. 2012 ). Yet school mathematics continues to play a significant role in perpetuating social inequity by alienating large numbers of students and disproportionately restricting access to higher education and better-paid jobs for disadvantaged learners. So why has mathematics education research had such limited success in challenging this situation? I argue in this section that a significant factor is the failure of most conventional research to take adequate account of the sociopolitical nature of school mathematics and the challenges and constraints facing teachers in the classroom.

Critical mathematics education challenges the myth that mathematics can be considered to be neutral and argues that agency, empowerment, critical understanding and sociopolitical engagement should all be key features of teaching and learning mathematics (Ernest and Sriraman 2016 ). It offers an alternative explanation for why conventional mathematics teaching has proved so resistant to change, regarding existing practice as indicative of the interests of those in positions of power (with privileged influence over policy decisions) in maintaining the subservience and exploitation of certain groups within society. Skovsmose ( 2011 , p. 9) argues that the exercise paradigm (see the previous section) helps prepare young people for “participating in work processes where a careful following of step by step instructions without any question is essential”. Gutstein ( 2006 , p. 10) contends that the current disempowering mathematics curriculum experienced by many reflects the economy’s need for “an ever-growing army of low-skilled, compliant, docile, pleasant, obedient service workers”. Critical mathematics education contends that research is sociopolitical in nature and claims by other researchers to objectivity and a lack of bias are merely denying the power relationships and ideologies that permeate the field (Valero 2004 ). Jorgensen ( 2016 ) argues that most mathematics education research fails to recognise social class as an issue which undermines its potential to enable all learners to enjoy the personal and economical rewards that success can bring. Williams and Choudhury ( 2016 ) warn against relying too much on studies that focus exclusively on effectiveness whilst ignoring the affective domain. They highlight how focusing on raising attainment in the short term can adversely affect dispositions towards learning mathematics, achievement and participation in the longer term.

Francis et al. ( 2017 ) highlight how the little research that does exist, which addresses the sociopolitical nature of mathematics education, is often ignored by policy-makers. They consider the high-profile case relating to ability grouping . They describe how the practice of setting students, by grouping together those with similar levels of prior attainment in homogenous teaching groups, has grown significantly in recent years, with 71% of secondary students in England set in mathematics. This is despite most research evidence suggesting that setting has no significant benefits on overall levels of attainment and a negative impact on outcomes for students in lower sets. Students are often assigned to sets on the basis of their behaviour, rather than their attainment, with limited opportunities for subsequent movement between groups (Wilkinson and Penney 2014 ). Those placed in lower sets commonly experience a “largely remedial (and boring) curriculum” (Hodgen and Marks 2009 , p. 31). Students from poorer backgrounds are invariably overrepresented in lower sets, where they are less likely to be taught by well-qualified teachers or to experience deep learning focused on conceptual understanding, thus further consolidating their disadvantage (Jorgensen 2016 ).

Francis et al. ( 2017 ) attribute the lack of impact of this research to a dominant discourse , i.e. a way of constituting knowledge that comes to be viewed by society as representing the “ natural order [which] needs no external evidence to support it” (p. 9). In this case, the dominant discourse views inequalities in outcomes as an inevitable consequence of innate differences in ability , and segregation is seen as a natural way of dealing with these differences. They argue that this discourse arises from “long-standing narratives in English culture” (p. 8) associated historically with a school system in which (selective) grammar schools and (fee-paying) independent schools exist alongside government-maintained comprehensive schools. This explains why schools often feel compelled to introduce setting to attract middle-class parents, seen as keen to secure advantages for their own children, in order to boost examination results.

Given the need to challenge such deeply rooted discourses, mathematics education research must pay careful attention to teachers’ perspectives, their classroom situations and the constraints they face (Bishop 1998 ). Much conventional research, however, is conducted in prototypical classroom situations which lack the typical challenges encountered by most teachers on a daily basis (Skovsmose 2011 ). Findings from such studies are often packaged and disseminated to schools for implementation. However, since they ignore institutional contexts and constraints, they have minimal success in challenging existing classroom practice. The blame for the continued lack of evidence - informed practice is then either placed on teachers for being reluctant to engage with the research (Oakley 2006 ; Sebba 2004 ) or on the poor quality of the research itself (Gough 2004 ). Watson et al. ( 2013 ) argue that, despite an abundance of research into effective learning of mathematics, this research is not made readily accessible to teachers. Sebba ( 2004 ) contends that teachers have such deeply entrenched beliefs and values relating to learning mathematics that they resist all attempts to transform their practice; hence, change is only possible through adopting more effective ways of managing policy implementation and imposing tighter control on teachers.

Whilst a minority of teachers are involved in carrying out research, the majority are expected to implement, unquestioningly, recommendations for changes in practice arising from (carefully selected) research which they know very little about. These expectations are often presented as targets associated with performance management procedures in schools, resulting in teachers distrusting new research findings, seen as promoting a political agenda or as tacit monitoring (Hammersley 2004 ; Thomas 2004 ). Given the need for research which challenges current orthodoxies, teachers need to develop their capacity to reflect critically on existing practice in relation to research evidence, rather than relying on adopting what works protocols (Winch et al. 2013 ). Cordingley ( 2013 ) argues that, by engaging in collaborative enquiry facilitated by external experts, teachers are encouraged to take risks and explore why, and in which contexts, certain practices are effective. Leat et al. ( 2014 ) distinguish between engaging with research, as a body of knowledge , and engaging in research, as a social practice . They argue that teachers who engage successfully both in and with research are more likely to generate new insights and perspectives, affect powerful changes in their practice and become more critical of existing policies and practice.

Collaborative research involving teachers and researchers

There has been a resurgence of interest recently in close - to - practice research involving teachers and researchers working together to address problems in practice. It is widely acknowledged that such research, due to significant involvement of teachers, has the potential to impact on professional learning and to challenge established practice (Myhill 2015 ). However, the biggest challenge facing close-to-practice research is the common perception that it lacks rigour and is limited in scale, prompting BERA ( 2017 ) to commission a review into its quality. Robutti et al. ( 2016 ) report that most practitioner-led collaborative research neglects to theorise collaboration and relies instead on theories relating to a community in which it is assumed to take place. One collaborative research methodology that has recently proliferated worldwide is lesson study , its popularity attributable to its widespread use over many years for professional development of teachers in high-performing jurisdictions in East Asia (Takahashi and McDougal 2016 ). Lesson study models incorporate characteristics that can facilitate effective professional learning, i.e. “it is sustained over substantial periods of time, collaborative within mathematics departments/teams, informed by outside expertise, evidence-based, research-informed, and attentive to the development of the mathematics” (Wake et al. 2016 , p. 245). However, these models appear to have developed organically, through the collaborative practice of teachers, and only recently become a focus for study by academic researchers (Takahashi and McDougal 2016 ).

Unfortunately, much collaborative research fails to recognise the sociopolitical nature of school mathematics and lacks the critical element necessary for challenging existing practices and dominant discourses. Lesson study normally involves advance agreement of the intended outcomes which is used to determine the research questions. Japanese lesson study, for example, is associated with the implementation of a specific problem-solving pedagogy around which there is already shared consensus amongst mathematics educators (Takahashi and McDougal 2016 ). Whilst action research is not uncommon in schools, much of it is either technical , i.e. it aims to improve practice by achieving pre-defined outcomes, or practical , i.e. the focus is left open for practitioners to decide but the legitimacy of existing modes of practice is not questioned (Kemmis 2009 ). Critical reflection requires that teachers view their own practice as problematic and question the consequences of their actions in relation to wider historical, cultural and political values and beliefs (Hatton and Smith 1995 ; Liu 2015 ). External stimulus is regarded as essential for promoting critical reflection in schools since, without this, collaborative enquiry is likely to merely perpetuate existing practice through the process of alignment with accepted norms (Jaworski 2006 ). Action research only becomes critical when it involves partners working together to “change their social world collectively, by thinking about it differently, acting differently, and relating to one another differently” (Kemmis 2009 , p. 471).

In the next section, I propose that participatory action research (PAR) offers the potential to bring about changes in practice where more conventional research has failed to do so. I argue that PAR presupposes a genuine partnership between teachers and researchers, recognises the sociopolitical nature of research and can be conducted in a rigorous and systematic way that promotes critical reflection on existing practice and facilitates transformations in classroom practice.

• Participatory action research

Participatory action research (PAR) is a collaborative approach to research in which researchers aim to carry out research with practitioners (as research partners) rather than on practitioners (as research objects). In the context of schooling, it recognises how academic researchers , with their expertise in conducting research, and teacher researchers , with their in-depth knowledge of the classroom situation, each have a distinct, but essential, role to play (Atweh 2004 ). It aims to seek positive social change through generating knowledge that is of greater relevance to practitioners, whilst developing a deeper understanding of theory - in - practice amongst teachers (Brydon-Miller et al. 2003 ). It pays closer attention to teachers’ perspectives, the challenges and constraints they face and the opportunities they are afforded on a day-to-day basis. PAR is overtly emancipatory and rejects the notion that research can be objective and value-free, which distinguishes it from other forms of action research. Brydon-Miller and Maguire ( 2009 , p. 80) describe PAR as “… a systematic approach to personal, organizational, and structural transformation, and an intentionally and transparently political endeavour that places human self-determination, the development of critical consciousness, and positive social change as central goals of social science research”.

PAR methods have most commonly been applied to the fields of health (e.g. nursing and health promotion) and technology (e.g. agriculture and environment) and less so to education, where it is usually limited to informal education contexts and adult education (Thiollent 2011 ). With notable exceptions (including two described below), PAR studies in mathematics education appear rarely in research publications. Raygoza ( 2016 ) reports on a project in which she taught mathematics to students, who had recently experienced failure in the subject, in an urban school in Los Angeles (US) with high levels of social deprivation. She describes how her students explored social justice issues around food by conducting a school-wide survey and using the results to generate an argument for change. Her project was participatory in the sense that the students themselves took a lead in conducting the research, whilst developing their own critique of injustices relating to their situations. Andersson and Valero ( 2016 ) report on a project in an upper secondary school in Sweden, in which the authors (teacher and researcher, respectively) challenged traditional pedagogies and attitudes towards mathematics through the introduction of classroom projects involving societal issues. They demonstrated how changes in pedagogical discourses are possible on a school-wide basis. Both of these PAR studies were limited in scale, with the former restricted to a collaboration between teacher and students within one class and the latter located within a single school. Andersson and Valero ( 2016 ) adopted the critical research model of PAR, which is the same model that I chose to adopt for the TMSJ project and is described below.

Skovsmose and Borba ( 2004 ) offer a critical research model of PAR, resonating with critical mathematics education, built on the premise that the existing situation must be challenged by addressing “possibilities that can be imagined and alternatives that can be realised” (p. 214). They theorise how this might be accomplished through three key processes (Fig.  1 ) that are integral to their research design. Pedagogical imagination (PI) is the process of developing a critical understanding of the current situation (CS) and articulating an alternative vision, i.e. imagined situation (IS), by drawing on previous research findings, theories and teachers’ practical knowledge. Practical organisation (PO) involves cooperation between teachers, researchers, students and others in organising an arranged situation (AS), i.e. trying out ideas from the imagined situation, taking into account the realities and constraints of the current situation. Explorative reasoning (ER) involves analysing the outcomes from the arranged situation in order to better understand the current situation and to draw conclusions about the feasibility of the imagined situation.

Model of critical research (Skovsmose and Borba ( 2004 , p. 216)

This critical research model, through making explicit the key research processes, offers a systematic form of PAR that has the potential to challenge existing practice and prevailing discourses in mathematics education through its focus on the critical reflection of teachers. I report below on how the model was developed and applied in the design of the TMSJ project that aimed to develop alternative teaching approaches that engage and empower all learners.

The Teaching Mathematics for Social Justice (TMSJ) research project

I report below on the case of one participatory action research project, which demonstrates the potential of the methodological approach for bringing about transformations in classroom practice. In June 2013, as part of my doctoral studies, I established a research group together with five teacher researchers, Anna, Brian, George, Rebecca and Sarah (all pseudonyms). All five had previously completed an ITE programme (on which I was a tutor), were nearing the end of their first year as newly qualified secondary mathematics teachers and had expressed an interest in addressing issues of social justice in their classrooms. They all taught in multi-ethnic comprehensive schools in London, which shared many characteristics including having relatively high proportions of students who were eligible for free school meals, who spoke English as an additional language and who had statements of special educational needs.

Whilst there is an abundance of research literature around social justice in mathematics education, most of this is theoretical in nature and there are relatively few studies that focus on how social justice can be realised in the classroom (Duncan-Andrade and Morrell 2008 ; Wright 2015 ). The project therefore focused on the following research question: How can a commitment towards “education for social justice” amongst mathematics teachers be translated into pedagogy and classroom practices which promote such aims? A second aim of the research project was to explore the potential of a participatory action research (PAR) methodology for achieving this.

The research design was based upon Skovsmose and Borba’s ( 2004 ) critical research model (see the previous section) and was collaborative and participatory in nature. The current situation in this case was represented by conventional teaching approaches and dominant discourses in mathematics education. The imagined situation was based on the following conceptualisation of teaching mathematics for social justice (Wright 2015 , p. 27), which draws on the ideas of critical mathematics educators (see the “Disengagement and inequity in school mathematics” section), particularly Skovsmose ( 2011 ) and Gutstein ( 2006 ):

Employ collaborative, discursive, problem-solving and problem-posing pedagogies which promote the engagement of learners with mathematics;

Recognise and draw upon learners’ real-life experiences in order to emphasise the cultural relevance of mathematics;

Promote mathematical enquiries that enable learners to develop greater understanding of their social, cultural, political and economic situations;

Facilitate mathematical investigations that develop learners’ agency, enabling them to take part in social action and realise their foregrounds;

Develop a critical understanding of the nature of mathematics and its position and status within education and society (Wright 2015 ).

The arranged situation involved a series of teaching ideas and activities that the teacher researchers tried out in their classrooms as part of three action research cycles spread over the course of one academic year. In order to demonstrate that PAR can be conducted in a systematic and rigorous manner, particular attention was paid in the research design to the three key processes of the critical research model ( pedagogical imagination , practical organisation and explorative reasoning ) described below. The three key processes were applied in the collaborative spirit of PAR, with teacher researchers’ in-depth knowledge of the classroom being drawn upon in developing and trying out ideas, and my research expertise utilised to facilitate discussion around research literature, methods and analysis.

Pedagogical imagination in this case involved developing a critical understanding of existing practice and articulating what teaching mathematics for social justice might look like in the classroom. The first meeting of the research group focused on engaging with critical perspectives. Prior to the meeting, I asked teacher researchers to read the introductory chapter from a book about rethinking school mathematics from a social justice perspective (Gutstein and Peterson 2005 ). The reading was used to facilitate a discussion on how the ideas related to teacher researchers’ own experiences and their viability as alternatives to existing practice. In subsequent meetings, teacher researchers read, and presented for discussion, other relevant publications that I identified from the research literature.

Practical organisation involved cooperation between members of the research group in designing and developing activities and teaching ideas to try out in the classroom. This was the main focus for the second, fourth and sixth meetings of the research group. Teacher researchers were asked to present ideas from existing resources (e.g. Gutstein and Peterson 2005 ; Wright 2004 ) and discuss how these might be adapted for use with students, taking into account the constraints of the classroom. The group also discussed how to go about collecting evidence to help evaluate the success of the activities, deciding to employ student surveys as a key tool for doing so. This survey was administered by teacher researchers with their own classes immediately after each activity was tried out. The survey was trialled by teacher researchers during the first cycle, after which the wording was amended and a protocol agreed for how it should be introduced to students in subsequent cycles. The survey aimed to distinguish between students’ general dispositions towards learning mathematics and how they felt about the activities tried out through the project. The following two questions were posed: How do you feel about maths in general? What do you think about the maths we did today?

Explorative reasoning involved evaluating and reflecting on the activities tried out in the classroom in order to better understand existing practice and to discuss and develop further ideas. This was the main focus for the third, fifth and seventh meetings of the research group. Teacher researchers took the lead in reflecting on their own teaching, taking it in turn to present their evaluations and to invite questions from other members of the group. They made use of students’ feedback from the surveys, along with examples of students’ work and notes made in research journals (which were used by the teacher researchers and me to record thoughts and reflections during the project). Presentations were followed by a general discussion to inform the planning of future activities.

Reliability of research findings

Lincoln and Guba’s ( 2003 ) framework for ensuring trustworthiness of qualitative research findings was applied to the critical research model in order to further enhance the rigour of its design and the reliability of research findings. I describe below how the four aspects of the framework, i.e. credibility , confirmability , transferability and dependability , were adapted to the PAR research design in the case of the TMSJ research project.

Credibility involves ensuring the phenomena being observed are accurately represented. This was addressed through the prolonged engagement of teacher researchers over one academic year, iterative questioning during interviews which followed up on previous responses, reviewing initial findings from data analysis in subsequent research group meetings (similar to member checks in other forms of qualitative research) and comparing responses from student surveys, meetings, interviews and final reports to generate richer meaning (related to triangulating data) (Shenton 2004 ).

Confirmability means ensuring the findings are derived from the experiences of the researchers rather than any preconceived ideas and beliefs. This was addressed through focusing on reflexivity, e.g. by maintaining research journals. The transferability and dependability of the research, which enable readers of the research to judge the extent to which the findings are relevant to their own situations and to repeat the study if so desired, were established by providing thick descriptions of the context and design of the research (Shenton 2004 ). Such detailed descriptions, whilst too lengthy to include in this paper, can be found in my doctoral thesis (Wright 2015 ).

Data collection and analysis

Data from the research project were generated from audio recordings of meetings and a series of three individual semi-structured interviews, which I conducted with each teacher researcher. I adopted an empathetic approach to interviewing by establishing relationships of trust that enabled stories to be jointly constructed through dialogue (Fontana and Frey 2008 ). Interview questions were designed to prompt teacher researchers to reflect on the development of their thinking and practice over the course of the project. Hence, the initial and final interviews included such questions as: What does teaching mathematics for social justice mean to you? How do you think social justice relates to your current classroom practice? How do you think your classroom practice has changed, since the beginning of the project, in relation to social justice? Given the focus of the project on exploring the potential of PAR to transform classroom practice, questions were also focused on the plan–teach–evaluate cycles, e.g. Tell me a bit more about the first classroom activity that you tried. How will you approach the second classroom activity? Individually tailored follow-up questions were used to explore responses in greater detail, e.g. How did the classroom activity relate to “teaching mathematics for social justice”? How did the students respond?

At the end of the project, teacher researchers were invited to write a short report on their experiences of participating in the project, and the impact it had on their thinking, classroom practice and their students. The student surveys (see previous section) and the final reports were used as supplementary data. They did not form part of the main data analysis (described below) but were used to compare with its findings to generate richer meaning.

The audio recordings were transcribed and two separate thematic analyses were carried out on the transcripts using meaning condensation and meaning interpretation (Kvale and Brinkmann 2009 ). The first analysis focused on the development of teacher researchers’ thinking and classroom practice, whilst the second focused on the characteristics and processes of the critical research model of PAR. For each analysis, the text was broken down into initial units of meaning, based on statements from participants, and the meanings were summarised with descriptive text.

For the first thematic analysis, a category was then assigned to each unit of meaning using inductive coding , whereby the categories are derived from an initial reading of the data. Categories included those relating to views teachers expressed about school mathematics (e.g. “consideration of nature of school maths” and “seeing maths as a legitimate area for issues of social justice”) and teachers’ perceptions of themselves as professionals (e.g. “reasons for becoming a school teacher” and “interest in personal and professional development”). During the coding process, initial units of meaning were broken down further, so that only one category was assigned to each unit. Each unit was also assigned a property, giving more detail about its meaning, and a score from 1 to 5, giving an indication of the extent to which the unit was seen as affirming that property. To further illustrate this process, I provide an example below from the analysis of an interview transcript.

The following statement (by Rebecca) was identified as a unit of meaning: I think that’s what [Brian] was saying wasn’t it? It’s quite easy to define injustice, than what social justice … I don’t really know what social justice means. The meaning of this unit was summarised as: Easier to define social justice by defining injustice. Little thought given to TMSJ before project. This unit was assigned the code “PreEng T2”, “PreEng” representing the category “Previous engagement with teaching mathematics for social justice issues”, “T” representing the property “Large amount of thought given”, and “2” indicating a mostly negative assertion of this property, i.e. Rebecca had given relatively little thought to such issues previously.

The coding was then used to compare commonalities, differences and relationships between units of meaning, by grouping together units with similar codes and re-reading the data (whilst taking account of the context of the discussions in which they were situated), enabling themes to emerge (Gibson and Brown 2009 ). Such comparisons allowed meaning to be constructed from the teacher researchers’ stories, whilst avoiding loss of meaning through simplistic quantifying of codes. Initial findings from the thematic analysis were related back to the underlying theories in order to generate new analytical questions and give further meaning to the data (Jackson and Mazzei 2012 ). The findings from this analysis, which demonstrate how teachers were able to overcome constraints and transform their classroom practice, are presented in the next section.

The second thematic analysis of the data was carried out using categories derived deductively from the key research processes and characteristics associated with the critical model of PAR (described earlier). The purpose of this analysis was to consider the extent to which the TMSJ project was conducted in a collaborative, systematic and rigorous manner and to inform how the critical model of PAR might be applied to future research that seeks to challenge dominant discourses in mathematics education and transform classroom practice. The findings from this second analysis are presented in the penultimate section. Note that in the next two sections, I have focused on the findings that I consider most relevant to the argument developed in this paper. Further reports of the research findings can be found elsewhere (e.g. Wright 2016 , 2017 ).

Findings of the TMSJ project

Two themes emerged from the first data analysis which highlight the potential of the PAR methodological approach adopted for the TMSJ project for promoting critical reflection and bringing about changes in classroom practice.

Theme 1: developing student agency

All five teacher researchers reported a significant increase in students’ engagement with mathematics, associated with the teaching approaches and activities developed through the research project. They witnessed unusually high levels of enjoyment of, and interest in, learning about mathematics and social justice issues when these were tackled in lessons. The feedback from student surveys concurred with these observations:

In general I do not enjoy maths as I think I’m not very good at it. Today I enjoyed the maths lesson as I enjoyed finding out about fair trade and I liked seeing all the statistics of the money different people make from a bar of chocolate. (Rebecca’s Year 9 Student, survey response to Fair Trade activity).

Across all classes, a majority of students expressed greater enjoyment of mathematics lessons involving social justice issues. Many students, when asked about mathematics in general, described it as boring and irrelevant. They noticed a difference in the project activities and claimed these helped them to see more clearly applications of mathematics to real life: “I liked what we did today because it was something totally different. We learn more about the world like this, while using maths”. (Brian’s Year 8 student, survey response to Election activity). Increases in engagement were particularly noticeable amongst low-attaining students and those previously poorly motivated or badly behaved in lessons. One noteworthy example was a Year 8 girl in Anna’s bottom set who had previously exhibited such challenging behaviour across all subjects that she was in her last week before being moved to a special school. Despite this, her behaviour during a project on wealth distribution, in which students discussed in groups the fairest way to share out wages between workers in different jobs, was exemplary: “in terms of her enjoyment of the project … she was asking so many questions, she was putting forward so many views, she was working in a team. She was just like a dream child for the whole project” (Anna, Meeting 3).

The project highlighted the cases of a number of lower-attaining and previously disengaged students who demonstrated the most noticeable increases in motivation towards learning mathematics and positive responses towards the alternative practices adopted by the teacher researchers. However, the limited scale of the project meant that the long-term effect on these students’ attainment was not examined and increased engagement does not necessarily translate into higher attainment. As Norton ( 2017 ) points out, promoting authentic and engaging curricula, and focusing on everyday knowledge, can instil in students an expectation that mathematics should be fun and immediately relevant to real life, with an associated risk of discouraging them from engaging with more powerful mathematical knowledge. This was reflected in initial concerns amongst some students that, by engaging in social justice issues, they were not doing real mathematics: “It was fun. The presenting was fun and enjoyable. It was okay, but it wasn’t really relevant to maths” (Anna’s Year 9 student, survey response to Making a Change Project). Barrett ( 2017 ) uses the same argument to justify the need to make the links between academic and non-academic discourses clearer to students, the above example indicating that in the early stages of the project these links were not made clear enough. However, higher levels of engagement were accompanied by increasing agency amongst students, suggesting that the risk of focusing exclusively on everyday knowledge was minimised.

Developing students’ agency became the main focus for the teacher researchers in the third action research cycle, when they tried the Making a Change project with their classes. This project, inspired by Rebecca’s experiences of designing a similar activity in the first cycle, involved groups of students choosing an issue of interest to them, using mathematics to develop their understanding of the issue, and presenting an argument for a change they would like to see made. Students particularly welcomed the opportunities to explore links between mathematics and issues of interest to them, and to present their arguments: “They were all so passionate about the things they were presenting about, was the key thing, and the fact that they got to actually tell everyone what they found out” (Anna, Interview 3).Teacher researchers attributed the exceptionally high levels of engagement they witnessed with this activity to the autonomy granted to students in choosing their own issues and making their own decisions about how to research these: “I liked the presentation as I got to do something that I felt strongly about. It gave me a chance to express how I feel, also including maths to support my presentation” (Rebecca’s Year 9 student, survey response to Making a Change project). The Making a Change project concluded with students evaluating how effectively they used mathematics to support their argument, exemplifying the legitimisation of mathematical techniques presented by Straehler-Pohl and Gellert ( 2013 ) as an essential dimension of powerful knowledge.

The idea of developing mathematical agency originated from Rebecca’s engagement with the research literature (Gutstein and Peterson 2005 ) and appeared to be of growing interest to teacher researchers: “I think the agency thing was definitely something I hadn’t considered at the start. Like, I saw it more as applying maths to different situations, rather than using maths to actually change something” (Rebecca, Interview 3). The evaluations of lessons suggest it was also a new way of working (in mathematics) for most students. The success of the project activities in drawing links between social justice issues and mathematics demonstrated how students were able to generate new meaning by relating academic and non-academic discourse (Straehler-Pohl and Gellert 2013 ): “Today shows us about the unfairness farmers get and how we can help them by using fair trade” (Anna’s Year 7 student, survey response to Fair Trade activity). Concurrent increases in students’ mathematical engagement and agency over the course of the project demonstrate how students were able to gain an appreciation of the use value of mathematics without being denied access to powerful knowledge.

Theme 2: dominant discourses on ability and attainment

The teacher researchers recounted how the research project prompted them to reflect critically on their own classroom practice and to recognise dominant discourses within mathematics teaching. All five acknowledged an initial reluctance to try out new ideas and teaching approaches with students in bottom sets through concerns that they would respond negatively:

I know the way that I teach classes that are badly behaved is so structured, to make up for the fact that they can’t be left to their own devices for five minutes. … That kind of approach doesn’t really lend itself necessarily to an extended open activity, where they actually get to think more deeply about the things that are involved. (Rebecca, Interview 3)

Through recognising these tendencies, teacher researchers began to challenge their own preconceptions and to appreciate the significant benefits of alternative teaching approaches for these students: “I tried a few things with my bottom set and their motivation has just been so high in those particular lessons that I’ve had to very rarely like tell them to get on with things or to do things” (Anna, Interview 3). They began to question their prior assumptions about mathematical ability. Reviewing setting was not an explicit focus for the research project, since the teacher researchers were not in positions to influence policy on grouping students within their schools. Despite this, they voiced growing criticism of setting and increasingly questioned its validity and benefits, thus challenging discourses around ability on which setting is predicated (Francis et al. 2017 ): “I’d like to bring in Year 7’s, unsetted [mixed attainment groups], at some point, once I’d gained my school’s trust. This is if I even get a head of department job, but that’s the long-term goal” (Anna, Interview 2).

Another significant shift in thinking amongst teacher researchers was from viewing education as a meritocracy towards recognising structural causes of inequity. All five had chosen an ITE programme that deliberately placed them in schools in relatively deprived areas of London, on the premise that inequalities in educational outcomes can be addressed by focusing on raising the attainment of disadvantaged students:

I’ve chosen to teach in a school where it’s classed as a challenging school, because the kids stereotypically wouldn’t be expected to achieve very much. … So I think, in the sense of bringing about social justice through education, I’m involved in that just through being at this school. (Anna, Interview 1).

In accepting this discourse, equity-oriented teachers are compelled to work as hard as possible to enable some students from poorer backgrounds to achieve higher grades and hence realise higher aspirations, e.g. in gaining access to more prestigious universities. Bourdieu and Passeron ( 1990 ) argue that every such success merely gives credibility to the myth that the school system represents a true meritocracy. Engaging with research literature, and relating this to their own practice, encouraged teacher researchers to question previous assumptions: “People say ‘Well, we’ve got a good education system, you know, we live in a country where you can get wherever you want’. Well actually, people can’t, because of the barriers” (Anna, Interview 2). Brian began to articulate how, in order to be successful in mathematics, all students need to develop the personal and social skills necessary to take advantage of opportunities to learn. This resonates with Jorgensen’s ( 2016 ) contention that teachers should strive to enable working-class children to realign their values and behaviours with those recognised by the school.

Whilst observing the benefits of alternative pedagogies employed through the research project, the teacher researchers became increasingly aware of the constraints they faced in developing these pedagogies, through their schools’ excessive focus on high-stakes examination and monitoring of teachers. They described how the tendency of managers to carry out brief unannounced visits to classrooms (often referred to as learning walks ) resulted in pressure to make it appear as though students are working hard and making progress at all times:

I think it makes you less likely to take risks with your classes. If you know that there’s a chance that someone pops in, you’re more likely to do lots of very average lessons, than one lesson that could blow up in your face or it could go amazingly, because you know that you’d be judged on that one lesson. (Brian, Interview 1)

This might help to explain the prevalence of transmissionist teaching approaches, given the pressures on teachers to demonstrate students’ short-term gains in attainment rather than focusing on longer-term improvements in students’ dispositions towards learning mathematics (Williams and Choudhury 2016 ).

The teacher researchers observed that some higher-attaining students showed the least enthusiasm for alternative teaching approaches, perhaps reflecting that they felt comfortable with the status quo and the personal success they associated with it:

I think, if you are at the top end of the top set, you’ve put your hat on the fact that you get things right, and as soon as in maths it’s no longer about you getting the right numerical answer, you suddenly feel like things are not under your control any more, and you’re not top dog any more. (Brian, Interview 2).

Boaler ( 2009 ) highlights, however, how success is not necessarily accompanied by positive dispositions towards learning mathematics, with many high-attaining students failing to see its relevance to their future lives and choosing not to study mathematics beyond the compulsory stage. Thus, assigning greater emphasis to the use value of mathematics means that all students, not just the low-attaining, have something to gain.

One of the most interesting findings was that, despite its limited scale, the project generated a substantial amount of interest from other teachers not directly involved. The teacher researchers described how news of the positive impact of the activities on their students’ engagement and achievement spread quickly across their departments causing a rapid growth in interest from colleagues. George described this as the multiplier effect : “Success has bred more success, because if they’ve seen a lesson go well, then they want to teach it, and then their lesson goes well, and then it sort of spreads” (Rebecca, Interview 3). In response to demands for more information, all five teacher researchers disseminated ideas to colleagues who were keen to try these out themselves. Anna, Brian and Rebecca were invited to run training sessions for their departments and three schools adopted activities from the project for use with a whole year group. This suggests the existence of a wider group of teachers receptive to the aims of the research project beyond those volunteering themselves as teacher researchers.

How can PAR bring about transformations in classroom practice?

The findings of the TMSJ project presented in the previous section demonstrate how one research project, which adopted the critical research model of PAR, brought about significant changes in teacher researchers’ thinking and classroom practice and provided their students with a more engaging and empowering mathematical learning experience. In this section, I draw on the second thematic analysis (referred to earlier) to examine how the above was achieved through the application of the critical research model (Skovsmose and Borba 2004 ). I highlight the characteristics of PAR, in particular the three key processes and strategies for ensuring reliability (described earlier), that demonstrate its potential for bringing about transformations in classroom practice in a context in which more conventional approaches to research have had limited impact.

Pedagogical imagination

The TMSJ project was participatory in the sense that the teacher researchers played a leading role in deciding how the conceptualisation of teaching mathematics for social justice (described earlier) should be translated into practice through activities and teaching approaches to be tried out and implemented in the classroom. My role in the research group was largely facilitative, i.e. organising and publicising meetings, inviting suggestions for the agendas and chairing discussions, summarising and clarifying decisions taken by the group, writing up and circulating notes from the meetings. The teacher researchers acknowledged the pivotal role that I played in encouraging them to engage with theoretical ideas and research literature, which I selected as relevant to the aims of the project, by presenting papers to each other for discussion. This reinforced the importance of external stimulus in promoting critical reflection whilst conducting collaborative enquiry (Jaworski 2006 ). At the same time, teacher researchers appreciated the opportunities to take decisions themselves relating to the research design and articulated a growing interest in, and understanding of, research processes: “This is my only experience of any kind of research … I have learnt an awful lot about the process, as opposed to just what we’re researching” (Rebecca, Meeting 4).

Whilst the adoption of this pre-existing conceptualisation might be seen as contradicting the participatory principles of PAR (by deciding in advance the focus for the research), it was effective in promoting critical reflection, with teacher researchers describing how it helped them to recognise existing practice as problematic by relating it to the research literature (Hatton and Smith 1995 ; Liu 2015 ). This illustrates a careful balance that needs to be established within PAR between the academic researcher acting as an external stimulus, promoting a critical understanding of current practice, and the agency of teacher researchers in playing an integral role in the design and development of the research project.

Practical organisation

The teacher researchers emphasised how the mutual support provided by the group enabled them to overcome constraints and gave them the confidence to take risks in developing their own practice. A striking example of this was the way in which the group rallied around Rebecca when she became disheartened by frustrations she experienced during early trials of the Making a Change project: “It is quite useful having that kind of, I don’t know, support almost and being able to just tell someone exactly what happened and have their, kind of, outside view on it” (Rebecca, Interview 2). The reassurance, encouragement and constructive feedback she received led to her (and others) recognising the potential benefits of the activity, which subsequently became the main focus for the third action research cycle. Holding regular group meetings (seven over the course of one year) also provided the impetus required to maintain the momentum of the project:

And it’s also provided that additional incentive to do it, and to take the risk, because you know that you’re going to be asked to talk about it. But also you know you’re going to be allowed to talk about it in a way that says that messing up doesn’t matter. (Brian, Interview 3).

The teacher researchers took much of the initiative for organising the action research cycles, choosing which activities to try out in the classroom and evaluating the success of these activities during research group meetings.

Explorative reasoning

The project highlighted the importance of establishing a genuinely collaborative approach between academic researcher and teacher researchers. This necessitated the teacher researchers and I recognising and respecting each other’s expertise and appreciating how each other’s contributions strengthen the robustness of the research design (Atweh 2004 ). The existence of such a collaborative relationship was apparent in the discussions accompanying the development of the student surveys. My familiarity with research methods enabled me to present options for how feedback might be collected from students, whilst the teacher researchers’ detailed knowledge of the classroom context informed the joint decision taken to employ student surveys. Similarly, the formulating of questions and the development of a protocol for administering the surveys drew heavily on the range of expertise available within the research group.

The five teacher researchers were drawn from four different schools. They welcomed the opportunity to discuss ideas, share experiences and to collaboratively plan a series of activities with teachers from outside their own schools, highlighting how this exposed them to a wider range of perspectives and allowed them to be more open in evaluating their own teaching:

Through sharing resources when we all have a clear focus, and we’ve all agreed on what we’re trying to achieve, you cut through all the rubbish, really. And the discussions give you fresh ideas you might not have thought about. (Anna, Interview 3).

Potential problems with the credibility of the findings were considered by the research group and strategies agreed to mitigate against these. The possibility of bias arising in the student surveys, due to students’ responses being influenced by what they believed teachers wanted to hear, was minimised by emphasising to students (through the survey protocol) the anonymity of the survey and that the results would be used solely for the purpose of the research project. A similar bias was possible in the data generated from research group meetings and interviews which were based primarily on teacher researchers self-reporting on their perceptions of students’ engagement and achievement. However, the responses from the student surveys and the final reports written by teacher researchers appeared to resonate closely with the initial findings from the thematic analyses. The responses from the teacher researchers when reviewing these findings suggested they considered these to accurately represent the accounts given during meetings and interviews. Indeed, George described how hearing the findings presented back helped him to make more sense of the story he had narrated. This suggests that the strategies described above along with others described earlier (including prolonged engagement and iterative questioning ) ensured the credibility of the findings.

What has emerged clearly from these findings is the crucial role of the academic researcher within PAR in ensuring confirmability , i.e. that the research findings are derived from teacher researchers’ experiences. Reflexivity appears to be vital for the academic researcher in making effective use of her/his research knowledge and expertise to inform decision making, whilst facilitating teacher researchers’ agency in conducting the research. One way in which this was achieved in the TMSJ project was through me introducing Skovsmose and Borba’s ( 2004 ) critical research model of PAR and Lincoln and Guba’s ( 2003 ) framework for ensuring trustworthiness , and prompting discussions around how these could be used by the group to inform the research design and to conduct the research in a systematic and rigorous way. This requires a high degree of transparency in sharing, and encouraging teacher researchers to engage with, the methodology underlying the research.

It should be acknowledged that there were aspects of the research for which I took more direct responsibility, e.g. collecting and analysing data (from the research group meetings and interviews) and disseminating the findings from the TMSJ project. Whilst the method for analysing data was discussed at length in research group meetings, given the time constraints on teacher researchers, it was me that carried out the coding. I also took responsibility for conducting the interviews. Given the participatory nature of the research group meetings and the empathetic approach to interviewing, which both generated the data, I consider this to be in line with the principles of PAR. However, given the extent to which teacher researchers engaged with the research processes, involving them more in conducting interviews and coding data is something I would be keen to explore in future studies. Another aspect that might be explored in greater depth is the extent to which teacher researchers are able to develop and maintain changes in practice beyond the duration of the project and how these relate to the organisational constraints they encounter in schools (which is beyond the scope of the TMSJ project).

It is important that all researchers should learn from their involvement in PAR. Whilst the project demonstrated how teacher researchers benefited through developing a deeper understanding of theory in practice (Brydon-Miller et al. 2003 ), conducting this analysis has also enabled me to progress my own understanding of how theory relating to PAR relates to my own professional context as a teacher educator and academic researcher. This paper is an attempt to share the insight I have gained from the project into how PAR can be conducted in a way that promotes critical reflection and transforms classroom practice.

Conventional approaches to teaching, based on transmissionist pedagogies and adhering to an exercise paradigm (Skovsmose 2011 ), are well established in mathematics classrooms. They have proved resistant to change for many years, protected by powerful discourses that present inequities in school and society as inevitable. Mathematics education research, which too often fails to take account of the sociopolitical nature of school mathematics and constraints teachers face in the classroom, has had limited success in challenging these pedagogies and effecting changes in practice. Some pockets of innovative practice exist which challenge these discourses; however, the challenge for those seeking change is formidable.

Research that aims to challenge social inequity through transforming classroom practice must take account of the structural causes of inequity as well as the constraints faced by teachers who wish to develop more empowering pedagogies. Participatory action research (PAR), in which teachers and researchers work collaboratively in order to challenge the current situation, offers a methodology that has the potential to achieve both of these objectives. It satisfies commonly accepted criteria for the effective professional learning of mathematics teachers (Geiger et al. 2016 ), i.e. it involves engagement over a long period, is carried out within teachers’ own working environments and involves teachers reflecting critically on their practice. By exploring new pedagogies within the context of teacher researchers’ own classrooms, it can generate findings that are more relevant and applicable to other classroom situations and that take account of the constraints teachers face.

The Teaching Mathematics for Social Justice (TMSJ) research project demonstrated how teachers can challenge dominant pedagogical discourses in order to develop alternative teaching approaches that engage a wider range of students, whilst enhancing their mathematical agency and enabling them to develop powerful forms of mathematics knowledge. By reflecting on how a critical model of PAR was applied in the design of the TMSJ project, I have demonstrated how PAR can be conducted in a systematic and rigorous way in order to generate reliable and trustworthy findings. I have shown how teacher researchers can play a leading role in the design of classroom trials and in the development of research tools to evaluate their success. I have highlighted how the input from an academic researcher can provide the external stimulus necessary for teachers to challenge orthodox thinking through engaging with theories and research literature that advocate an alternative vision of mathematics education. Conversely, the active participation of teachers in the research process can provide an academic researcher with greater insight into the classroom context in which the research is situated. I have demonstrated how a group of mathematics teachers from a number of different schools can work together with an academic researcher to develop a critical understanding of their own situation and provide the mutual support necessary to identify and overcome the constraints they face in transforming their practice.

The project highlighted the existence of a wider group of teachers in schools, who are potentially interested in adopting more empowering approaches to teaching mathematics, but who may only be persuaded to devote time for trying these out once they have seen the positive outcomes achieved in colleagues’ classrooms. My experience as a teacher educator is that many teachers decide to join the profession out of a concern for issues of equity and social justice, but then lose sight of this aspiration once they become assimilated into its prevailing practices and discourses. This group of teachers might therefore be bigger than many people think. The substantial interest shown in the project by other teachers not initially involved underlines the potential of the critical research model of PAR to have a wider impact on classroom practice in a formal school setting through a bottom - up approach that involves collaborations between teachers and academics. It is the responsibility of critical mathematics educators, of whom I consider myself to be one of many, to initiate and facilitate such opportunities, to provide the external stimulus necessary for critical reflection to occur, and to develop models of PAR that enable such transformations in classroom practice to take place.

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Wright, P. Transforming mathematics classroom practice through participatory action research. J Math Teacher Educ 24 , 155–177 (2021). https://doi.org/10.1007/s10857-019-09452-1

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Published : 16 January 2020

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DOI : https://doi.org/10.1007/s10857-019-09452-1

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