Pirie-Kieren Theory Research Articles

Analysis of primary school students’ process of understanding about the concept of ratio: A view from the Pirie-Kieren theory

The understanding process of primary education students was analyzed when they solve tasks related to the concept of ratio. The study was based on the theoretical framework of Pirie and Kieren (1994). The methodology was qualitative with the case study method. The study was carried out in three stages: planning, development and analysis, using the field observation technique. Data collection was carried out through a task and an interview. The data were analyzed based on theoretical articulation. The results revealed that students lack the prior knowledge necessary to understand the concept of ratio. In conclusion, it can be noted that students do not present logical arguments to formalize the concept, and their understanding process is reduced to memorization or the use of mathematical strategies without understanding the relationship between the task and the mathematical concept.

Students' Learning Obstacles on Sequences and Series Viewed by Pirie Kieren's Theory

Mathematical understanding develops through eight layers, as described by Pirie and Kieren: primitive knowing, image making, image having, property noticing, formalizing, observing, structuring, and inventizing. However, evidence indicates that many students encounter obstacles that hinder their progression through these layers. This study aims to identify and describe the learning obstacles students face in understanding sequences and series, utilizing Pirie and Kieren's theoretical framework. A descriptive qualitative research design was employed, purposive sampling to select participants from 30 Year 11 students in Malang, Indonesia. Data were gathered through tests and interviews and analyzed based on indicators of learning obstacles and the corresponding layers of mathematical understanding outlined by Pirie and Kieren. The findings reveal that many students experience significant difficulties in noticing and formalizing layers within the property. These challenges are attributed to inadequate foundational knowledge (ontogenic conceptual obstacles) and a lack of structured opportunities for developing deeper mathematical understanding (ontogenic instrumental, didactical, and epistemological obstacles). The results underscore the need for further research to address these learning barriers by focusing on enhancing students' foundational knowledge and designing educational experiences that foster the growth of mathematical understanding.

The Impact of the Pirie-Kieren Theory on Developing Fraction Understanding in Third-Grade Students

This study aims to examine the impact of applying the Pirie-Kieren theory to enhance third-grade students' understanding of fractions in Kosovo. Fractions are a foundational concept in mathematics, essential for both mathematical competency and general education. However, comprehending fractions remains a major challenge for students, educators, and parents worldwide, including in countries with well-established traditions in mathematics education. By incorporating the Pirie-Kieren theory into the design of instructional programs and textbooks, the content is structured in a coherent and logical progression. This method ensures that each learning unit builds on previously acquired knowledge, thereby transforming present learning into a strong foundation for future understanding. In Kosovo, where the introduction of fractions begins in the third grade, our study utilized an experimental design with a control group, involving a total of 148 students - 80 from experimental classes and 68 from control classes - across three schools in different regions. We started by conducting a detailed analysis of curriculum components and textbooks related to fractions, identifying several didactic-methodological shortcomings, including disorganized content, inadequate visual representations, and material that was not age-appropriate. In the experimental phase, we created detailed lesson plans for teachers and worksheets for students in the experimental groups, ensuring they were in accordance with the principles of the Pirie-Kieren theory. These materials were refined with input from experts and reputable organizations such as UNESCO, the State Department, NCTM, and OECD. The study also designed a specialized test to evaluate outcomes in both the experimental and control groups. The results were analyzed using various statistical methods to compare the performance of the two groups. The study findings reveal that teaching and learning fraction-related content through the Pirie-Kieren theory/model is significantly more effective than traditional instructional methods.

Layers of Student Understanding Based on Pirie Kieren's theory in Solving Story Problems in Terms of Cognitive Style

Objective: Understanding lines and angles serves as the foundation for further mathematical topics such as trigonometry, geometry, calculus, etc. Students may struggle with advanced mathematical reasoning and problem-solving without a firm grasp of these basic concepts. This study aims to investigate how students' mathematical understanding processes are based on the folding back theory. Method: The research utilizes a qualitative approach with descriptive eksploratif design. Two subjects were selected from 28 seventh-grade students, each representing the field-dependent and field-independent cognitive style. Data was collected through mathematical comprehension tests, GEFT tests, and interviews. Data were analyzed through data reduction, data presentation, and verification stages, with each subject being interviewed to verify the processes. Results: Based on the research results, students in the field-independent cognitive style were much more active and better understood the problem-solving process than those in the field-dependent. However, both subjects still required learning assistance. Novelty: This research explores the folding back theory in the mathematical understanding process based on cognitive styles, whereas previous studies have mainly focused on mathematical comprehension abilities. Therefore, further research would benefit from using instructional media to better engage students in understanding the material.

Elementary school students’ layers of understanding in solving literacy problems based on Sidoarjo context

Elementary school students' difficulties in solving literacy problems are caused by students' lack of understanding of mathematical concepts. Using a specific regional context in literacy problems is essential to analyze layers of understanding elementary school more deeply. Based on Pirie-Kieren's theory, the study aims to analyze layers of understanding elementary school students in solving literacy problems in the context of Sidoarjo. The theory consists of eight understanding layers: primitive knowing, image making, image having, property noticing, formalising, observing, structuring, and inventing. A qualitative approach with a case study type was used as the research method. The research participants were 26 fifth-grade elementary school students. The determination of the research subjects was done by purposive technique. In this, the emphasis of determining the subject is taken on three categories of understanding by having one student each in the low, medium, and high categories. The results show that the layers of understanding of elementary school students are at the highest, namely observing. In addition, literacy problems in Sidoarjo contexts can overstimulate students' understanding. The context of learning should be used in Sidoarjo. Future research recommendations suggested using the regional context as a background for literacy problems in learning or problem-solving; this is needed to explore and develop layers of understanding to a higher level.

Conceptual Understanding of Students with Special Needs in Two-dimensional Shapes Based on Pirie Kieren's Theory

This study aims to describe the conceptual understanding of two-dimensional shapes in students with special needs based on Pirie Kieren's theory in terms of learning motivation. This research is a case study with the research subject is 1 students with special needs of class IV at SDN Pademawu Timur V The results of this study are low learning motivation with layers of understanding on the square starting from Image making then head to property noticing to ineffective folding back to image making and do ineffective folding back again to primitive knowing. In solving the square problem, the subject doubled back to ineffective image-making and primitive knowing. Meanwhile, in the rectangular problem, the subject does one folding back to image making. There are similarities in solving square and rectangular problems, for example, the subject describes the two-dimensional shape first before image making and folding back because they don't know how to calculate the area of a square and rectangle. From the similarities and differences in the thinking flow of students with special needs in squares and rectangles, educators are expected to be able to pay attention to their understanding abilities and learning motivation before carrying out learning.

The Growth of Students' Function Limit Concepts Understanding in Solving Controversial Problems Based on Pirie Kieren's Theory

Almost all students understand the limit of a function only up to an intuitive definition and have difficulty understanding the concept of a limit function formally. This study aims to describe the growth of student understanding of functions limit concept in solving controversial problems based on Pirie Kieren's theory. There were twelve Calculus class students in the short semester as participants. The students selected were those who had taken calculus courses. Students are given the task of solving controversial problems to understand the concept of limit functions. There was only one student who showed a growing understanding of the concept of the limit of a function and was interviewed for further exploration. This research is a qualitative descriptive research. Therefore, the researchers analyzed the results of students' work through data reduction, data presentation, and conclusion drawing. The result shows that through controversial problems, students' understanding grows to an inventising level. However, students did ‘fold back’ at the observing level. At this level, students look at or re-read their notebooks to recall previously owned concepts. For further research it is suggested that researchers can design a learning process that can help grow student understanding through controversial problems.

A study on the use of collaborative learning to enhance mathematical understanding among elementary students

This study investigated the use of collaborative learning to enhance mathematical understanding among elementary students. This study will give an overview on the usage of collaborative learning in mathematical teaching and learning as an interactive and interesting alternative approach that helps in enhancing mathematical understanding of learners. Mathematics may be regards as hard and difficult subject by most learners as they tend to be learning this subject in more traditional approach. This study focuses on the objective to study the interactions of the students during the collaborative learning activities, to investigate students’ perception and attitude towards collaborative learning as well as the effectiveness of collaborative learning in enhancing student’s mathematical understanding. This study employed the qualitative approach by combining the Communication Study Framework by Sfard and Kieren (2001) and The Pirie-Kieren Theory (1992,1991). The data collected from selective respondents were analysed through the recorded videos of the mathematics lessons on selected sub-topic. From the data, the respondents indicated active interactions during the tasks, and it also indicated that by having collaborative learning, respondents have positive perceptions and attitudes based on their active participations during the group discussions. From the data as well, the respondents’ portrayed appropriate level of mathematical understanding based on the selected topic.

Using Prior Knowledge to Advance Students’ Mathematical Understanding: Geometric Translation as a Function

Matematik eğitimi araştırmacıları ve öğretim programı geliştiricileri, matematik öğretiminin öğrencilerin ön bilgileri dikkate alınarak tasarlanması gerektiği konusunda hemfikirdir. Bu araştırmanın amacı, öğrencilerin herhangi bir matematiksel konuyla ilgili matematiksel anlamalarını geliştirmek için, ön bilgilerin öğretim dizaynında nasıl kullanılabileceğine dair bir yaklaşım sunmaktır. Konu olarak, okul matematiği ve matematik öğretmen adaylarının eğitimi için önemli konularından birisi olan öteleme seçilmiş ve Pirie-Kieren teorisinde yer alan geriye katlamalar ile öğretim müdahaleleri temel alınarak bir eylem araştırması dizayn edilmiştir. Katılımcılar, bir devlet üniversitesinin Matematik Öğretmenliği programının dördüncü sınıfında öğrenim gören 28 matematik öğretmen adayıdır. Araştırmanın verileri, öteleme dersleri başlamadan önce uygulanan ön teste katılımcıların verdiği yazılı cevaplar, dersler boyunca gerçekleşen matematiksel tartışmalara yönelik kayıtlar, ders gözlem notları ile dersi yürüten eğitimcilerin ders öncesi veya sonrası görüşmelerinde alınan notlardan oluşturmaktadır. Dersler boyunca, öğretmen adaylarının ön bilmeleri üzerinde çalışmalarını sağlayan geriye katlamaları teşvik eden öğretim müdahalelerin, adayların ötelemeyi bir fonksiyon olarak anlamlandırmalarına yardımcı olduğu belirlenmiştir. Araştırmanın, öğrencilerin matematiksel anlamalarını desteklemek amacıyla ön bilmeleri referans alarak öğretim tasarlamak isteyen matematik eğitimcilerine fikir vereceği düşünülmektedir.

Profile of the Concept Understanding of Two-Dimensional Figure Based on Pirie Kieren's Theory Reviewed from Learning Motivation in Elementary School

This study aims describe understanding concept of two-dimensional figure based on Pirie Kieren’s theory in terms of learning motivation. This study uses a qualitative approach with a case study type of research. The results found that there were differences in the flow of students' understanding of high, medium, and low learning motivation in solving questions. When solving square questions, students with high learning motivation do effective folding back and arrive at inventing, students with motivation are doing infective folding back and not inventing, and students with low motivation do not fold back and do not get inventing. When solving rectangular problems, high-motivated students did ineffective folding back and did not arrive at inventing, moderately motivated students did not fold back and arrived at inventing, and low-motivated students did effective folding back and arrived at inventing. The similarity is that they both start from primitive knowledge. Keywords: concept understanding, Pirie Kieren Theory, two-dimensional figure. Abstrak: Penelitian ini bertujuan untuk mendeskripsikan pemahaman konsep bangun datar berdasarkan teori pirie kieren ditinjau dari motivasi belajar. Penelitian ini menggunakan pendekatan kualitatif dengan jenis penelitian studi kasus untuk memaparkan secara terperinci pemahaman siswa berdasarkan teori pirie kiren. Hasil yang ditemukan terdapat perbedaan alur pemahaman siswa motivasi belajar tinggi, sedang dan rendah dalam menyelesaikan soal bangun datar. Saat menyelesaikan soal persegi siswa motivasi belajar tinggi melakukan effective folding back dan sampai pada inventising, siswa motivasi sedang melakukan inffective folding back dan tidak sampai inventising, siswa motivasi rendah tidak melakukan folding back dan tidak sampai inventising. Saat menyelesaikan soal persegi panjang siswa motivasi tinggi melakukan ineffective folding back dan tidak sampai pada inventising, siswa motivasi sedang tidak melaukukan folding back dan sampai pada inventising, siswa motivasi rendah melakukan effecting folding back dan sampai pada inventising. Persamaannya yaitu sama-sama dimulai dari primitive knowing. Kata kunci: pemahaman konsep, Teori Pirie Kieren, bangun datar. DOI: http://dx.doi.org/10.23960/jpmipa/v23i3.pp1135-1148

Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames

The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner’s growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner’s growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner’s mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.

Folding back in students’ construction of mathematical generalizations within a dynamic geometry environment

In this work, we studied the cognitively recursive process of generalizing relying on data collected from a 2-week constructivist teaching experiment where a group of middle school students were working together to explore and generalize properties of geometric transformations in a dynamic geometry environment. In doing so, we focused on the role of reflection and outcomes of reflective work of learners on their progress towards construction of mathematical generalizations. Relying on the construct of folding back in the Pirie–Kieren theory, we present categories and subcategories that describe sources, forms, and outcomes of students’ reflections in the course of construction and refinement of mathematical generalizations. We then discuss the roles of teacher intervention and technology in motivating and supporting students’ folding back actions. The results of the study not only elaborate on the construct of folding back in the Pirie–Kieren theory but also provide a useful analytical tool to examine learners’ reflective practices.

Teacher Interventions for Advancing Students’ Mathematical Understanding

The relationship between teacher interventions and students’ mathematical thinking has been the subject of inquiry for quite some time. Using the Pirie–Kieren theory for dynamic growth in mathematical understanding, this study documents teacher interventions that support students’ growth toward developing a general understanding of a mathematical idea in a designed learning environment. By studying the interactions of seven middle school students and the teacher-researcher working on a two-week unit on geometric transformations within a dynamic geometry environment, this study identified nine major categories of teacher interventions that support and extend students’ investigations of mathematical ideas around geometric transformations. The typology of teacher interventions reported in this study provides a cognition-based framework for teacher moves that extend and advance students’ mathematical understanding.

Characterizing Learners’ Growth of Geometric Understanding in Dynamic Geometry Environments: a Perspective of the Pirie–Kieren Theory

The emergence of interactive mathematics software in the early 1990shas invited mathematics education researchers to consider ways that dynamic geometry environments (DGEs) might influence K–16 learners’ growth of geometric understanding. This article examines the applicability of the Pirie–Kieren theory to trace this growth. Data for this article came from a series of task-based interviews that aimed at examining the relationship between The Geometer’s Sketchpad(GSP) usage and the development of geometric knowledge about compositions of geometric transformations. Data analysis showed that the Pirie–Kierentheory provides a useful tool to characterize, at the micro-level, a learner’s process of gaining a particular geometric understanding of a composition of geometric transformations in this DGE. By supporting the participants to transform geometric objects into their desired form, and to become acquainted with their properties or structures, tools in this DGE facilitated not only their transition between different levels of understanding, but also the to-and-fro movements within each specific level of understanding. The presence of a DGE challenges mathematics education researchers to reconsider the Primitive Knowing level of the Pirie–Kieren theory, and to attend to the difference between a visual image brought to mind by a GSP diagram and a mental image developed from manipulating the diagram.

The 9th grade students’ mathematical understanding in problem solving based on Pirie-Kieren theory

The purpose of this study is to analyse the mathematical understanding in 9th grade students’ problem solving based on Pirie-Kieren theory. The Pirie-Kieren theory provides a framework to evaluate the students’ mathematical understanding in solving mathematical problem. The layers of mathematical understanding in Pirie-Kieren theory consist at primitive knowing, image making, image having, property noticing, formalising, observing, structuring, and inventizing. This study designed according to case study toward two students who are studying at one of primary school in Bandung district. The participants in this study were selected among voluntary students by considering their mathematics interesting and recommendation from the mathematics teacher. The data was collected by testing and interviewing. One problem solving item was posed for the mathematics test. The interview of this study was collected by through a semi-structured interview. The obtain data was descriptive analysis with qualitative approach based on Pirie-Kieren theory. The result show that two of participant was started working at image having layer and they have placed at formalising layer. The participants have lacked at primitive knowing and image making according to Pirie-Kieren theory.

Analysis of Student’s Understanding of Exponential Concept: A Perspective of Pirie-Kieren Theory

Solving mathematical problem is frequently used by employing exponential concept, such as logarithm. The aim of the research is to analyze students’ understanding of exponential concept based on Pirie-Kieren theory. The participant of the research is one female student and one male student at the 11th grade. The data collection was conducted by giving two tasks and interviewing with semi-structure interview. The result showed that both students can solve the first task systematically with different process. The processes are correct although the last expression is not simple. For primitive understanding level, they had exponential basic knowledge. They could explain exponential definition by their own statements. At image making and image having level, they had different ideas to solve problem and they were able to deliver their solutions. At property noticing level, they had competences to identify and differentiate the exponential properties for multiplication and division. At formalizing level, they are able to express these properties to mathematics symbol. However they had no algebraic ability for proving exponential properties. At the second task, they couldn’t reach observing level. They answered the problem incorrectly. Therefore, it can be concluded that the female student and male student’s understanding have already reached formalizing level in the exponential concept.

Folding back, thickening and mathematical met-befores

The study reported here explores how teachers can encourage students to build mathematical understandings that are connected to their prior experiences and knowledge. We illustrate this through considering data extracts taken from the classroom of one high school teacher, as he works with a Grade 12 class on the concept of vectors in three-dimensional space. Drawing on elements of the Pirie-Kieren theory for the dynamical growth of mathematical understanding we consider how the deliberate and purposeful teaching action of encouraging folding back can offer a way for students to engage with problematic mathematical met-befores and so promote understandings that are both more general in nature and conceptually connected.

The Analysis of Understanding Factorial Concept Processes of 7th Grade Students who have Low Academic Achievements with Pirie Kieren Theory

The purpose of this research is to analyze the understanding factorial concept processes of 7th grade students who have low academic achievements with Pirie-Kieren theory. The research participants are composed of two 7th grade students who are studying at a public primary school in the İnegol Province of the city of Bursa. This research has been designed according to case study design since it was carried out with two students who had low academic achievements in a constructive learning environment. The research participants were selected among voluntary students by considering their mathematics achievement test scores and their grades stated in the previous year’ reports. Mathematics achievement test was composed of 15 questions that were covering pre-cognitive knowledge of the factorial concept. Educational games, group work, puzzles, worksheets and activities designed for the objectives in the factorial concept were used in the research. The data of the research was collected by through a semi-structured interview, participant observation and document analysis for each student in the study group. The obtained data was analyzed by using a descriptive analysis method. According to the findings, students who have low academic achievements were able to construct concepts difficulty by interpreting activities and they reached formalising level according to Pirie and Kieren theory. In the light of the findings of this study, the researchers have developed suggestions.

Examining Students’ Mathematical Understanding of Geometric Transformations Using the Pirie-Kieren Model

The content of curriculum designs and innovations made in the field of mathematics education in the last decade have been shaped with emphasis on the need for to have understanding while learning mathematics. For example, the recently updated curriculum for secondary mathematics in Turkey indicates that learning environments which not emphasize meaning or do not provide with an opportunity or possibility to create meaning from the mathematics that is being learned cannot properly meet the expectations of teaching (Ministry of Education [MoNE], 2013, p. 1). Similarly, according to the National Council of Teachers of Mathematics ([NCTM], 2000), students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge, (p. 20). How do then learn mathematics with understanding?Hiebert and Carpenter (1992) remind us that the major goal of mathematics education is to ensure mathematical understanding in the learning environment, and they indicate that researchers should focus on finding the answer to the question of how and in what ways make sense of mathematics. As widespread rhetoric, mathematical understanding is vital for teachers, researchers, and curriculum developers (Mousley, 2005, p. 553). When the nature of mathematical understanding is characterized, researchers can focus on studies that analyze students' understandings in detail, curriculum developers can design the curriculum according to the growth of mathematical understanding, and teachers can organize their teaching goals based on this growth (MacCullough, 2007). Researchers have endeavored to identify the complex nature of mathematical understanding with respect to the learning theories which they support and the interpretations they have made of the term understanding (Meel, 2003). An important part of this effort includes the work done by researchers to describe the development of students' mathematical understanding via mental processes based on the constructivist approach.One should consider the individual experiences, perceptions, and interactions carry out in their environment when analyzing the of mathematical understanding from the constructivist perspective (Pirie & Kieren, 1992). Many researchers have looked at learning from this perspective and provided distinct theories to characterize mathematical understanding (i.e., Davis, 1984; Sfard, 1991; Sierpinska, 1994; Skemp, 1978). One of these theories is the representation theory. This theory promotes the idea that understand mathematics as much as they can make sense of the different representations of mathematical concepts and build connections among these representations (Goldin, 2003; Hiebert & Carpenter, 1992; Janvier, 1987). Goldin (2003) defines the term representation as configuration of signs, characters, icons, or objects that can somehow stand for, or 'represent' something else (p. 276). Students work with representations of mathematical objects while dealing with mathematics across variety of situations (Duval, 2006). They can learn mathematics with understanding if they can accurately apply different representations of mathematical concept and construct the relationships between these representations (Lesh, Post, & Behr, 1987). Another theory that analyzes mathematical understanding from the constructivist perspective is the Pirie-Kieren theory, which was presented by Susan E. Pirie and Thomas Kieren in 1989. Pirie and Kieren (1994) define mathematical understanding as a whole, dynamic, leveled but non-linear, transcendently recursive process (p. 166). This theory emphasizes that the growth of mathematical understanding is not linear; by contrast, it improves dynamically with back and forth movements between mathematical ideas.Mathematical Understanding LevelsPirie and Kieren (1994) developed model in order to represent ideas while they were building the theory of the growth of mathematical understanding (see Figure 1). …

Images and the Growth of Understanding of Mathematics-for-Working

Abstract This article considers the nature and growth of mathematical understanding within the context of the workplace-training classroom. In doing this, it draws on elements of the Pirie-Kieren Theory for the Dynamical Growth of Mathematical Understanding, in particular the notions of “Image Making” and “Image Having.” These theoretical ideas are elaborated to allow the more appropriate description of how the understanding of construction industry apprentices is observed to grow in their training as they engage with tasks with significant mathematical components. Through doing this we illustrate how the Pirie-Kieren Theory can be used to describe the growth of understanding of “mathematics-for-working” (influenced by the work of Ball and Bass (2003) on “mathematics-for-teaching”), and identify some of the key elements in this growth.