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The Role of Experiential Learning on Students’ Motivation and Classroom Engagement

Yangtao kong.

1 School of Education, Shaanxi Normal University, Xi’an, China

2 Faculty of Educational Science, Shaanxi Xueqian Normal University, Xi’an, China

Due to the birth of positive psychology in the process of education, classroom engagement has been flourished and got a remarkable role in the academic field. The other significant determining factor of success in education is motivation which is in line with classroom engagement. Moreover, based on the constructivist approach, experiential learning (EL) as a new method in education and a learner-centric pedagogy is at the center of attention, as a result of its contributions to improving the value of education which centers on developing abilities, and experiences. The current review makes an effort to consider the role of EL on students’ classroom engagement and motivation by inspecting its backgrounds and values. Subsequently, the efficacy of findings for academic experts in educational contexts is discussed.

Introduction

It is stated that a basic causative factor in the general achievement of learners studying in higher education is learners’ engagement ( Xerri et al., 2018 ; Derakhshan, 2021 ). It is extensively approved that learners who are actively participating in the learning progression and take interest in their academic education are more likely to achieve higher levels of learning ( Wang et al., 2021 ). Therefore, higher education institutions encourage learners to use their capabilities, as well as learning opportunities and facilities that enable them to be actively engaged ( Broido, 2014 ; Xie and Derakhshan, 2021 ). Moreover, students’ dissatisfaction, boredom, negative experiences, and dropping out of school are in part due to the low engagement in academic activities ( Derakhshan et al., 2021 ). It has been demonstrated that engagement is, directly and indirectly, related to intelligence, interest, motivation, and pleasure with learning outcomes within many academic fields ( Yin, 2018 ). Likewise, engagement is a construct that is shaped from the multifaceted relations of perceptions, feelings, and motivation which is corresponding to the progress of self-determination theory in the motivation realm ( Mercer and Dörnyei, 2020 ). Besides, the student’s motivation is a significant factor in cultivating learning and consequently increasing the value of higher education because the more the learners are motivated, the more likely they can be successful in their activities ( Derakhshan et al., 2020 ; Halif et al., 2020 ).

From a psychological point of view, motivating learners and engaging them in the classroom are closely related ( Han and Wang, 2021 ); nevertheless, motivation consists of factors that are psychological and difficult to observe, while engagement involves behaviors that can be observed by others that it is not simple to notice and estimate learners’ motivation ( Reeve, 2012 ). In other words, educators cannot concretely understand the fulfillment of their learners’ basic mental necessities and enthusiasm for learning ( Reeve, 2012 ). Nonetheless, Reeve asserted that in contrast to motivation, learners’ engagement by all accounts is a phenomenon that is distinctive and can nearly be noticed. Generally, educators can impartially consider whether or not a specific learner is engaged in the class exercises, such as problem solving.

As a reaction to the traditional teaching approach that is teacher-centric ( Che et al., 2021 ) and following the inclination to expanding interest in a more unique, participative learning atmosphere, educational organizations are orienting toward learning approaches that cultivate students’ involvement, interest, and dynamic participation. EL is a successful teaching method facilitating active learning through providing real-world experiences in which learners interact and critically evaluate course material and become involved with a topic being taught ( Boggu and Sundarsingh, 2019 ). Based on the teaching theory of Socrates, this model relies on research-based strategies which allow learners to apply their classroom knowledge to real-life situations to foster active learning, which consequently brings about a better retrieval ( Bradberry and De Maio, 2019 ). Indeed, engaging in daily activities, such as going to classes, completing schoolwork, and paying attention to the educator, is all indicators of classroom engagement ( Woods et al., 2019 ). Moreover, by participating in an EL class paired with relevant academic activities, learners improve their level of inherent motivation for learning ( Helle et al., 2007 ) and they have the opportunity to choose multiple paths to solve problems throughout the learning process by having choices and being autonomous ( Svinicki and McKeachie, 2014 ). EL is regarded as learning by action whereby information is built by the student during the renovation of changes ( Afida et al., 2012 ). Within EL, people become remarkably more liable for their learning which regulates a stronger connection between the learning involvement, practices, and reality ( Salas et al., 2009 ) that are key roles in learning motivation.

To make sure that the learners gain the required knowledge and get the factual training, it is equally important to give them time to develop their ability to use their knowledge and apply those skills in real-world situations to resolve problems that are relevant to their careers ( Huang and Jiang, 2020 ). So, it seems that they would like more hands-on training and skills development, but awkwardly, in reality, they generally just receive theoretical and academic education ( Green et al., 2017 ). In addition, as in today’s modern world, where shrewd and high-performing people are required, motivation and engagement should be prioritized in educational institutions as they are required features in the learning setting while they are often overlooked in classrooms ( Afzali and Izadpanah, 2021 ). Even though studies on motivation, engagement, and EL have been conducted so far; however, based on the researcher’s knowledge, just some have currently carried out systematic reviews about the issue and these studies have not been all taken together to date; therefore, concerning this gap, the current mini-review tries to take their roles into account in education.

Classroom Engagement and Motivation

As a three-dimensional construct, classroom engagement can be classified into three types: physical, emotional, and psychological ( Rangvid, 2018 ). However, it is not always easy to tell whether a learner is engaged because observable indicators are not always accurate. Even those who display signs of curiosity or interest in a subject or who seem engaged may not acquire knowledge about it. Others may also be learning despite not displaying any signs of physical engagement ( Winsett et al., 2016 ).

As an important component of success and wellbeing, motivation encourages self-awareness in individuals by inspiring them ( Gelona, 2011 ). Besides, it is a power that manages, encourages, and promotes goal-oriented behavior, which is not only crucial to the process of learning but also essential to educational achievement ( Kosgeroglu et al., 2009 ). It appears that classroom motivation is influenced by at least five factors: the learner, the educator, the course content, the teaching method, and the learning environment ( D’Souza and Maheshwari, 2010 ).

Experiential Learning

EL, developed by Kolb in 1984, is a paradigm for resolving the contradiction between how information is gathered and how it is used. It is focused on learning through experience and evaluating learners in line with their previous experiences ( Sternberg and Zhang, 2014 ). The paradigm highlights the importance of learners’ participation in all learning processes and tackles the idea of how experience contributes to learning ( Zhai et al., 2017 ). EL is a method of teaching that allows learners to learn while “Do, Reflect, and Think and Apply” ( Butler et al., 2019 , p. 12). Students take part in a tangible experience (Do), replicate that experience and other evidence (Reflect), cultivate theories in line with experiences and information (Think), and articulate an assumption or elucidate a problem (Apply). It is a strong instrument for bringing about positive modifications in academic education which allow learners to apply what they have learned in school to real-world problems ( Guo et al., 2016 ). This way of learning entails giving learners more authority and responsibility, as well as involving them directly in their learning process within the learning atmosphere. Furthermore, it encourages learners to be flexible learners, incorporate all possible ways of learning into full-cycle learning, and bring about effective skills and meta-learning abilities ( Kolb and Kolb, 2017 ).

Implications and Future Directions

This review focused on the importance of EL and its contributions to classroom engagement and motivation. Since experiential education tends to engage a wider range of participants who can have an impact on the organization, employees, educators, leaders, and future colleagues, it is critical to maintain its positive, welcoming atmosphere. The importance of EL lies in its ability to facilitate connections between undergraduate education and professional experience ( Earnest et al., 2016 ), so improving the connection between the university and the world of work ( Friedman and Goldbaum, 2016 ).

The positive effect of EL has actual implications for teachers who are thinking of implementing this method in their classes; indeed, they can guarantee their learners’ success by providing them with the knowledge required in performing the task as following the experiential theory, knowledge is built through converting practice into understanding. Based on the literature review, the conventional role of the teacher shifts from knowledge provider to a mediator of experience through well-known systematic processes. Likewise, teachers should encourage learners by providing information, suggestion, and also relevant experiences for learning to build a learning milieu where they can be engaged in positive but challenging learning activities that facilitate learners’ interaction with learning materials ( Anwar and Qadir, 2017 ) and illustrates their interest and motivation toward being a member of the learning progression. By learners’ dynamic participation in experiential activities, the teacher can trigger their ability to retain knowledge that leads to their intrinsic motivation and interest in the course material ( Zelechoski et al., 2017 ).

The present review is significant for the learners as it allows them to model the appropriate behavior and procedures in real-life situations by putting the theory into practice. Indeed, this method helps learners think further than memorization to evaluate and use knowledge, reflecting on how learning can be best applied to real-world situations ( Zelechoski et al., 2017 ). In the context of EL, students often find activities challenging and time-consuming which necessitates working in a group, performing work outside of the classroom, learning and integrating subject content to make decisions, adapt procedures, compare, and contrast various resources of information to detect a difficulty at one hand and implement that information on the other hand to form a product that aims to solve the issue. Participation, interaction, and application are fundamental characteristics of EL. During the process, it is possible to be in touch with the environment and to be exposed to extremely flexible processes. In this way, education takes place on all dimensions which cover not only the cognitive but also the affective and behavioral dimensions to encompass the whole person. Learners enthusiastically participate in mental, emotional, and social interactions during the learning procedure within EL ( Voukelatou, 2019 ). In addition, learners are encouraged to think logically, find solutions, and take appropriate action in relevant situations. This kind of instruction not only provides opportunities for discussion and clarification of concepts and knowledge, but also provides feedback, review, and transfer of knowledge and abilities to new contexts.

Moreover, for materials developers and syllabus designers to truly start addressing the learners’ motivation and engagement, they could embrace some interesting and challenging activities because when they can find themselves successful in comprehending the issue and being able to apply their information to solve it; they are not only more interested to engage in the mental processes required for obtaining knowledge but also more motivated and eager to learn. More studies can be conducted to investigate the effect of EL within different fields of the study courses with a control group design to carry out between-group comparisons. Besides, qualitative research is recommended to scrutinize the kinds of EL activities which make a more considerable effect on the EFL learners’ motivation and success and even their achievement.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

This study was funded by the Projects of National Philosophy Social Science Fund, PRC (17CRK008), and the Projects of Philosophy and Social Science Fund of Shaanxi Province, PRC (2018Q11).

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Original research article, the effectiveness of experiential learning in teaching arithmetic and geometry in sixth grade.

experiential learning methodology case study

  • 1 Department of Mathematics Education, School of Education, Can Tho University, Can Tho, Vietnam
  • 2 Duc Tri Secondary School, Ho Chi Minh City, Vietnam

Many educators and policymakers worldwide have noticed the burgeoning field of experiential learning in the twenty-first century. Learning theory and practice together is beneficial to education in general, and mathematics education in particular, because it enables students to realize their full potential for knowledge and skill, and it connects the two aspects of knowledge. A focus on the cross-cutting and critical role of experience activities within the framework was emphasized in Vietnam’s general education program in mathematics, released in 2018 and included views on the content and methods of teaching and learning mathematics in the country. Experiential learning in mathematics was studied to see if this method could positively help students participate, increase their motivation and interest in learning, and impact their math outcomes. A series of pedagogical experiments with 29 sixth-grade students on arithmetic and geometric topics was conducted to confirm the research goals. Students were required to develop solutions to real-world problems related to their studying subjects. The experimental and control classes are subjected to a pre-test and a post-test design. Mixed methods, including qualitative and quantitative analysis, are handled by the statistical data processing software (SPSS) program and the results of observations and surveys of learners’ opinions. The results were found that experiential learning activities positively influenced math learning attitudes and student achievement progress in the classroom.

Introduction

The educational trend of the twenty-first century is student-centered, experiential, technology-based, and question-based learning and empathic and understanding ( Habib et al., 2021 ). According to Vietnam’s General Education Curriculum for 2018, the math program focuses on application, linking to practice or educational activities. It is not just about implementing math learning topics and projects; it is also about organizing math learning games, clubs, forums, seminars and contests, and other practical and experiential activities in mathematics education that reflect this. In these activities, learners should be allowed to put their knowledge into practice through creative means ( Ministry of Education and Training, 2018 ). Thus, students can learn and be creative while exploring and putting their knowledge into practice in real-world contexts through experiential teaching activities ( Ministry of Education and Training, 2018 ). Experiential learning is essential ( McCarty et al., 2018 ). Many mathematical concepts have been studied and taught experientially by students at the high school and university levels, including the equation of a circle ( Tong et al., 2020 ), function continuity and angle between two planes ( Davidovitch et al., 2014 ), mathematics with statistics ( Venkatraman et al., 2019 ), arithmetic ( Mayoral-Rodríguez et al., 2018 ), some advanced algebraic topics ( Wynn, 2018 ), as well as mathematics and science ( Roberts et al., 2016 ). Also, according to Pambudi’s (2022) research, elementary students’ motivation and achievement in geometry can be attributed to outdoor learning methods. For mathematics education in Vietnam and worldwide, research on mathematical experiential activities should be promoted to allow students to experience positive emotions, exploit their personal experiences, and mobilize their knowledge and skills to perform assigned tasks or solve real-world problems. As a result, learners transform their experiences into new knowledge, understanding, and skills, thereby promoting their creative potential and adaptability to the future life, environment, and career.

Literature Review

The concept of experiential learning.

Experiential learning has been studied by many educators in a wide range of fields, including Kolb (1984) , is “…the process by which knowledge is created through the transformation of personal experience” [ Kolb (1984) , as cited in Mutmainah et al. (2019) , Cotič et al. (2020) ]. According to Dewey (1938), the experience can be divided into two types: the entire experience and the universal experience. An experience is simply a collection of random activities or events someone has joined. In contrast, universal experience results from a methodical, self-reflective survey that considers prior knowledge and future predictions [Dewey, 1938; as cited in Breunig (2017) ]. Davidovitch et al. (2014) agree that experiential learning can be divided into personal experiences in life and events and educational programs. The second group’s interpretation necessitates thorough preparation and a lengthy process to conclude ( Davidovitch et al., 2014 ). As such, experiential learning is a form of active learning in which students are actively involved in learning through direct participation, supported by experience, analysis, and reflection ( Mutmainah et al., 2019 ; Habib et al., 2021 ). Furthermore, the author Voukelatou (2019) claims that students are the driving force behind the learning process and that the effectiveness of learning is influenced by the student’s “learning style” and “thinking”. While participating in various activities, students are encouraged to think critically and creatively, investigate, inquire, and make decisions, according to Mutmainah et al. (2019) . Their participation in learning opportunities for students directly results in the development of the necessary skills and knowledge that will support them in succeeding in their future studies [Atherton, 2009; cited in Chesimet et al. (2016) ], a concentration on maximizing students’ potential ( Tong et al., 2020 ; Mc Pherson-Geyser et al., 2020 ). The Design–Instruction–Assessment–Learning model has been remixed by Heinrich and Green (2020) to promote high-quality experiences for both learners and instructors.

Characteristics of Experiential Learning

Students are more likely to persevere in an active learning role if exposed to an authentic experience [Dewey (1938), cited in Behrendt and Franklin (2014) , Cotič et al. (2020) ]. An experiential learning framework can be successful if each student is directly involved in the experience by carrying out tasks, as Venkatraman et al. (2019) stated. Ultimately, each student must respond rationally to any feedback they receive by transforming their analytical skills experience into higher-order thinking strategies.

According to Kolb and Kolb (2017) , experiential learning is characterized by a unique dynamic between educators, students, and the content they are studying. Hence, teachers and students can gain firsthand knowledge of the subject. It is transmitted to them, but they are also responsible for creating it themselves. From there, all subjects can directly participate in the subject experience when using this method, which is similar to previous methods in that they can do so. Depending on the experience’s design and implementation, a wide range of viewpoints on the subject will be expressed.

According to Voukelatou (2019) , experiential learning is based on students’ thoughts, feelings, and openness during the educational process. Student-teacher collaboration is also important in teaching and learning because it allows teachers to better engage with and understand the material. Among the ways, question-and-answer, discussion, role play, case study and model interviews, educational tours, brainstorming, confrontation, expert interviews and exercises, group work, art education, and debate are all examples of experiential teaching techniques that can help students participate actively, interact and communicate more effectively ( Voukelatou, 2019 ; Canino et al., 2021 ). According to Vietnam’s General Mathematics Education Program issued in 2018, it is possible to engage a wide range of students in hands-on math activities by implementing topics and projects focused on the practical application of mathematics; organizing math learning games, math clubs, forums, seminars, and competitions ( Ministry of Education and Training, 2018 ). In their study, the term “mathematical debate” was coined by Davidovitch et al. (2014) . For that reason, this is an exercise where students work in groups to solve high-level problems and present their solutions to the class and other groups of students. The most important part of emulation is learning about and critiquing the various options available to members.

Thus, experiential and project-based learning (PBL) are closely related. Moreover, cooperative and collaborative learning are also closely related ( Burrell et al., 2017 ; Scogin et al., 2017 ; Cline et al., 2020 ). According to author Larmer (2015) , project-based learning is defined as experiential activities linked to oriented and open-ended problems and questions, real-world application of content and skills, and student-centered learning. Students create productions, presentations, or performances that address issues underlying questions. Additionally, cooperative learning is one of the ways to organize group work to improve learning effectiveness and student achievement by organizing how students interact and participate in achieving goals together [Zaitou (2003), as cited in Hossain and Ariffin (2018) , Algani and Alhaija (2021) ]. Because of these features, these methods of instruction can help foster an environment where experiential learning can have fruitful results. Also, it is believed that the flipped classroom approach, which is based on the experiential learning theoretical perspective, has gained preliminary validation in the system course environment ( Chen, 2021 ).

The Cycle of Experiential Learning

There is no denying that many educational institutions have examined hands-on experience in teaching. In addition, the research of authors Kolb (2014) , Breunig (2017) , Hsu (2019) , Cotič et al. (2020) , Lamya et al. (2020) , and Mc Pherson-Geyser et al. (2020) in general education and other fields mentioned the application of this model. According to Kolb (1984) , the four stages of experiential learning are individual experience, reflective observation, abstract conceptualization, and active experimentation ( Muro and Terry, 2007 ; Cotič et al., 2020 ). Chesimet et al. (2016) provide the following explanation of the model described above: Starting with a specific experience (e.g., a traumatic event), students then reflect on their experiences from a variety of perspectives (observation). Students build theories or models (conceptualization) from their reflections in order to conduct experiments and act on their findings (experimentation) ( Chesimet et al., 2016 ). Learning through and from experience is described by Kolb (2014) as a process of (1) engaging in individual experiences, (2) observing and reflecting, (3) forming knowledge and testing concepts in new situations, (4) applying and testing concepts ( Kolb, 2014 ). There is also a five-step experiential learning process put forth by that comprises of the following steps: setting up an experiential situation; sharing it with others; putting it into practice; generalizing it; and finally, applying it. Student skill development and knowledge application are encouraged through active, experiential learning, as proposed by this model ( Davidovitch et al., 2014 ).

Benefits of Experiential Learning

It has long been known about the positive effects of experiential learning on educational outcomes, particularly in the field of mathematics education ( Avelino et al., 2017 ; Mutmainah et al., 2019 ). The quality and effectiveness of learning can be improved by experiential learning ( Weinbern et al., 2011 ; Mayoral-Rodríguez et al., 2018 ; Wynn, 2018 ) motivating learning ( Venkatraman et al., 2019 ).

Furthermore, Venkatraman et al. (2019) show that experiential learning positively impacts students’ mathematical creativity. The study of Chesimet et al. (2016) found that the experiential learning method is more effective than traditional teaching and learning methods in enhancing students’ mathematical creativity. As a result, students who engage in experiential learning can better express their creativity in mathematics and develop their critical thinking skills. To help students improve their problem-solving skills, researchers suggest incorporating hands-on activities into the classroom. A study by Mwei (2017) and Manfreda and Hodnik (2021) found that providing students with the opportunity to resolve real-life problems impacted their mathematical problem-solving abilities. It has been found that experiential activities in mathematics help improve students’ knowledge and understanding of math and active learning activities that reduce the burden on the curriculum. Especially, students who complete tasks requiring a thorough understanding of the lesson and openness to confronting unfamiliar problems benefit from this ( Davidovitch et al., 2014 ). Mayoral-Rodríguez et al. (2018) found that experiential learning can teach mathematics.

The use of experiential learning in the classroom has numerous advantages for educators. Teachers must consider whether their teaching methods are in harmony with the skills they want to teach their students or if they need to change them ( Wang, 2006 ). Consequently, according to Pittaway and Cope (2007) , teachers are encouraged to abandon a traditional approach in favor of one that emphasizes hands-on learning opportunities through experiential learning. When teachers and schools engage in experiential activities, they are more likely to create effective educational programs, foster an educational and cultural climate for students, and create a positive learning environment ( Tong et al., 2020 ). Besides, using liminality as a lens to examine experiential learning activities provides a new perspective on their impact on individuals, institutions and society ( Amigó and Lloyd, 2021 ).

Drawbacks of Experiential Learning

Despite extensive research into experiential learning and its educational benefits, its application in the classroom remains limited and subject to a set of rules and guidelines. As Kolb and Kolb (2017) found when researching experiential learning in higher education, some challenges can be traced back to similar issues in the educational system. Secondary school students are learning math through hands-on activities. According to the authors, experiential learning should cover all four modes of the learning cycle and apply to all learning situations in class and real-life situations. As a result, the disparity between theoretical courses and hands-on activities hurts both types of learning. Students’ actions in learning projects are not integrated with the conceptual reflections and analyses in the classroom. Experiential learning programs are considered ancillary and only prepare students for low-level professional development in the fixed-duration system.

On the other hand, the teacher focuses on teaching higher-level knowledge ( Kolb and Kolb, 2017 ). According to authors, Cranton (2011) and Tong et al. (2020) found obstacles to implementing experiential learning in the classroom due to time constraints, class structure, and the number of students. Besides, learning content, textbooks, student participation requirements, and grades are all factors that contribute to the difficulties that students face in the classroom. Another factor is that educators and students come from various cultural backgrounds ( Giroux, 2015 ). Another concern Darling-Hammond (2016) expressed was a lack of compatibility between learning content and pedagogy of experiential learning in the traditional teaching program.

Evaluation of Experiential Learning

In an educational setting, learning, and assessment are inextricably linked; therefore, it is critical to determine whether learning serves as the foundation for assessment or vice versa students, according to educators, are concerned about the purpose of assessments, whether or not students assess what they learn, and whether or not course experiences should be taken into account when developing assessments. When evaluating students’ performance, this includes considering fairness and the public interest. On the other hand, research shows that evaluating students’ knowledge and skills must be in harmony with the aspects above ( Venkatraman et al., 2019 ). For the author, this assessment offers a chance to integrate information that students have been taught about cognition, attitude, and psychology. The author mentions cognitive aspects like knowing, comprehending, putting into practice, and synthesizing and evaluating [Payne, 1997; as cited in Venkatraman et al. (2019) ].

While students are engaged in active classroom learning activities, practical assessments can be made. There are various ways to conduct student evaluations, from informal to more formal ones. First, teachers must determine which skills will be tested to develop an appropriate assessment strategy ( Venkatraman et al., 2019 ).

A self-assessment tool called the Kolb Educator Role Profile (KERP) was developed by Kolb and Kolb (2017) to help teachers evaluate their teaching methods from the perspective of the experiential learning cycle. For teachers, there are four roles: facilitator, expert, standard-setter and coach, according to the KERP model. Using this model, teachers can better understand the available types of instruction, the responsibilities of teachers and students, and how to make the best possible decisions in specific circumstances.

Theoretical Framework

The process of teaching based on experiential learning.

According to the findings, a five-stage process for designing and organizing experiential mathematics education is proposed. These steps are illustrated in Figure 1 .

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Figure 1. The process for designing and organizing experiential mathematics education.

Stage 1: Decide on a subject matter of study. Students’ characteristics, basic conditions, and educational objectives are used to determine the characteristics of mathematics in the general education curriculum in mathematics. Teachers in each school work together to identify and develop educational themes.

Stage 2: Aim to organize the experience. The teacher chooses experiential teaching organizations by the learning topic and classroom conditions. The teaching process can be made more efficient by combining different organizational forms. There must be a clear connection between the objectives, content, organizational structure, and activities. Determine the objectives and implementation methods for each task to ensure the project’s success.

Stage 3: Preparation is a key when it comes to academic success. The instructor will gather all necessary supplies and equipment for the hands-on learning session during this step. Basic knowledge is taught and put into practice through hands-on activities.

Stage 4: Experiential activities should be organized. New knowledge will be clarified based on the comprehension of related knowledge; students work with teachers to practice and gain experience. Students learn in school by applying, honing existing skills, and acquiring new knowledge and skills due to new experiences.

Stage 5: Verify and assess the outcomes. Making appropriate assessment tools and criteria for students’ abilities and overall experience outcomes helps teachers understand the level of achievement in each student’s abilities and overall experience outcomes. Teachers can use various assessment methods at this stage to determine whether or not the lesson has been completed successfully.

Experience-based learning processes have been studied by Kolb (1984) and educators on the application of experiential learning in mathematic, all of which have resulted in the proposal of organizational processes. There are six distinct steps of organizing experiential activities in mathematics instruction for students’ experiential learning.

Step 1: Describe what the experience is all about. The teacher gives a brief description of the experience before beginning the lesson by naming it and introducing it.

Step 2: Set up a plan for the time here. Instructors experiment with students, guiding them and asking for their feedback on their performance. It is common practice for students to work in groups to practice what they have learned in their own lives. The teacher will monitor each student’s participation and progress throughout the class, assisting as needed.

Step 3: Feedback, discussion, and evaluation are all encouraged. The teacher does the organization of students to present practice results. The teacher arranges for groups to meet and exchange ideas about working and accomplishing their goals. The teacher organizes students to analyze the data and draw conclusions about their findings.

Step 4: Plan out the ideas. Teachers allow students to draw from their own experiences in the classroom and then sum up their findings with a final statement. The instructor notes the discussion and then calls time on the class.

Step 5: Apply. The teacher helps students put what they have learned into practice in other settings. After the experiment, the teacher helps students identify any behavioral changes they may have made and provides additional opportunities to apply or discuss what they have learned with others.

Step 6: Summarize. The teacher’s responsibility is to provide feedback and assign homework based on the lesson’s content.

Arithmetic and Geometric Topics in the Vietnamese Mathematics Curriculum and Textbooks

Teaching through outdoor activities consists of two main directions in the Math curriculum in Vietnam. The first direction concerns experience to form new knowledge. It is the process by which students directly work with learning objects, observe, analyze, predict, and connect existing knowledge to discover and form new mathematical knowledge. Indeed, that knowledge can be a new concept, a new formula or a theorem, a way of proving under the direction and organization of the teacher. The other direction is practical activities and math experiences for students, such as: Conducting math learning topics and projects, especially topics and projects on applying mathematics in real life; organizing math games, math clubs, forums, seminars, competitions on math; publishing a wall newspaper (or internal magazine) on mathematics; exchanges with gifted students in math, exhibition ( Lykke et al., 2021 ), creative dance ( Payne and Costas, 2021 ), simulation decision-making games ( Kuczera, 2021 ), folk stories ( Menon, 2021 ), and school field trips ( Behrendt and Franklin, 2014 ). These activities will help students apply accumulated knowledge, knowledge, skills, and attitudes; help students initially identify their capacity and forte to orient and choose a career; create some basic competencies for future workers and responsible citizens. Also, learning topics create opportunities for students to recognize their talents and interests, develop interest and confidence in learning mathematics, develop mathematical competence, and explore mathematics-related problems throughout life. Some activities oriented by the program to organize experiences for students are as follows:

(1) Get familiar with savings deposits and bank loans; calculate loss, profit, and outstanding balance; practice calculating interest rates in savings deposits and loans.

(2) The invoice should make payment, or the change should be calculated when making a purchase. Practice keeping track of the income and expenses, and keep invoices on hand if needed.

(3) Apply statistical knowledge to read and understand Grade 6 History and Geography tables.

(4) Collect and represent data from a few real-life situations; for example, collect local temperatures at a certain time in a week to make comments about time changes of local weather for the week.

(5) Put symmetry into practice: folding paper to create shapes with symmetry axis or center of symmetry; collecting shapes in nature that have a center of symmetry or have an axis of symmetry; searching for videos of centered, axial symmetry in the natural world.

(6) Apply the concept of three straight points into practice, such as planting trees in a straight line and placing objects in a straight line.

(7) Apply formulas for calculating area and volume in practice. Measure and calculate the surface area, calculate the volume of objects related to the learned shapes.

Life and nature are reflected in the arithmetic and geometric sequences taught in the 6th-grade math program. When teaching these topics, the learning method can be put into practice. Aside from that, the Vietnam Mathematics General Education Program (2018) emphasizes several requirements for the organization of practical activities and experiences in the teaching of the topic of arithmetic and geometric sequences, which include: Mathematical concepts such as arithmetic and geometry are emphasized heavily in the new 6th grade Math curriculum, and they are covered in numerous periods throughout the school year. Hence, students become more open to the world of numbers, sets, calculations, and problems that they encounter in their daily lives when they study arithmetic in high school. To overcome these problems, they begin to reason, analyze, compare, and synthesize to find solutions to problems and situations they find themselves in. Meanwhile, geometry study progresses, with particular attention paid to the edges and corners of visual and metrological geometry. Using activities such as collage, drawing, and experimenting can help students apply what they have learned about ants in a natural and non-coercive manner outside of the classroom.

According to the General Education Program in Mathematics 2018, the content of calculations with natural numbers includes the requirement to deal with real-world problems that arise as a result of performing the calculations in question (for example, calculating the shopping money, calculating the number of goods purchased from the amount already available). For this reason, students must have the opportunity to apply their learning in real-world situations. As a starting point, students should become familiar with spending and finance, think about balancing needs and wants, gain a more in-depth understanding of the value of money and labor, and learn how to manage their money effectively and appropriately organize their lives.

One of the objectives of visual geometry in the General Education program of mathematics in grade 6 about rectangles, rhombuses, parallelograms, and the isosceles trapezoid is to cope with some real-world problems associated with calculating the perimeter and area of the special shapes mentioned above in the previous paragraph. Because of this, the content of knowledge about perimeter and area is extremely appropriate for students to experience and contribute to developing students’ capacity. Although students need to understand the construction of formulas, they must also calculate the perimeter and area of special shapes. Therefore, the educators want to design and organize a knowledge-forming experiment that will involve developing formulas to calculate the area of geometrical objects. Because of this, the researchers propose that experiential learning be applied to the topic of arithmetic and geometric topics to improve teaching quality, spark student interest in learning, and support students in creating a more positive attitude toward mathematics. Vietnamese math curricula and textbooks strongly emphasize experiential math activities because they are regarded as innovative teaching methods associated with socio-constructivist teaching methods. Because of this, it requires teachers to have digital competences ( Pozo-Sánchez et al., 2020a ).

Evaluation of Students’ Activities in Experiential Learning

Students may be asked to grade their performance on the criteria in Table 1 .

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Table 1. Criteria for students to self-assess.

Teachers evaluate students based on the criteria and levels of assessment in Table 2 .

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Table 2. Criteria for teachers to evaluate students.

The Study’s Purpose, as Well as the Research Questions That Will Be Addressed

Ultimately, this research aimed to determine the effectiveness and feasibility of incorporating experiential learning methods into the teaching of arithmetic and geometric topics in sixth grade. The following are the questions that the research will address:

1. When students read the 6th-grade textbook, how do they learn about arithmetic and geometric topics?

2. Can experiential learning help students learn more effectively and achieve better results?

3. When students are instructed through experiential learning, how has their participation, motivation, and attitude toward mathematics changed?

Materials and Methods

Participants.

The research team provided training in experiential learning to 30 volunteers, all of whom worked as substitute teachers in their spare time. A teacher was selected because she demonstrated proficiency in implementing the fundamental principles of the experiential learning model while instructing the experimental class of 29 students. In addition, the first names of the students in the experimental groups were coded with the letters S01–S29. According to tradition, a teacher who had not been trained used a conventional model to instruct a class of 27 students who served as the control group. Parents of students were notified in advance of their children’s participation in the experiment, and they were allowed to express their concerns. In this study, the students enrolled are 6th graders at a Duc Tri secondary school in Ho Chi Minh City in Vietnam (from September 29, 2021, to October 29, 2021). Especially, school districts in the city had closed their doors, and students had to attend online classes because of the Covid-19 pandemic, which significantly impacted educational activities at the time of the research.

Data Collection and Analysis

The research looks at classes formed by the school rather than regrouping random samples, so it uses a quasi-experimental approach. The quasi-experiment was conducted similarly to the studies on Sumirattana et al. (2017) , Yuberti et al. (2019) , and Chusni et al. (2022) to examine how the collected data might differ from testing a hypothesis. The data collection process is shown in Table 3 .

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Table 3. Quasi-experimental study design.

The data was gathered using the first year’s average scores (rather than the pre-test results), the post-test results, and responses from a survey of students. After completing the post-test, an evaluation was made of how much the students had learned about problem-solving abilities. The test instrument contained three items related to the arithmetic and geometric topics covered during this study. The pre-test contains three items based on real-world problems in construction and grocery shopping that require students to apply their newly acquired arithmetic and geometry knowledge to complete. The questions on conceptual comprehension were adapted from previous state-level trial examinations to meet the needs of the researchers who used them. According to Anderson Taxonomy, the test question items were also created. Also devised by researchers, the test’s scoring method was given a rubric by the team. The instrument and rubric were reviewed and scored by three mathematics teachers with over 10 years of classroom experience and two mathematics lecturers who were subject experts on arithmetic and geometry to determine their facial and content validity. The study by Yuberti et al. (2019) provided data that was confirmed to be extremely reliable.

The data was analyzed quantitatively with statistical data processing software (SPSS) 22 software and qualitatively with a qualitative analysis tool. Experiential learning-based treatment was effective by conducting qualitative assessments before and after each intervention. According to the paired t-test method, it was hypothesized that the average score of students in the experimental class would differ from the average score of students in the control class. The qualitative assessment results were used to analyze students’ worksheets, which evaluated students’ abilities to identify problems and resolve them in a real-world context.

Experimental Design

Based on the learning outcomes of the experimental and control classes, the researcher team and teacher collaborated to develop lesson plans that covered the arithmetic and geometric topics in the context of the experiential learning model application and its applications. Three distinct periods are proposed for the experimental lesson plan, which are as follows: the new lesson period, the practice-and-consolidation period, and the test period. Finally, students completed a post-test and a survey about their overall experience with the program. To evaluate the effectiveness of the pedagogical experiment, both quantitative and qualitative data were collected.

About Quantitative Analysis

Table 4 shows the tests using Shapiro-Wilk distributions to see if the scores before and after the classes were normally distributed. It can be concluded from the data processing results obtained using the SPSS 20 software that the two data scores have Sig values greater than 0.05, indicating that the two test scores obtained before and after the experiment are normally distributed.

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Table 4. Shapiro-Wilk test normally distributed pre-test and post-test.

Before the experiment, the t-test independent of the experimental class and the control class was used to determine level equivalence between the experimental class and the control class and the difference in mean score value between the experimental class and the control class after the experiment (2-tail). The formula developed by Cohen et al. (2005) is used to determine the extent to which the experimental class influences the mean score difference between the experimental class and the control class between two groups. The magnitude of the impact is indicated by the level of influence (ES), which is a percentage.

In order to assess attitudes, the student survey statements include a total of six items on a Likert scale with five levels, as follows: The following are the possible responses: strongly disagree, disagree, neutral, agree, strongly agree. A set of questionnaires to survey students after the experiment about their attitudes toward experiential lessons in the experimental class was developed based on a 5-level Likert scale and SPSS results. Correspondingly, the researchers concluded that the scale meets the requirements of internal reliability with a suitable variable-total correlation coefficient (not less than 0.3). The Cronbach’s alpha coefficient of the post-test questionnaire is greater than 0.7; specifically, this coefficient equals 0.871. With a variable-total correlation coefficient that is appropriate, the researchers also concluded that the scale meets the requirements for external reliability (not less than 0.3). Additionally, students in the experimental class were asked to answer the additional question “Do you have a different opinion about the class?” to express their opinions about the lessons taught. Besides, the criteria in Tables 1 , 2 were used to clarify the student worksheets further.

Pre-test Results

As previously described in the data collection and analysis section, a pre-test was administered to both the experimental and control classes to ensure that the two classes were on the same level of performance. These are the pre-test results that have been statistically processed and presented in Tables 5 , 6 .

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Table 5. Descriptive statistics of pre-test scores.

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Table 6. Independent t -test of pre-test results.

Table 5 shows that the mean of the experimental and control classes were 6.83 and 6.85, respectively, with no statistically significant differences between them. To test for variance differences between the experimental and control groups, the Levene test in Table 6 yielded Sig values of 0.601 > 0.05. The independent t-test revealed that the difference between the mean scores of the two classes was statistically insignificant (Sig = 0.957 > 0.05). As a result, the experimental and control classes’ mathematical learning levels can be related well.

Post-test Results

Table 7 shows that the experimental and control classes had mean values of 7.66 and 6.26, respectively, indicating a statistically significant difference between the two groups. The Levene test in Table 8 shows Sig = 0.501 > 0.05, indicating no variance difference between the two groups. The results of the independent t-test reveal the significance of the result. Because the difference in mean score between the two classes was statistically significant (two-tailed), the difference in mean score between the two classes was 0.005. So the null hypothesis was rejected, and the alternative hypothesis was accepted as the conclusion. Notably, the experimental students appear to have outperformed the control students in overall academic achievement, based on the two classes’ average scores. The mean standard deviation has been calculated as 0.74 based on the data. It is between 0.50 and 0.79 on Cohen’s scale, indicating a moderate effect size. In conclusion, experiential learning has a moderate impact on students and helps them learn more efficiently.

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Table 7. Descriptive statistics of post-test scores.

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Table 8. Independent t -test of post-test results.

To test the hypothesis, the study used a 0.05 significance level in Table 9 . This resulted in the value of 0.003 < 0.05. The evidence rejected the hypothesis because the value applied fell within the rejection domain. Pre- and post-test scores were significantly different for the experimental class. The findings indicate that students’ learning efficiency increased in the experimental class compared to before the experiment. After a successful intervention is promoted, students perform better academically. Before and after the experiment, the correlation test results show a correlation between experimental class scores on both tests with a Sig significance level (2-tailed) less than 0.05, as shown in Table 10 . Table 11 shows a Pearson correlation coefficient of 0.659, which indicates a significant correlation.

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Table 9. Pair samples test of the experimental group.

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Table 10. Pair samples correlations of the experimental group.

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Table 11. Pair samples statistics of the experimental group.

With only one student in the experimental group, the post-test revealed that the control group had an unusually high percentage of students below average, accounting for 18.5% with five students, whereas the control group had only one student, accounting for 3.9%. With 13.8% of the students in the experimental class scoring a 10, the experimental class had more students scoring a 10. Because of the difference in performance between the experimental and control classes at milestones 7, 8, and 9, it can be concluded that many students in the experimental group performed exceptionally well. Meanwhile, most students scored between 5 and 6 points in the control class.

During the post-test, the control group saw a decrease in the percentage of students who received excellent scores and an increase in the number of students who received scores below the average, indicating that students were still having difficulty in solving real-world problems and applying their newly acquired knowledge. Compared to the pre-test, the experimental class tended to increase the number of students who received good grades; specifically, many students who received good grades of 8 or higher saw an increase of more than 10%. The experiment discovered that most students in the experimental class were more engaged and enthusiastic about real-world issues than their counterparts in the control group. The wide variety of mathematical applications found in diverse fields of study has led to the discovery that there is a correlation between the attraction of real-world problems and mathematical interest.

Results of a Survey of Student Opinion and Observation

It was observed that most students in the experimental group studied very actively and enthusiastically. More specifically, they expressed an interest in gaining practical experience as smart diners by enthusiastically contributing ideas and participating in group discussions to handle the given situation and choose the most cost-effective option. As a result, students better understand how mathematics can be applied in everyday life and progress after the experimental class.

The student survey results on Google Form conducted after the lesson demonstrated that students began to enjoy learning mathematics, with more than 75.9% of students completely agreeing and 20.7% of students agreeing in the question “I love to study maths more” following the experiment. Regarding question 6, “I want to learn similar experiential lessons,” in the experiment, 62.1% of students were completely in agreement, and 27.6% of students wanted to learn through hands-on experiences.

To better understand what students were experiencing after the lesson, the researchers also asked question 7 when evaluating students’ attitudes. Student responses to the lesson they had just completed were sought through this activity, which was designed to express their thoughts and feelings about the lesson they had just completed. Based on this, the researchers can see how much students enjoy creating situations and having the opportunity to connect knowledge and experience in order to overcome real-world math problems.

Some answers to the question “Do you have a different opinion about the class?” are as follows.

Student S01: “I really enjoyed today’s class because she made the lesson easy to understand and absorb. I hope you will do classes like this because it makes me feel interesting, fun, and receptive.”

Student S09: “This class is very fun because it gives me much knowledge and helps me love math more.”

Student S15: “The class was very fun, and the teacher spoke very interestingly.”

Student S26: “Tea teacher teaches very easy to understand.”

In regards to the individual assessment of students: based on the criteria established by the researchers and the results of the individual assessment of the students, it was concluded that the students were always enthusiastic and responsible and that they participated in the organization and management of the group with 34.5% of students; the majority of learners performed at a satisfactory level, and the evaluation points were at 3, 4, and 5 in the remaining criteria such as the spirit of cooperation, respect, and responsibility; the majority of students performed at a satisfactory level, and the evaluation points are. The test results after the experience period show that the students have made positive changes. As a result of this study, students’ learning outcomes in mathematics have been oriented toward assessment based on competence with the combination of different assessment methods, including learners’ self-assessments.

When it comes to group assessment, the groups worked quite actively, had lively discussions, and many students were able to manipulate technology quickly and effectively; they shared the screen on their own, prepared PowerPoint presentations, vivid and attractive videos, and shared the screen when giving a presentation with others on their own. It was decided to use the given criteria to evaluate the group evaluation results in content experiment 1. It was discovered that 41.4 and 27.3% of students participate in group activities, respectively, and that 6.9% of students participated in group activities only rarely because they had device problems when learning online, thereby reducing their participation.

In the arithmetic lesson, the students were aware that they were expected to observe, analyze, and consider making appropriate choices for the problem’s requirements. For instance, the students were presented with cases to select the most appropriate one for their needs and circumstances. From there, the students would better understand the value of money, learn how to manage money effectively, and learn how to organize their lives.

As a result of the design project “My Dream City” in geometric lessons in Figure 2 , the researchers discovered a great deal of students imagination in designing and creating ideas, such as designing a city for residents who were bunnies or cities of fun and strength or designing a city for residents who would use utility apps during payment, among other things. As well as demonstrating knowledge through experiential activities and the spirit of cooperation, learners could connect their learning to the design of a great city filled with interesting and modern facilities.

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Figure 2. Product of student work.

This finding demonstrates that incorporating exercises into group activities in the classroom had numerous benefits. For starters, it increased the practicality of learning while still in school. During their time at the school, students were exposed to a great deal of theoretical information about the subject matter they were studying. Indeed, an exercise associated with real-life situations that are related to mathematics assisted students in applying theory to practice more effectively. Furthermore, it encouraged students to take the initiative, be creative, and most importantly, be excited about their learning process. The lecture method imparted one-way theoretical knowledge, whereas active exercises assisted students in applying the knowledge they gained to analyze situations or devise solutions based on the theories they had learned. As a result, students became more involved and interested in the learning process, and nothing stood in the way of their ability to be creative when confronted with a challenge.

Additionally, this method aided the learners in developing teamwork and other skills. As soon as assignments were given out, students were divided into groups to work on them. They had to be given specific tasks to complete the work as a team. As a result, students’ teamwork abilities improved an important skill for them during the learning process and later in their educational careers. Aside from that, skills in analysis, presentation, and problem-solving were developed while defending the group’s points. Teaching students to work in groups provided teachers with diverse experiences and solutions that they could use to enrich their lessons and other students’ lessons.

The teacher’s evaluation: Due to the time constraints, the researchers only conducted group evaluations based on a few criteria. By the evaluation results, up to three groups were performing at a high level, scoring highly on the criteria of positivity, enthusiasm for group discussions, and coordination among group members. All five groups consulted teachers on a satisfactory level. However, only two of the participants made the necessary progress, and only two gave good presentations in the experiment. Groups in the experiments were split into seven, with four groups meeting all the requirements, while three did not sound all that great. As a result, mathematics teachers in schools should employ the same tools used to assess competence: mind maps, a criteria sheet for evaluation, and research products created by students during learning activities. In particular, the math test questions are designed to assess the ability to apply previously learned knowledge and skills to solve real-world problems. Generally speaking, this is considered to be one of the most significant characteristics of the learner’s competency assessment procedure.

The majority of the time, students actively and enthusiastically participated in activities, gaining skills such as mathematical communication, mathematical modeling, mathematical thinking, and reasoning in real-world situations. Aside from that, the design and organization of mathematical experiential activities aided students in developing necessary qualities through the delivery of content, messages, and integration that teachers had communicated. Specific to group activities, personalization was encouraged, whether in person or online learning, regardless of the setting. When students participated in hands-on activities that were related to arithmetic and geometry topics, their learning outcomes had shown to be better. Organizing activities in various situations provided the teacher with more hands-on experience. Aside from that, the teacher gained a better understanding of the difficulties and advantages faced by students and have launched appropriate activities as soon as possible, promoting students’ abilities and qualities while also fostering stronger relationships between students and teachers, among other things.

Following the experiment, the researchers were able to obtain results that were consistent with the goals that had been established. In line with the authors’ research ( Mayoral-Rodríguez et al., 2018 ; Wynn, 2018 ), additional results significantly impact student knowledge acquisition, comprehension of mathematical sequences, and application of that knowledge from experiential activities were planned and implemented. Researchers hypothesized that partly because of the short duration of the experiment, the students did not adjust well to the new learning method, which helped explain the average impact of the effect on the students’ average score on the test. Although the effect was not particularly large, this was a promising indication of the positive impact of experiential learning on student achievement in mathematics. Analyzing student work and tests revealed that their analytical and computational abilities had improved, as had their capacity for applying what they had learned in the classroom to the real world ( Davidovitch et al., 2014 ; Mayoral-Rodríguez et al., 2018 ). Students’ interest and motivation to learn increased due to these activities observed in the classroom during experiential activities ( Weinbern et al., 2011 ). Those findings can be explained by assuming that students’ learning activities were actively engaged due to participating in group work activities. These activities included instructing students on the skills of assigning tasks, debating, and reaching consensus while working in teams ( Weinbern et al., 2011 ; Venkatraman et al., 2019 ). In addition, it was documented that the cooperative learning method improved students’ academic performance in mathematics ( Algani and Alhaija, 2021 ).

Furthermore, the student survey results revealed that students had a positive attitude toward hands-on mathematics experiences related to arithmetic and geometric topics. This outcome is also in line with what was discovered by Pambudi’s (2022) investigation. This author has concluded that using outdoor learning methods to teach geometry to elementary students positively impacts their motivation and learning achievement. Based on students’ responses to survey questions, it appears that they were aware of the importance of experiential activities in the formation of knowledge and their ability to apply newly acquired knowledge to real-world problems. At the same time, students could see how far they had come in terms of mathematical reasoning and real-world problem-solving ( Manfreda and Hodnik, 2021 ). Accordingly, students could provide valuable feedback through this outcome. After this feedback, students believed that this new learning method was effective enough to participate actively in future experiential activities to gain additional knowledge ( Habib et al., 2021 ). This result also explains why many students expressed an interest in using this method in future lessons.

It has been discovered through observation and analysis of experimental teaching and assessment results that teaching through experiential activities is highly effective and feasible. After being exposed to experiential learning activities associated with two topics in arithmetic and geometry, it has been demonstrated that the experimental class achieves higher test scores than the control class. The students in the experimental class had a positive attitude toward learning and were eager to learn about lessons that included content-related math experiences that they could apply in their real-life situations. As a result of students’ active and enthusiastic participation, they developed skills in mathematical communication, modeling, thinking, and reasoning in real-life situations. Additionally, through the content, messages, and integration those teachers convey to students, the design and organization of experiential activities aid students in developing necessary qualities that they will need in the future. More specifically, personalization is encouraged in group activities regardless of the setting, face-to-face or online. Also, students’ learning outcomes improve due to their involvement in experiential activities. Students benefit from the knowledge and skills teachers have gained from organizing activities in a variety of settings.

Furthermore, teachers better understand the difficulties and advantages students face from this position. It appears that they can offer appropriate activities quickly, promote students’ abilities and qualities, and build stronger bonds between students and teachers. In addition, experiential math activities generate a great deal of information and data from students’ observations and observations. Teacher digital competencies are also required as a result of this in order to effectively analyze and manage data, which includes product analysis and evaluation, as well as student learning outcomes ( Pozo-Sánchez et al., 2020b ).

Experiential activities in math assisted students in developing personal qualities and competencies. From here, they gained the ability to adapt to various living, learning, and working environments, adapt to the changes that modern society brings; and organize their lives, work, and management. Moreover, they can develop an interest in a career related to mathematics and make decisions about choosing a future career; develop a training plan to meet the requirements of this career, and contribute to society as productive citizens. Consequently, the mathematics program must be open and flexible in order for educational institutions and teachers to choose the content actively, methods of instruction, location of operations, and hours of operation that are appropriate for their particular circumstances and conditions. the principle of ensuring educational goals and requirements for quality and competence at every level and in every classroom.

In addition to the findings, the research has some limitations. Time spent in experiments was short; activities had to be planned to fit into an already-short learning program, which was constrained in its duration. As a result of some students’ inability to adjust to the new way of doing things, some initial confusion has been on their part. Thus, some of these students’ educational outcomes are negatively impacted. Several other studies have also found that this is a problem ( Cranton, 2011 ; Kolb and Kolb, 2017 ).

These findings and limitations suggest that future research on experiential learning in mathematics should consider long-term planning that includes both inside and outside classroom activities and an interdisciplinary approach. Mathematical topics in algebra, calculus, statistics, and probability can be studied at various levels of study. When combined, experiential, problem-based, and project-based learning should be used greatly in educational settings. Another trend to consider is incorporating technology elements into the classroom, becoming increasingly popular as science and technology advance ( Tran et al., 2020 ).

The studies on improving teachers’ theoretical and practical knowledge of experiential learning are unnecessary to improve the effectiveness of this method for teaching mathematics and general education. According to this point of view, Jay and Miller (2016) present three models of teacher training programs that assist students in being fostered in the theory and practice of experiential learning, which is consistent with this viewpoint. The authors identify the most generalizable aspects of these programs, identify the factors that lead to the breakdown of theory and practice, and propose more sustainable models. Math projects, STEM, and competitions can all benefit from the integration of arithmetic and geometry, which is another area of study worth exploring further in the future.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by Institutional Ethics Committee of the School of Education at Can Tho University in Vietnam. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

Thanks to all of the students who took part in this investigation, as well as the faculty and staff who assisted with it.

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Keywords : arithmetic, experiential learning, geometry, mathematics achievement, students’ attitudes

Citation: Uyen BP, Tong DH and Lien NB (2022) The Effectiveness of Experiential Learning in Teaching Arithmetic and Geometry in Sixth Grade. Front. Educ. 7:858631. doi: 10.3389/feduc.2022.858631

Received: 20 January 2022; Accepted: 07 March 2022; Published: 28 April 2022.

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Copyright © 2022 Uyen, Tong and Lien. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Duong Huu Tong, [email protected]

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Case Study in Experiential Learning - From Chaos to Order: Sensemaking with the Interactive Timeline Tool in Architecture and Civil Engineering Studies

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Many students have a difficult time memorizing and retaining facts that seem in-consequential to their everyday lives. Experiential learning techniques can make learning more interactive and meaningful, and thereby easier for students to comprehend and retain the material long after the course has ended. This paper presents a longitudinal case study of an interactive teaching method developed for the history of architecture courses over several years (2012–2019), which are compulsory for civil engineering, architecture and landscape architecture students. The professions related to the field of architecture are creative in nature, therefore learning methods based on experience and visual memory are very suitable to teach within these professions.

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Nutt, N., Salmistu, S., Meitl, C., Karu, K. (2021). Case Study in Experiential Learning - From Chaos to Order: Sensemaking with the Interactive Timeline Tool in Architecture and Civil Engineering Studies. In: Auer, M.E., Rüütmann, T. (eds) Educating Engineers for Future Industrial Revolutions. ICL 2020. Advances in Intelligent Systems and Computing, vol 1328. Springer, Cham. https://doi.org/10.1007/978-3-030-68198-2_8

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Experiential and Case-Based Learning

Case-based teaching.

Case-based teaching strategies use real-life examples to offer a shared learning experience. It may be difficult for students to experience real-world situations together. These scenarios, provide a common “experience” so that students can solve problems, make decisions, and generally think critically together. Many case studies are stories, designed to engage students in research and analysis of a specific problem or set of problems. Case studies tend to work well in the online/hybrid learning environment.

Examples and Resources

  • National Center for Case Study Teaching in Science, University of Buffalo  – A comprehensive site for all scientific disciplines, links to numerous articles on case study teaching in science plus an extensive collection of cases.
  • Teaching with the Case Method from Carleton College   – Several examples of teaching with case studies.

Major, A., & Viswanathan, R. (2019). Create a case method group activity to engage students in critical thinking. In A. deNoyelles, A. Albrecht, S. Bauer, & S. Wyatt (Eds.),  Teaching Online Pedagogical Repository . Orlando, FL: University of Central Florida Center for Distributed Learning.  https://topr.online.ucf.edu/create-case-method-group-activity-engage-students-critical-thinking/ .

Experiential Learning

Experiential learning is an activity-oriented strategy rooted in experiences. Personalized reflection on experiences and the formulation of plans to apply learning to other contexts are critical factors. Experiential learning is effective in providing opportunities for students to engage and apply academic understanding through hands-on experience. There are many methods and tools that can be useful when employing experiential instruction such as simulations, field experience, games, storytelling, and surveys.

Field Experience

Field experience is an excellent way to bring real-world experiences back to a course. Students are often asked to document their experiences and observations and share reflections. For example, students majoring in art history frequently visit local museums to view examples of artwork presented in class.

Games/Simulations

Games and simulations allow learners to practice skills, acquire knowledge and learn concepts while having fun. Tools such as Kahoot can be used by faculty in order to conduct formative assessment in the form of an online, in-class game. In addition, technologies such as SimCity  provide virtual environments for students to explore, like  Center of the Cell and the many simulations found on  Phet  can offer students experiences that might be impossible in real life.

Role-playing gets students to explore acting out different scenarios or characters. For example, in a business class students may act as a buyer or seller of a specific product. In doing so, students develop a better understanding of the concepts they’ve learned by testing them out.

Ertmer, P. A., & Koehler, A. A. (2014). Online case-based discussions: Examining coverage of the afforded problem space.  Educational Technology Research and Development , 62( 5 ), 617-636.  https://doi.org/10.1007/s11423-014-9350-9

Kolb, D. (1984).  Experiential learning: Experience as the source of learning and development . Englewood Cliffs, NJ: Prentice Hall. Retrieved from  https://www.pearson.com/us/higher-education/program/Kolb-Experiential-Learning-Experience-as-the-Source-of-Learning-and-Development-2nd-Edition/PGM183903.html

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Experiential Learning From a Pilot Study of Insider Research: Interviewing International Doctoral Students

  • By: Myra C. Y. Lee
  • Product: Sage Research Methods Cases Part 1
  • Publisher: SAGE Publications Ltd
  • Publication year: 2018
  • Online pub date: December 17, 2018
  • Discipline: Education
  • Methods: Pilot studies , Cross-national research , Research design
  • DOI: https:// doi. org/10.4135/9781526474797
  • Keywords: Australia , experiential learning , foreign students , students Show all Show less
  • Online ISBN: 9781526474797 Copyright: © SAGE Publications Ltd 2019 More information Less information

The research design of a study involves continual refinement through experiential processes. This case study demonstrates the importance to novice researchers of conducting a pilot study to test the feasibility of the study’s research design and practice data collection (in this instance, interviewing skills). The case study is based on an investigation of the international transition experiences of doctoral students of Chinese heritage studying in Australia. As a fellow doctoral researcher, I had assumed an insider position when interacting with my participants. The study did not proceed as anticipated and problems with trustworthiness emerged that could have eventually led to failing the doctoral degree. Resulting from the pilot study, I re-evaluated my researcher stance and in particular, my underlying epistemological positions. Through four reflections, the common issues experienced by novice interviewers are presented and advice is suggested to improve interviewing skills. The lessons learnt from the pilot study and remedial actions taken that saved the study are also described. In sum, the assertion made in the case study is that while conducting a pilot study may seem time-consuming, it is a vital opportunity for experiential learning that can save novice researchers from encountering future misadventures. Thus, conducting a pilot study is an integral part of shaping research design and represents a hallmark of good methodological practice.

Learning Outcomes

By the end of this case, students should be able to

  • Understand the importance of conducting a pilot study when designing studies with unclear parameters
  • Differentiate between the different interview approaches
  • List the different epistemological positions underpinning each interview approach
  • Describe the common issues facing novice interviewers
  • Evaluate the need for refining the study’s research design and deciding what changes to make

Research design is seldom formulated in a linear manner. In the first year of my doctoral studies, I attended many study support seminars conducted by professors or students who were more advanced in their research studies in the hopes of avoiding pitfalls. I expected to develop an understanding of what research involved, what it meant to conduct research, and who I was as a researcher. In particular, one professor’s advice proved visionary. He opined that the thesis represented an account of everything that went well in a study and elaborated, “No one writes about their failures or what went wrong, otherwise, you invite too much critique from the examiners. Instead, you go back and fix things up and try again.” He added that the thesis should read as a “success story,” an uninterrupted piece of writing that glosses over the disasters, although these disasters and the remedial actions should be alluded to in the Limitations.

Lulled into a false sense of competence from attending seminars and extensively reading research design textbooks, I had assumed that such disasters would never befall me. I was unprepared for the complexities of formulating a feasible research design. Research design turned out to be a messy process that involved the back and forth trialling and refining of methods, which Berg (2004) calls the Spiralling Research Approach.

One of the first tasks in research design involves identifying the research problem and according to Pryor (2010) , doctoral thesis examiners are looking for significant and original contributions to knowledge, which usually occur through identifying gaps or reinterpreting existing knowledge. Identifying a gap, my study investigated the under-studied population of pan-Chinese international students in the under-studied area of doctoral study support. I wanted to explore their transition experiences when relocating from one sociocultural context and immersing in another. At that time, I had neglected to consider that because the parameters of a gap study are ill-defined, a pilot study is required as Persaud (2010) asserts. Schreiber (2008) explains that a pilot study is a feasibility or small-scale exploratory study that can form part of a larger study. Although conducting pilot studies can be time-consuming, the consequences of persisting with an ill-conceived study are dire and may ultimately even result in failing the doctoral degree.

Essentially, a pilot study illustrates Kolb’s (1984) Experiential Learning Cycle which comprises the four sequential stages of concrete experience , reflective observation , abstract conceptualisation , and active experimentation . In short, Kolb’s experiential learning suggests gaining actual experiences and engaging in reflections to distill realistic lessons that are incorporated into the next experience. Such lessons cannot be learnt from any textbook or seminar. This case study is loosely based on Kolb’s experiential learning cycle. Beginning with a background of the study’s context, the thinking that underpinned the shaping of the research design is discussed. Next, research practicalities are elaborated before examining the method in action through four reflections of common issues experienced by novice interviewers. The practical lessons are shared and concluded.

Context of the Study

According to the United Nations Educational Scientific and Cultural Organization (UNESCO), international students from China have increasingly been going abroad to study over the last decade, and their top destinations include the United States, the United Kingdom, and Australia ( UNESCO Institute for Statistics, 2018 ). As an under-studied cohort when I began my doctoral studies, the pilot study aimed to explore the experiences and influences of international postgraduate research students of Chinese heritage as they transitioned to study in Australia and in particular, their career decision-making process.

With a background in career guidance, I was drawn to investigating the impact of culture on careers, and in particular, whether a cultural “Chineseness” would affect career decisions. Tu (2005) describes Chineseness as a common historical and cultural identity found in Cultural China, which consists of China (including its Special Administrative Regions of Hong Kong and Macau) and its periphery of overseas Chinese societies steeped in Confucian traditions including Singapore and Taiwan. The pilot study drew on a pan-Chinese sample of 12 students across these societies that were recruited through student social networks. Eventually resulting from the pilot study for reasons that will become apparent, the target population was narrowed to a homogeneous cohort of international students from China.

Shaping the Research Design

Before crafting the research design, Higgs (2001) advises that the researcher needs to consider, first, the study’s research paradigm which is its philosophical underpinning, and second, their researcher stance. In relation to the research paradigm, Brannen (2004) asserts that the choice of research design and method should depend on the research paradigm as well as the nature of the research investigation. Aligned with the nature of my investigation which was to explore lived experiences, interpretivism was chosen which views knowledge as an interpretation of the social world through the construction of meaning ( Higgs, 2001 ). Congruent with interpretivism, I adopted a qualitative methodology to comprehend the sociocultural context in which my participants resided. Yin (2016) explains that qualitative methodologies assist in the investigation of social thinking and behaviour through the in-depth examination of how individuals make sense of their social structures and negotiate their social roles.

Related to researcher stance, Schensul (2012) advises that when crafting the research design, aside from methodological issues such as formulating research questions and selecting appropriate theoretical frameworks, researchers should also consider their position on data collection. Blaikie (2007) explains that this stance or researcher positionality concerns how researchers intend to approach their participants, for example, whether to approach participants as insiders or outsiders, and whether to approach them as experts or as learners. As an overseas Chinese born outside China but who identifies as culturally Chinese and speaks Mandarin as a second language, I have assumed an understanding of Chineseness. As such, I believed that I was an insider to my participants by virtue of our common Chinese heritage. Simultaneously, as an insider who was studying a doctoral degree outside of my home country, I was also a fellow learner.

Resulting from the pilot study, the study’s research paradigm and my researcher stance were refined. In realising that interpretivism encompasses a wide range of epistemological positions which explain the nature and derivation of knowledge ( Higgs, 2001 ), I developed a more nuanced understanding of epistemology and shifted toward a more “structured” version of interpretivism.

Research Practicalities

Prior to seeking ethical clearance, the participant selection criteria were devised. To avoid imposing overly restrictive criteria in case I could not find enough participants, the criteria included students enrolled in Research Higher Degrees (RHD), that is, the Masters by Research and Doctor of Philosophy (PhD) programs. I would then have the leeway during recruitment to seek out only PhD students from societies included in Tu’s (2005) Cultural China. As such, the participant selection criteria were (1) born outside Australia, (2) of Chinese descent, and (3) currently enrolled in RHD programs in Australia.

After ethics approval, I considered which networks were easily accessible and would readily yield suitable participants. This exemplifies convenience sampling which Creswell (2012) defines as a sampling technique that selects participants who are easily accessible and willing to participate. Furthermore, I employed purposeful (or sometimes called purposive ) sampling, which according to Creswell, is the deliberate selection of participants and sites to investigate a phenomenon. Participants were primarily recruited through two sources: my social network of doctoral students as well as placing advertisements on the social media sites of university student associations. Eventually, 11 students were recruited through my personal networks and only one student through the student associations. Of the 11 students, two were acquaintances and the rest were introduced either by mutual friends or existing participants. Recruiting new participants from the recommendations of existing participants is called snowball sampling ( Creswell, 2012 ). The sample saw nearly equal representation from societies included in Cultural China: Taiwan, Singapore, and China (including Hong Kong). All were enrolled in PhD studies in a university in Australia.

Method in Action

Semi-structured interviews were utilised, which Qu and Dumay (2011) describe as a form of qualitative interview where interviewers have the flexibility to explore emerging areas of interest by deviating from a prepared protocol that guides the interview structure. Qu and Dumay state that semi-structured interviews lie in the middle of a continuum. On one end, unstructured interviews are conversational and shaped by the interview context. Open-ended interview protocols are employed to explore the interviewee’s perspective. At the opposite end of the continuum, structured interviews contain prepared closed-ended questions that limit the number of responses. Below, I elaborate on what went well and what did not go well in the interviews.

In terms of what went well, I followed the advice of King and Horrocks (2010) and prepared for the interviews in three ways. First, to facilitate audio recording, interviews were arranged in conducive and mutually agreeable locations such as meeting rooms that were quiet and away from distractions. Privacy was crucial as many of my participants were interviewed on campus where they could be identified by other colleagues ( Lee, 2014 ). Furthermore, interviews were held after office hours or on weekends with minimal human traffic. Second, I prepared an interview protocol and sought the feedback of my two PhD supervisors. Third, I prepared for contingencies such as bringing a backup recorder and a notepad for handwritten notes should the main recorder malfunction (which happened during one interview). The notes proved very helpful during transcription as a reminder of the interview conditions by documenting the general line of questioning (e.g., provided clues when a word was inaudible) and the progress of the interview (e.g., supplemented the recording when participants gave important non-verbal cues).

Regarding issues that arose, in the bustle and excitement of interview preparation, I had neglected one key element: my interviewing skills. As a former recruiter, I had assumed some expertise in interviewing without realising that the nature of job and research interviews was different. To make matters worse, I had not pre-tested the interview protocol believing that semi-structured interviews afforded flexibility to change questions and explore emerging themes.

It is not unusual for novice interviewers to encounter issues during interviews and three sets of authors offer guidance: Josselson (2013) , King and Horrocks (2010) , and Qu and Dumay (2011) . I outline four issues experienced as a novice interviewer: stifling the discourse, forced rapport-building, interviewer bias, and managing interview responses and interactions. Collating advice from the three sets of authors, I reflect on excerpts from the interview transcripts. All names are fictitious to protect participant identities.

Reflection 1: Stifling the Discourse

King and Horrocks (2010) suggest using probes to obtain rich descriptions by elaborating (asking for more information), clarifying (explanation of specific words or terms that interviewees used), and completing (encouraging interviewees to finish their stories or opinions). The interviewer should remain unobtrusive and allow interviewees to drive the narrative, for instance, by not interrupting or completing their sentences, or asking questions out of curiosity. Importantly, questions and probes must be carefully phrased. Interviewers should avoid asking four types of questions: (1) multiple or double barrel questions which cause confusion as two or more questions are asked together and interviewees do not know which one to respond to, or more often, only address one question; (2) one-sided questions that only explore one side of an issue, and which appear as a rapid series of questions that confirm or disconfirm the interviewer’s prior assumptions; (3) leading or closed questions that anticipate a particular answer (usually a “yes” or “no”); and (4) loaded questions that contain assumptions.

In the following excerpt with Mary, she was narrating the process of selecting a university major. I disrupted and redirected her flow because I was eager to steer the conversation toward a discussion of her interests.

Mary: I was forced to choose that [major]. (Laughs) Because you know, in mainland China, we have entrance examination to universities … and I got a relatively (emphasises) high score … but I chose the wrong one [university]. (Laughs) I mean I choose …

Interviewer (interrupts): Actually what do you mean “chose the wrong one?”

Mary: Oh, the wrong one … before I graduated [from senior high], students would choose the university before they know their final score. So they choose [based on a] rough calculation of their scores. So it was fair or reasonable. But when I graduated, things changed. Students would know their scores [before choosing] the universities… . A lot of students would choose the safe [choice]. I did the same … for example, I chose the university in which the enrollment score was 20 or 30 points lower than mine but …

Interviewer (interrupts): Couldn’t you re-choose?

Mary: Actually I had several universities listed. But if you can’t go to the first choice, your [score] for the second will be reduced [by a] further 20 points. And if you can’t get [into your] second choice, your marks [for your third] will be reduced by a further (emphasises) 20 points… . So if you can’t get [into your] first choice, you may have no other choice and finally, you have to wait for the universities to arrange. That means if this university doesn’t have enough [enrolled] students, they go to the big (emphasises) pool to find the students who have not been enrolled in any university but their scores are higher than the [university’s] entrance [examination] score so …

Interviewer (interrupts): So did you do that? Did you put yourself in that pool to appeal to that first university that you wanted to go to?

Mary: Automatically (emphasises) … This is not (emphasises) something you can choose. Yeah, you will be automatically put in that (laughs) big pool and then you wait to be chosen by the universities … Maybe it’s so hard for you to understand but it’s a common practice in mainland China.

Mary’s annoyance and frustration were obvious and she became dismissive, probably because she felt that I had not even attempted to understand her. In addition, as Mary and some other participants were second language speakers, I thought that I was being helpful in offering responses and options. In retrospect, the effect of constantly interrupting and asking multiple questions in quick succession seemed like an interrogation and she finally yielded and gave up. A better clarification technique would be active listening as explained in the next reflection.

Reflection 2: Interviewer Bias

The interviewer must maintain neutrality and avoid any verbal cues that imply partiality such as offering opinions or praise. If solicited, the interviewer should generalise or normalise the interviewee’s opinion or experience. A judgmental attitude may inadvertently lead interviewees to suppress their true opinions or feelings, and either offer opinions that they think the interviewer is fishing for, or state socially acceptable opinions. Similar to the first reflection, leading questions should be avoided as interviewees may feel coerced into following the interviewer’s direction or are distracted from their original direction. Interviewees may also become defensive if they perceive that the interviewer does not agree with or support them.

When interviewing Molly, I was keen to understand the influences underpinning her interest in science. She had followed a close friend to select Chemistry as her undergraduate major so that they could attend classes together.

Interviewer: I’ve very curious about the interest. I mean other than the friend, there must (emphasises) have been something …

Molly: Hmm, it’s just when I started doing it, I seemed to be doing way better? Because when I was younger … I would fail subjects but I didn’t know why but when I went to Chemistry, my grades just seemed to get better? … I could never understand why, and maybe that’s why I did [choose Chemistry]. Or maybe because it’s just interesting and … I really don’t know.

Interviewer: But I guess you would have done pure science in secondary school?

Molly: Yeah, I did pure [science and arts] subjects… . But when I went to junior college … it wasn’t pure science or pure arts again. But then I dropped out of Chemistry because I didn’t go for lessons and wasn’t happy and …

Interviewer (interrupts): Because I guess you weren’t interested in it?

Molly: Ah I did like Chemistry? The reason why I choose this junior college was because there was Chemistry and History. I really liked Chemistry but I also liked History but no junior college offered both.

Molly seemed forced to justify her major because I implied that following a friend was a trivial reason. I also disrupted her flow and insisted that her choices resulted from her subject interests, which was an imposition of my interpretation. She seemed coerced into offering a credible reason about getting good grades and following her interests, which seemed to satisfy me even though she seemed unconvinced.

Furthermore, to satisfy my curiosity, I shifted the discussion from her interest in Chemistry to her subject choice in secondary school which was irrelevant. I had interpreted Molly’s influences based on my own schooling experiences. The interviewer should employ empathetic and active listening through summarising and paraphrasing with the same language that the interviewees use (mirroring), without putting words in their mouth or adding their own interpretations. A better probe would have been to mirror her words: “Oh, what did you find interesting?”

Reflection 3: Forced Rapport-Building

The interviewer should strive to create positive feelings and build rapport early on in the interview. Once an environment of trust has been established, interviewees are relaxed and feel safe to share. In this excerpt, Vaughn had earlier shared that he intended to return to Taiwan after his PhD studies to look after his parents. I asked him about finding work and suggested that he might return to his former employer.

Interviewer: You used to work in a hospital so will you be contacting your ex-boss?

Vaughn: Err yes, I think that I will let him know I’m going back to Taiwan, but the hospital is located in City A, and my parents are living in another part of Taiwan. So I think it’s very hard to go back to the original hospital.

Interviewer: City B?

Vaughn: No, in my hometown, City C.

Interviewer: Oh. So that means that when you were working in that hospital, you were actually away from your parents?

Vaughn: Yeah, yeah, so …

Interviewer: So how come last time you could work away from your parents, and now you want to go back to [your hometown]?

Vaughn: Because in those days, they had the ability to take care of themselves. But now, they’re getting old. I think the best (emphasises) is to live with your parents and look after them.

Interviewer: OK, very filial of you.

Vaughn: Pardon?

Interviewer: [repeats in Mandarin]

Vaughn: I don’t think that way, I just think … because they provide some benefit for me, they allowed me to study overseas. I’m 36 [which] means I need to have a family. I’m single and can study overseas without any family pressure. So I think I’m lucky… . They gave me that chance [to] carry on my dream, so after that, I think I can [repay] my parents.

With Vaughn, first, I tried being helpful by suggesting potential employment opportunities and praising him. Aside from introducing interviewer bias as discussed in the previous reflection, I thought that offering praise would help create a positive atmosphere. Second, from my literature review, I sensed that the theme of filial piety was emerging. Confucianism describes filial piety as a sacred duty to look after one’s aged parents to repay them for the debt of upbringing. I tested this theme with Vaughn, which is also an imposition of my interpretation and should have been avoided.

Furthermore, interviewers should avoid contributing their own stories which can distract the interviewee. In this excerpt, Sarah was collecting data outside Australia and I had asked about her access to research resources. We discussed different libraries to borrow books and she asked for assistance.

Sarah: The only text that I probably might need to look for eventually is looking at Asian culture and … (laughs). Maybe I could ask you more about this.

Interviewer (laughs): Sure!

We then spent a few minutes discussing our respective studies and I helpfully suggested alternate resources, which shifted the conversation away from exploring her transition experiences. After this, we moved on to a new question regarding her post-PhD plans. On one hand, my suggestion of doctoral resources helped in establishing rapport, by building a sense of “insiderness” with Sarah regarding our common doctoral challenges. On the other hand, I had been distracted and had also distracted her. A more skilled interviewer would have returned to the interview protocol and continued probing her transition experiences.

With the excerpts of Vaughn and Sarah, although I had intended to assume a learner stance, in fact, I was assuming an expert stance by imposing my expertise as a career counselor and PhD student who was more advanced in her studies. Lagesen (2010) argues that to build rapport, interviewers should seek to reduce the power distance between themselves and their interviewees. Lagesen elaborates that interviewers should not expect to “mine” (p. 129) information from the interviewee but should engage in a mutual “trade” (p. 129) of information. Rapport building should be natural, as Sarah’s excerpt illustrates, where the request for help occurred during the course of conversation. In contrast with Vaughn, I imposed my assumed expertise, which effectively reinforced our power distance. In both instances, I should have persisted with the learner stance by acknowledging that I knew little about their individual situations but was keen to learn more, and then humbly and patiently seek clarification.

Reflection 4: Managing Interview Responses and Interactions

The three earlier reflections show that much occurs in the interviewer’s mind during an interview. Due to interviewer fatigue, the same questions may be asked in different ways or incorrectly phrased. With the following three participants, my intention was to determine their attendance at orientation programs and their impressions. Noting each participant’s personal circumstances, I customised the questions. International students are encouraged to attend an orientation program specially organised for them but Australian students do not need to attend an orientation. Furthermore, international students who speak English as a second language are also entitled to attend free English lessons. Finally, the graduate school also runs writing and reading skills workshops for PhD students.

Eric had migrated to Australia as a child and was educated in the Australian education system. I asked him about the graduate school workshops: “When you started university here, did you attend any of the orientations [e.g.,] teaching you how to write?”

Vaughn had completed a year of PhD studies in the United Kingdom before giving up. He subsequently arrived in Australia to restart his PhD studies. He spoke English as a second language. I queried: “Did you go for the international student orientation? … What about … English? Because I heard they also have English classes?”

Similar to Vaughn, Inoua was an international student who had arrived in Australia for the first time. However, unlike other participants, his wife accompanied him to Australia. Even though English was his second language, I had forgotten to ask about English classes as I was pre-occupied with finding out about his wife. My questions were: “When you first arrived in Australia, did you attend any of those orientations for international student? … Do they allow family members to participate? Did your wife attend?”

In the above examples, the same experience of orientating to PhD studies in Australia was asked in different ways based on participants’ life situations that only became apparent during the course of the interviews. For instance, I sometimes asked about attending English classes but neglected to at other times, and could not ask first language speakers about attending such classes which would be deemed insulting. Likewise, some students had migrated to Australia at a very young age or others had arrived in Australia for undergraduate studies. They were accustomed to the Australian education system and would have experienced the transition to PhD studies very differently from those who had just arrived in the country and were undergoing cultural shock.

The varied life situations revealed that my interview protocol was insufficiently comprehensive and had neglected areas to probe such as participants’ experiences of the different types of orientations. This highlights the importance of trialling the interview protocol to check for depth of coverage. In addition, a more experienced interviewer would have obtained demographic information to discern the participants’ life situations, for instance, during informal conversations when arranging the interview appointments. Aside from establishing rapport, these conversations would help the interviewer to tailor the questions to different life situations while maintaining sufficient coverage.

Practical Lessons Learned

The four reflections concern trustworthiness which qualitative studies emphasise. According to Lincoln and Guba (1985) , the quality criteria for qualitative studies differ from quantitative studies. Regarding the former, trustworthiness involves whether readers can trust the findings of a research study, due to steps that the researcher may have taken to bracket researcher bias and to check that researcher interpretations have faithfully documented the interviewees’ constructions. To achieve trustworthiness in semi-structured interviews, Morse (2015) suggests the use of several measures including (1) “thick description” (p. 1218) which relates to collecting sufficient data until saturation is achieved, that is, data collected from later participants overlap with existing data and no new data emerge; (2) bracketing researcher bias which involves acknowledging the introduction of unconscious bias that may skew the findings or re-confirm the researcher’s pre-conceived notions and assumptions. As researcher bias can never be eliminated, Alvesson (2003) urges researchers to reflexively consider their impact on the social world of the interviewee during the interview, by being mindful of its influence during analysis and employing healthy “skepticism” (p. 28) when writing up the findings; and (3) negative case analysis where both sides of an issue should be explored and addressed (e.g., Did the participant feel happy or sad? Why? If she felt sad, what would have made her happy? If she felt happy, what would have made her sad?).

From the four reflections, three lessons were learnt that augmented my main study in several ways to improve trustworthiness. First, an understanding of epistemology is necessary. As Reflections 2 and 3 showed, my researcher bias crept in when assuming an insider and expert position. Cognisant of being friendly and building rapport to create a positive and safe environment to share in, I thought that I was being helpful by providing options and paraphrasing. Also in Reflection 3, I was checking for emerging themes. However, I had mistakenly assumed a romanticist position by going wildly off-script when I should have assumed a localist position but increased the interview structure by asking more consistent questions or probes. Alvesson (2003) explains that underlying epistemological positions influence qualitative interviews. On one end of the spectrum, the romanticist position seeks a deep understanding of the participant’s inner world. This is achieved through conversation with the interviewee, which aligns with unstructured interviews. In the middle of the spectrum lies a localist position, which views interviews as a collection of situational data specific to that participant in that particular social context and is congruent with semi-structured interviews. On the opposite end of the spectrum, the neopositivist position describes an attempt to find a true reality, which offers interviewees several response options and is consistent with structured interviews.

The second lesson learnt is related to the first. Evident from Reflections 1 and 4, I felt a loss of control during the interviews, which Josselson (2013) explains, was due to my anxiety. In thinking that I was allowing my interviewees to dictate the direction, I was ironically controlling the direction. My mind was cognitively overloaded with the script of upcoming questions while ensuring that I probed deeply. The interviewer has to ensure that the interview protocol is adhered to, formulate effective and appropriate follow-up probes, ensure that the interview is on track, while continuing to build rapport. Simultaneously, logistical issues need attention, such as ensuring that the recorder is still functioning or listening out for potential distractions in the environment. For novice interviewers, the interviewer’s mental load can be overwhelming and lead to inattentiveness. Exacerbating the issue, interviewees become frustrated or annoyed when the interviewer does not seem to be listening, and some interviewees then transform into difficult and unresponsive interviewees.

Resulting from the first two lessons, a more structured approach to semi-structured interviewing was adopted for my main study. Interview questions were supplemented with multiple probes to cater to different possible situations. This removed my mental load as I simply adhered to the interview protocol and chose the relevant probe. To complement the comprehensive preparation of probes, I also pre-tested the protocol to check for depth of coverage and breadth of emergent issues. For interview practice, I selected two international Chinese students who spoke English as their second language. After their interviews, I requested feedback on my interviewing style and more importantly, their comprehension of the questions. Based on their answers, the questions were amended and more probes were written to cover unanticipated areas.

The third lesson pertains to sample selection. Reflection 4 revealed the sample’s diversity. To simplify data collection for a novice interviewer, the sampling selection criteria were tightened to produce a more homogeneous sample by focusing on international PhD students from China. The main study’s selection criteria were students who were (1) born in China, (2) had never studied abroad, (3) had arrived for the first time in Australia to commence PhD students, and (4) were presently enrolled in full-time PhD studies in Australia. Furthermore, to help in comprehending participants’ life situations before the interview and adapting the probes accordingly, a short demographic survey was circulated with the information sheet and consent form to potential participants, which would also facilitate shortlisting. This survey captured details including birthplace, time spent in Australia, enrollment status, study discipline, and year of study. Similarly, informal conversations were held at the start of the interviews but before switching on the audio recorder to build rapport and comprehend life situations.

And Finally, a Twist in the Tale …

The introduction mentioned that research is seldom designed in a linear and progressive manner but involves continual advancing and retreating to produce the final desired result. As a final revelation, I have neglected to share that the present case study was not initially conceived as a pilot study. Instead, it was originally part of the main study and I had enthusiastically embarked on interviewing while expecting a smooth ride. During data analysis, unforeseen difficulties emerged due to the lack of coverage in some areas and insufficient probing. My two very experienced PhD supervisors dug into the interview data and asked me to critique my transcripts by reflecting on the interviewing style and interviewee reactions, similar to the process undertaken in the four reflections. Half a year was wasted and I had to restart ethics clearance and data collection for the new main study. Had I persisted with the original research premise presented here, I might have failed the thesis examination because the method would not have been deemed sufficiently trustworthy.

In sum, my experiential learning cycle involved the process of experiencing, reflecting, distilling lessons, and applying them to a new context. Textbooks could not have sufficiently prepared or taught me such lessons. As Schreiber (2008) asserts, pilot studies are the hallmark of a sound methodology because they provide the opportunity to take corrective or preventive actions as well as refine the research problem, while allowing novice researchers to practice their data collection skills. With the benefit of hindsight, I would have trialled the interview protocol and practiced my interviewing skills with up to three participants, before conducting analysis to confirm the feasibility of the data. Had I known about the importance of conducting a pilot study beforehand, a lot of time, headache, and heartache could have been prevented.

Exercises and Discussion Questions

  • 1. What types of research studies might necessitate the use of pilot studies? In terms of your research study, what might be the advantages and disadvantages of conducting a pilot study?
  • 2. Name and explain some common issues with interviewing that novice interviewees may experience. What might be the consequences of each issue?
  • 3. Differentiate between the different interview approaches: structured, semi-structured, and unstructured interviews. Describe the type of research study that might suit each interview approach.
  • 4. Describe the epistemological positions underlying interviews. Thinking about your research study, which epistemological position might suit yours and why?

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A comprehensive guide on Experiential Learning and its tools

Considerable advancement in the domain of science and technology is the most visible characteristic that describes our life in the modern-age. We’ve made tremendous progress as human beings and this progress is a continuous process. This progress encompasses a diverse range of functions. One such function is learning. Learning as an activity has witnessed great changes over the period of time. It has evolved and accommodated a lot of elements of modernity into itself. Traditional forms of learning are being gradually replaced by modern ones albeit not in their entirety. One such form of learning that has gained prevalence over the years is experiential learning. As the word suggests, it has a lot to do with experiencing or practicing with an aim to learn rather than learning through theory conceptualisation or rote learning.

experiential learning methodology case study

Experiential Learning- An overview

Experiential Learning thrives on a first-hand learner experience. This experience helps the user to derive learning through facilitating the accumulation of knowledge and skills by performing, analysing and reflecting on a particular activity. Post reflection, the adjustments are made to further refine the learnings right on the next attempt.

Experiential Learning is particularly an important tool in the domain of Learning and Development. On par with other domains, the L&D industry has also observed a shift in mindset with the advent of technological advancements. The mounting usage of experiential learning serves as a testimony to this change, whereby an employee-centric approach is now being focussed upon as against the practices in the past.

While experiential learning isn’t just a product of these modern advancements, the rising importance and usage of experiential learning is certainly a change that’s driven through these advancements. Experiential learning was used in the past. It has evolved and its extent is widening as ever!

The Kolb’s Model

The Kolb’s Model, formulated by David Kolb is one of the most famous experiential learning models. It classifies experiential learning into four stages-

  • Concrete Experience
  • Reflective Observation
  • Abstract Conceptualisation
  • Active Experimentation

According to this model, learners gather knowledge through a combination of grasping and transforming experience.

In the first stage, the Concrete Experience stage, they gain experience by trying something out. This can be anything that builds up an instance of an experience.

Example- After observing your friend riding a bike, you decide to ride a bike yourself. When you finally ride, it amounts to a concrete experience.

The second stage or the Reflective Observation stage makes them go through the reflections of the experience gathered in the first stage. The learners reflect upon to learn from their experience. Analysing their experience in terms of positive practices and scope of improvements is the essence of this stage.

Example- You analyse your experience of riding a bike. ‘What are the key takeaways? Was there anything that went wrong? If yes, then what? What were the positives that need to be continued?’

The third stage of Abstract Conceptualisation is the stage of planning for the next experience. The recommendations through the reflection stage are taken into account for enhanced performance in the future.

Example- You found a problem in terms of shifting the gears at the right time. Jotting down this point to be kept in mind in the next stage so as to improve, is the main objective of this step.

Finally, the fourth stage or the Active Experimentation stage is the platform where the learner gets a chance to experiment with their improved plan of action.

Example- You decide to work on your gear-shifting in your next attempt so as to further improve from the previous stage. You execute the modifications and, in the process, learn the changed approach through active experimentation.

This is Kolb’s model of experiential learning- It’s essentially a ‘Learning by doing’ approach. Many experiential learning-based training interventions are built up on the basis of this model.

Experiential Learning Tools

Experiential Learning delivers learning through a diverse set of tools/methods and techniques. These include age-old tools, at the same time, modern ones laced with high-quality advancements are also used as a part of many learning and development interventions in the 21st century.

Here’s a description of some tools that are used to deliver experiential learning to the candidates-

Simulations

Simulations are experiential learning tools that create a simulated environment of the actual workplace. The Airline industry is one of the best examples of using simulations for training pilots wherein an environment of the aircraft in air is generated. The trainee pilots get to work in the exact situation as a pilot in the cockpit. It’s amazing to have such simulations, right? You certainly don’t expect a pilot to face tough situations in the air without any prior experience!

Such simulations are rising in prominence in other industries too.

Case Studies

Case Studies help in presenting a situation or a scenario in front of the learner. The learner gets to decide the course of action under a particular case study. Active Reflection of case studies are vital as they make the learner aware of the actions that they need to take under a similar situation that might arise in the future. The experience of dealing with a particular situation proves to be quite handy for the learner.

Field Visits

Field visits are actual visits to designated areas of work such as a factory, a plant or an Industrial unit etc. A lot of knowledge including basic information about sophisticated machineries can be gathered through observation during a field visit.

Role Plays include learning through a self-depiction of a particular case or an instance. It is a more personal form of experiential learning where indulgent learners step into the shoes of characters, taking a particular course of action to learn from a case. It is an extension of case study form of experiential learning wherein, the learners enact the case, in addition.

Management Games

Management Games are modern forms of experiential learning. A lot of game-based activities are used by several organisations as a captivating intervention that increases the engagement of the learners. Management Games are highly advantageous for inculcating personal and organisational skills such as team work, problem-solving etc.

On the Job Training

On the Job Training or OJT is an age-old experiential learning method. It is well-used in several industries where learners simultaneously learn as they work.

Apart from these discussed tools, there are several other tools and methods of experiential learning. These are used with respect to the requirement of the client organisation, particularly governed by the nature of work and the industry.

Experiential Learning- the way to go!

There are plenty of reasons that support the fact that experiential learning is the way to go in the Learning and Development function. These are discussed as follows-

The human brain is much more capable of retaining first-hand information that’s learnt through an active self-participation rather than an information that is learnt through rote methods. As a result, the trainees learn better under experiential form of learning.

Experiential learning is much more engaging and relatable than traditional forms of learning. It enhances user attention. In addition, it also adds to the user-motivation to learn. Modern methods of experiential learning are equipped to provide a great experience to the user.

Experiential learning is highly relevant to the training objectives. It is to-the-point and as a result, it reduces the information overload by eliminating unnecessary information.

The outcomes under an experiential learning program are better than those from the traditional ones. The learners not only learn in a better way, they’re also better off to apply the learnt concepts in the real-world work situation since they’ve been learning through an application-based program.

Experiential learning is vital than ever in the 21st century. As previously discussed, the organisational mindset has evolved towards a heightened focus on user-centricity. The users of a training program of any organisation are empathised upon. The employee turnover is high and hiring costs involve significant expenditure. As a result, it’s better to use methods that motivate a user or a learner towards the accumulation of skill-sets that result in the practical implementation while reducing the rates of attrition. This is ensured through an experiential learning program.

Under an experiential setting, there’s a great scope of self-learning. Self-learning is result-oriented since it provides a method to refine yourself. Continuous refinements lead to improvements that go a long way to enhance productivity in the actual workplace environment.

With low attention spans along with rising range of redundancies, experiential learning comes out as a panacea to the ineffective training interventions. It certainly deserves the appreciation in terms of rising usage, not only for the organisational benefit but also for individual growth of the employees through valuable skill-sets additions, particularly important in these times of immense competition.

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Experiential Learning

This resource provides an overview of experiential learning, a process where students learn through hands-on experiences and reflection. It explains how experiential learning works, highlighting the integration of knowledge, activity, and reflection. It gives examples of various forms of experiential learning, including internships, service learning, and undergraduate research and introduces guidance on how it can be implemented.

Experiential learning is an engaged learning process whereby students “learn by doing” and by reflecting on the experience. Experiential learning activities can include, but are not limited to, hands-on laboratory experiments, internships, practicums, field exercises, study abroad, undergraduate research and studio performances.

Well-planned, supervised and assessed experiential learning programs can stimulate academic inquiry by promoting interdisciplinary learning, civic engagement, career development, cultural awareness, leadership, and other professional and intellectual skills.

Learning that is considered “experiential” contain all the following elements:

  • Reflection, critical analysis and synthesis.
  • Opportunities for students to take initiative, make decisions, and be accountable for the results.
  • Opportunities for students to engage intellectually, creatively, emotionally, socially, or physically.
  • A designed learning experience that includes the possibility to learn from natural consequences, mistakes, and successes.

How does it work?

Kolb’s (1984) cycle of learning depicts the experiential learning process (see figure below).  This process includes the integration of:

  • knowledge—the concepts, facts, and information acquired through formal learning and past experience;
  • activity—the application of knowledge to a “real world” setting; and
  • reflection—the analysis and synthesis of knowledge and activity to create new knowledge” (Indiana University, 2006, n.p.).

Figure-V2

What does experiential learning look like?

Experiential learning  has the following elements  ( Association for Experiential Education , 2007-2014):

  • Experiences are carefully chosen for their learning potential (i.e. whether they provide opportunities for students to practice and deepen emergent skills, encounter novel and unpredictable situations that support new learning, or learn from natural consequences, mistakes, and successes).
  • Throughout the experiential learning process, the learner is actively engaged in posing questions, investigating, experimenting, being curious, solving problems, assuming responsibility, being creative, and constructing meaning, and is challenged to take initiative, make decisions and be accountable for results.
  • Reflection on learning during and after one’s experiences is an integral component of the learning process. This reflection leads to analysis, critical thinking, and synthesis (Schon, 1983; Boud, Cohen, & Walker, 1993).
  • Learners are engaged intellectually, emotionally, socially, and/or physically, which produces a perception that the learning task is authentic.
  • Relationships are developed and nurtured: learner to self, learner to others, and learner to the world at large.

During experiential learning,  the facilitators role is to :

  • Select suitable experiences that meet the criteria above.
  • Pose problems, set boundaries, support learners, provide suitable resource, ensure physical and emotional safety, and facilitate the learning process.
  • Recognize and encourage spontaneous opportunities for learning, engagement with challenging situations, experimentation (that does not jeopardize the wellbeing of others) and discovery of solutions.
  • Help the learner notice the connections between one context and another, between theory and the experience and encouraging this examination repeatedly.

Some forms  of experiential learning include (Indiana University, 2006; Moore, 2010):

  • Internships  – A more broad term used to describe experience-based learning activities that often subsume other terms such as cooperative education, service-learning or field experiences.  It is often a credit-bearing, free-standing activity in a student’s field of interest not connected to a theoretical course.  It is usually assessed by a faculty member and supervised by an employer who is not a faculty member. The student may work with practicing professionals, complete a project, attend public events, interview and observe constituents and employees.  The student may or may not be paid for this experience.  When attached to a classroom course, a student may spend several hours a week volunteering in an agency, supporting co-curricular activities, shadowing a professional in the field, or observing people in their natural environments. Key to this form of experiential learning is some type of guided reflection. The mission of this experience may be to support the integration of theory and practice, explore career options, or foster personal and professional development.
  • Service learning  – This term is used to denote optional or required out-of-classroom community service experiences/projects attached to courses or a separate credit bearing experience.  The location may be the broader community outside the university or one embedded in co-curricular activities. In these experiences, students participate in an organized service activity that meets identified community needs and reflect on the service activity to better understand course content and gain a broader appreciation of the discipline and an enhanced sense of civic responsibility.
  • Cooperative education  – Mostly a part of professional programs, students gain practical relevant work experience over a period of multiple terms that intersperse their coursework.  Students alternate work and study, usually spending a number of weeks in study (typically full-time) and a number of weeks in employment away from campus (typically full-time). Alternatively, cooperative education may occur when students simultaneously attend classes part-time and work part-time during consecutive school terms in an intentionally planned and coordinated way. Students receive academic credit for cooperative education when the experiences meet the criteria for credit (i.e., faculty supervision, reflective components, evidence of learning). The purpose of these programs is to build student’s career skills and knowledge.
  • Clinical education  – This is a more specifically defined internship experience in which students practice learned didactic and experiential skills, most frequently in health care and legal settings, under the supervision of a credentialed practitioner.  It is often is a separate credit-bearing course tied to a related theoretical course or a culminating experience after a sequence of theoretical courses.
  • Student teaching  –  This experience is specific to students in pre-professional and pre-service teacher education who are gaining required and evaluated experience in supervised teaching.
  • Practicum  –  A relative of the internship, this form of experiential learning usually is a course or student exercise involving practical experience in a work setting (whether paid or unpaid) as well as theoretical study, including supervised experience as part of professional pre-service education.
  • Undergraduate research experience  – Students function as research assistants and collaborators on faculty projects.
  • Community-based research  – Faculty and students cooperate with local organizations to conduct studies to meet the needs of a particular community.  Students gain direct experience in the research process.
  • Field work  – Supervised student research or practice carried out away from the institution and in direct contact with the people, natural phenomena, or other entities being studied. Field work is especially frequent in fields including anthropology, archaeology, sociology, social work, earth sciences, and environmental studies.
  • Study abroad  – Students usually engage in courses at higher education institutions in another country.  The experiential learning component is the cultural immersion which provides novel challenges for navigating living in a new place.  The coursework connected to a study abroad can also include internships and service-learning experiences.

Research on experiential learning

Expand to view research.

Ambrose, S. A., Bridges, M. W., DiPietro, M., Lovett, M. C., & Norman, M. K. (2010).  How learning works: 7 research-based principles for smart teaching.  San Francisco, CA: Jossey- Bass.

Association for Experiential Education. (2007-2014). Retrieved from  http://www.aee.org/ .

Bass, R.  (2012, March/April). Disrupting ourselves: The problem of learning in higher education.  EDUCAUSE Review, 47(2).

Boud, D., Cohen, R., & Walker, D. (Eds.). (1993). Using experience for learning.  Bristol, PA: Open University Press.

Indiana University. (2006). Experiential learning notations on Indiana University official transcripts. Retrieved from http://registrar.iupui.edu/experiential-learning.html.

Kolb, D. A. (1984).  Experiential learning: Experience as the source of learning and development. Englewood Cliffs, NJ: Prentice-Hall.

Lave, J., & Wenger, E.  (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge University.

Linn, P. L., Howard, A., and Miller, E. (Eds). (2004). The handbook for research in cooperative education and internships. Mahwah, NJ: Lawrence Erlbaum Associates.

Moore, D. T. (2010). Forms and issues in experiential learning. In D. M. Qualters (Ed.) New Directions for Teaching and Learning (pp. 3-13). New York City, NY: Wiley.

Schon, D. (1983).  The reflective practitioner:  How professionals think in action.  New York City, NY: Basic books.

The University of Texas at Austin College of Natural Sciences. (2013). Freshman Research Initiative Retrieved from  http://cns.utexas.edu/fri .

Wurdinger, D. D., & Carlson, J. A. (2010).  Teaching for experiential learning:  Five approaches that work.  Lanham, MD: Rowman & Littlefield Education.

You may also be interested in:

Service learning, active learning, embodied learning: teaching and learning with reacting to the past, student engagement part 2: ensuring deep learning, assessing learning, project-based learning, universal design for learning resources, student engagement part 1: focusing on the emotional aspects of learning.

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Learning Methodologies

From the case method, to coaching and projects, to experiential learning and simulations, our teaching methodologies engage participants in  dynamic discussions  about real-life business challenges. On their own, with a peer or in teams, participants are invited to reflect on these challenges and extract insights to apply to their work.

The case method

The case method

The highly dynamic and hands-on case study method serves as one of our main learning methodologies. Faculty complement case studies with other learning methodologies, including lectures, action-based learning, simulations and coaching.

The case method asks students to put themselves in the shoes of the manager: How can I boost flagging sales? How can the company expand its revenue streams given its new competitive environment? What incentive policies might work in an industry with high turnover?

In contrast to lecture-based teaching methods, here students do most of the talking. Professors facilitate and guide the discussion, asking questions and eliciting participation from the entire class to enrich the discussion with contrasting viewpoints, different industry experiences and varied cultural backgrounds.

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Action Learning

An approach to solving real problems that involves taking action and reflecting upon the results, which helps improve the problem-solving process, as well as the solutions developed by the team.

https://www.iese.edu/wp-content/uploads/2022/03/Experiential-Learning.jpg

Experiential Learning

Learning through experience in local contexts to create memorable learning experiences that, facilitated by our faculty, can drive development by sparking insights outside the classroom setting.

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Simulations (or Business Games)

Fully interactive exercises that replicate real-life situations that may be faced in the business world.

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Online Learning

A learning experience that can be accessed remotely, and consists of a directed learning process comprised of content and some way to measure achievement.

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Work with a coach to increase self-awareness, improve performance and achieve explicit goals through a change in behavior.

https://www.iese.edu/wp-content/uploads/2022/03/mentoring.jpg

Sharing of knowledge and advice with another person who has experience different from one’s own.

Learning methodologies at IESE Business School

  • Open access
  • Published: 07 February 2024

Genomic prediction using machine learning: a comparison of the performance of regularized regression, ensemble, instance-based and deep learning methods on synthetic and empirical data

  • Vanda M. Lourenço 1 ,
  • Joseph O. Ogutu 2 ,
  • Rui A.P. Rodrigues 1 ,
  • Alexandra Posekany 3 &
  • Hans-Peter Piepho 2  

BMC Genomics volume  25 , Article number:  152 ( 2024 ) Cite this article

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Metrics details

The accurate prediction of genomic breeding values is central to genomic selection in both plant and animal breeding studies. Genomic prediction involves the use of thousands of molecular markers spanning the entire genome and therefore requires methods able to efficiently handle high dimensional data. Not surprisingly, machine learning methods are becoming widely advocated for and used in genomic prediction studies. These methods encompass different groups of supervised and unsupervised learning methods. Although several studies have compared the predictive performances of individual methods, studies comparing the predictive performance of different groups of methods are rare. However, such studies are crucial for identifying (i) groups of methods with superior genomic predictive performance and assessing (ii) the merits and demerits of such groups of methods relative to each other and to the established classical methods. Here, we comparatively evaluate the genomic predictive performance and informally assess the computational cost of several groups of supervised machine learning methods, specifically, regularized regression methods, deep , ensemble and instance-based learning algorithms, using one simulated animal breeding dataset and three empirical maize breeding datasets obtained from a commercial breeding program.

Our results show that the relative predictive performance and computational expense of the groups of machine learning methods depend upon both the data and target traits and that for classical regularized methods, increasing model complexity can incur huge computational costs but does not necessarily always improve predictive accuracy. Thus, despite their greater complexity and computational burden, neither the adaptive nor the group regularized methods clearly improved upon the results of their simple regularized counterparts. This rules out selection of one procedure among machine learning methods for routine use in genomic prediction. The results also show that, because of their competitive predictive performance, computational efficiency, simplicity and therefore relatively few tuning parameters, the classical linear mixed model and regularized regression methods are likely to remain strong contenders for genomic prediction.

Conclusions

The dependence of predictive performance and computational burden on target datasets and traits call for increasing investments in enhancing the computational efficiency of machine learning algorithms and computing resources.

Peer Review reports

Rapid advances in genotyping and phenotyping technologies have enabled widespread and growing use of genomic prediction (GP). The very high dimensional nature of both genotypic and phenotypic data, however, is increasingly limiting the utility of the classical statistical methods. As a result, machine learning (ML) methods able to efficiently handle high dimensional data are becoming widely used in GP. This is especially so because, compared to many other methods used in GP, ML methods possess the significant advantage of being able to model nonlinear relationships between the response and the predictors and complex interactions among predictor variables. However, this often comes at the price of a very high computational burden. Often, however, computational cost is less likely to present serious challenges if the number of SNPs in a dataset is relatively modest but it can become increasingly debilitating as the number of markers grows to millions or even tens of millions. Future advances in computational efficiencies of machine learning algorithms or using high-performance or more efficient programming languages may progressively ameliorate this limitation. Given their growing utility and popularity, it is important to establish the relative predictive performance of different groups of ML methods in GP. Even so, the formal comparative evaluation of the predictive performance of groups of ML methods has attracted relatively little attention. The rising importance of ML methods in plant and animal breeding research and practice, increases both the urgency and importance of evaluating the relative predictive performance of groups of ML methods relative to each other and to classical methods. This can facilitate identification of groups of ML methods that balance high predictive accuracy with low computational cost for routine use with high dimensional phenotypic and genomic data, such as for GP, say.

ML is perhaps one of the most widely used branches of contemporary artificial intelligence. Using ML methods facilitates automation of model building, learning and efficient and accurate predictions. ML algorithms can be subdivided into two major classes: supervised and unsupervised learning algorithms. Supervised regression ML methods encompass regularized regression methods, deep, ensemble and instance-based learning algorithms. Supervised ML methods have been successfully used to predict genomic breeding values for unphenotyped genotypes, a crucial step in genome-enabled selection [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ]. Furthermore, several studies have assessed the relative predictive performance of supervised ML methods in GP, including two ensemble methods and one instance-based method [ 5 ]; four regularized and two adaptive regularized methods [ 6 ]; three regularized and five regularized group methods [ 9 ] and several deep learning methods [ 1 , 2 , 3 , 4 , 8 ]. However, no study has comprehensively evaluated the comparative predictive performance of all these groups of methods relative to each other or to the classical regularized regression methods. We therefore rigorously evaluate the comparative predictive performance as well as the computational complexity or cost of three groups of popular and state-of-the-art ML methods for GP using one simulated animal dataset and three empirical datasets obtained from a commercial maize breeding program. We additionally offer brief overviews of the mathematical properties of the methods with emphasis on their salient properties, strengths and weaknesses and relationships with each other and with the classical regularization methods. While we offer a somewhat comprehensive review of genomic prediction methods with a specific emphasis on ML, our contribution extends to showcasing novel findings derived from comparative assessments of ML techniques across both real and simulated datasets.

Besides ML methods, Bayesian methods are also becoming widely used for genomic prediction [ 3 , 8 , 10 ]. So, even though our goal is not to provide an exhaustive review of all genomic prediction methods, we offer two Bayesian methods for benchmarking the performance of the ML methods.

The rest of the paper is organized as follows. First we present the synthetic and real datasets. Second, we detail the methods compared in this study. Next, the results from the comparative analyses of the data are presented. Finally, a discussion of the results and closing remarks follow.

Simulated (animal) data

We consider one simulated dataset [ 9 ], an animal breeding outbred population simulated for the 16-th QTLMAS Workshop 2012 (Additional file 1 ). The simulation models used to generate the data are described in detail in [ 11 ] and are therefore not reproduced here. The dataset consists of 4020 individuals genotyped for 9969 SNP markers. Out of these, 3000 individuals were phenotyped for three quantitative milk traits and the remaining 1020 were not phenotyped (see [ 9 ] for details). The goal of the analysis of the simulated dataset is to predict the genomic breeding values (PGBVs) for the 1020 unphenotyped individuals using the available genomic information. The simulated dataset also provides true genomic breeding values (TGBVs) for the 1020 genotypes for all the traits.

As in [ 9 ], to enable model fitting for the grouping methods, markers were grouped by assigning consecutive SNP markers systematically to groups of sizes 10, 20, ..., 100 separately for each of the five chromosomes. Typically, the last group of each grouping scheme has fewer SNPs than the prescribed group size. Table 1 summarizes the simulated phenotypic data and highlights differences in the magnitudes of the three simulated quantitative traits \(T_1\) , \(T_2\) and \(T_3\) .

Real (plant) data

For the application to empirical data sets, we use three empirical maize breeding datasets produced by KWS (breeding company) for the Synbreed project during 2010, 2011 and 2012. We first performed separate phenotypic analyses of yield for each of the three real maize data sets to derive the adjusted means used in genomic prediction using a single stage mixed model assuming that genotypes are uncorrelated (Additional file 4 , S1 Text). The fixed effect in the mixed model comprised a tester (Tester) with two levels, genotypic group (GRP) with three levels, Tester \(\times\) GRP and Tester \(\times\) GRP \(\times\) G (G=genotype). The random factors were location (LOC), trial (TRIAL) nested within location, replicate (REP) nested within trial and block (BLOCK) nested within replicate. The fitted random effects were LOC, LOC \(\times\) TRIAL, LOC \(\times\) TRIAL \(\times\) REP, LOC \(\times\) TRIAL \(\times\) REP \(\times\) BLOCK, Tester \(\times\) GRP \(\times\) SWITCH2 \(\times\) G1 and Tester \(\times\) GRP \(\times\) SWITCH1 \(\times\) G2. SWITCH1 and SWITCH2 in the last two effects are operators defined and explained briefly in the supplementary materials (Additional file 4 , S1 text; and Additional file 5 , Section 1) and in greater detail in [ 12 , 13 ]. All the three maize datasets involved two testers and three genotypic groups. Accordingly, prior to genomic prediction, we accounted for and removed the effect of the tester \(\times\) genotypic group (GRP) effect from the adjusted means (lsmeans) of maize yield (dt/ha) by computing the arithmetic mean of the lsmeans for the interaction of testers with GRP for the genotyped lines. This mean was then subtracted from the lsmeans for each tester \(\times\) GRP interaction term. The resulting deviations were subtracted from the lsmeans of the individual genotypes corresponding to each Tester \(\times\) GRP interaction. This enabled us not to consider the Tester \(\times\) GRP effect in the genomic prediction model.

For all the years, every line was genotyped for 32217 SNP markers. A subset of the SNP markers with non-zero variances were split into groups of sizes 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. Groups were defined by systematically grouping consecutive and spatially adjacent markers, separately for each of 10 chromosomes (Additional file 4 , S2 Text). All the checks (standard varieties) and check markers were deleted prior to genomic prediction. More details specific to the three datasets follow (Table 2 summarizes the number of genotypes in the training and validation datasets). The true breeding values are not known in this case.

For each of the 2010, 2011 and 2012 datasets, the genotypes or test crosses were genotyped for 32217 SNPs and randomly split into 5 parts (folds) for 5-fold cross-validation (Additional file 4 , S3 Text & S4 Text). The random splitting procedure was repeated 10 times to yield 10 replicates per dataset. The total number of genotypes and the number of individuals assigned to the training and validation sets for each dataset are provided in Table 2 .

Table 3 summarizes the KWS phenotypic data for 2010, 2011 and 2012. Each data split for each year (2010, 2011 and 2012) contained approximately 20% of the phenotypic observations and was obtained using stratified random sampling using the algorithm of [ 14 ]. The strata were defined by the combinations of the two testers and three genotypic groups.

In this section we describe the four supervised ML groups of methods.

Regularized regression methods

Consider the general linear regression model

where \(y_i\) is the i -th observation of the response variable, \(x_{ij}\) is the i -th observation of the j -th covariate ( p is the number of all covariates), \(\beta _j\) are the regression coefficients (unknown fixed parameters), \(\varepsilon _i\) are i.i.d. random error terms with \(E(\varepsilon _i)=0\) and \(var(\varepsilon _i)=\sigma ^2_e\) , where \(\sigma ^2_e\) is an unknown random variance, and n is the sample size. The ordinary least squares estimator of \(\varvec{\beta }=(\beta _0,\dots ,\beta _p)'\) , which is unbiased, is obtained by minimizing the residual sum of squares (RSS), i.e.,

This estimator is typically not suitable when the design matrix \(\textbf{X}\) is less than full rank ( \(\textbf{X}\) has a full rank if the number of its linearly independent rows or columns \(k=\min (p,n)\) ) or is close to collinearity (i.e., the covariates are close to being linear combinations of one another) [ 15 ]; problems that are frequently associated with \(p>>n\) .

In genomic prediction (GP) one is interested in estimating the p regression coefficients \(\beta _j\) so that genomic breeding values of non-phenotyped genotypes can be predicted from the fitted model. The response variable \(\textbf{y}\) is often some quantitative trait and the \(\beta _j\) ’s are the coefficients of molecular markers spanning the whole genome, usually Single Nucleotide Polymorphisms (SNPs). Because in GP typically \(p>>n\) , the ordinary least squares (OLS) estimator breaks down and thus other methods for estimating \(\varvec{\beta }\) in ( 1 ) must be sought. Indeed, the increasingly high dimensional nature of high-throughput SNP-marker datasets has prompted increasing use of the power and versatility of regularization methods in genomic prediction to simultaneously select and estimate important markers and account for multicollinearity [ 5 , 6 ].

Without loss of generality, we assume, consistent with the standard practice in regularized estimation where a distance-based metric is used for prediction, that the response variable is mean-centered whereas the covariates in ( 1 ) are standardized, so that

Regularized regression methods minimize a non-negative loss function (RSS or other) plus a non-negative penalty function. Standardizing the covariates prior to model fitting ensures that the penalty is applied evenly to all covariates. Mean-centering the response and the covariates is usually done for notational simplicity but also eliminates the need to estimate the intercept \(\beta _0\) .

After the penalized models have been fit, the final estimates are obtained by back transformation to the original scale by re-introducing an intercept ( \(\beta _0\) ). In particular, for a mean-centered response \(\textbf{y}\) and standardized predictor \(\textbf{X}^{\varvec{*}}\) , predictions are obtained by

with \(\widehat{\varvec{\beta }}^*=(\widehat{\beta }^*_1,\dots ,\widehat{\beta }^*_p)\) , the regression coefficients from the model fit with the mean-centered response \(\textbf{y}\) and standardized covariates \(\textbf{X}^{\varvec{*}}\) , \({\textbf{X}}^*_j=(x_{1j},\dots ,x_{nj})'\) the j -th covariate and \(\beta _0=\bar{\textbf{y}}\) . One can also choose to predict using the original predictor \(\textbf{X}^{\varvec{*}}\) without standardization. In that case one should back transform the \(\widehat{\beta }^*_j\) to the original scale and consider

with \(\widehat{\beta }_j=\widehat{\beta }^*_j/s_j\) , \(s_j=\sqrt{n^{-1}\sum \limits _{i=1}^nx_{ij}^2}\) the standard deviation of the j-th covariate \({\textbf{X}}^*_j\) and \(\beta _0 = \bar{\textbf{y}}- {\tilde{\textbf{X}}} \widehat{\varvec{\beta }}\) , where \({\tilde{\textbf{X}}}_j=(m_j,\dots ,m_j)'\) is a vector of size n with \(m_j\) being the mean of the j -th covariate \({\textbf{X}}^*_j\) .

The primary goal of regularization methods is to reduce model complexity resulting from high dimensionality by reducing the number of predictors in the model. This is achieved by either shrinking some coefficients to become exactly zero, and so drop out of the model, or shrinking all coefficients to be close to zero and each other but not exactly zero. Ideally, a desirable estimator of \(\varvec{\beta }\) should (i) correctly select the nonzero coefficients with probability converging to 1 (i.e. with near certainty; selection consistency ) and (ii) yield estimators of the nonzero coefficients that are asymptotically normal with the same means and covariances that they would have if the zero coefficients were known exactly in advance ( asymptotic normality ). An estimator satisfying these two conditions is said to possess the oracle property [ 16 , 17 ].

For the remainder of the paper, we assume that \({\textbf {X}}\) is a \(n\times p\) marker matrix (e.g., with the genotypes \(\{aa,Aa,AA\}\) coded as \(\{0,1,2\}\) or \(\{-1,0,1\}\) for p biallelic SNPs under an additive model) with \({\textbf {X}}_j\) denoting the j -th SNP covariate and \(\varvec{\beta }=(\beta _1,\dots ,\beta _p)\) denoting the unknown vector of marker effects. Table 4 (upper half) summarizes the methods discussed in this sub-section.

Bridge-type estimators

The most popular regularization methods in genomic prediction include ridge regression (RR; [ 18 ]), the least absolute shrinkage and selection operator (LASSO; [ 19 ]) and the elastic net (ENET; [ 20 ]). All these methods are special cases of the bridge estimator [ 15 , 21 ] given by

where the regularization parameter \(\lambda\) balances the goodness-of-fit against model complexity and the shrinkage parameter \(\gamma\) determines the order of the penalty function. The optimal combination of \(\lambda\) and \(\gamma\) can be selected adaptively for each dataset by grid search using cross-validation (CV; if the focus is on predictive performance) or by information criteria (e.g., AIC or BIC; if the focus is on model fit). Bridge regression automatically selects relevant predictors when \(0<\gamma \le 1\) , shrinks the coefficients when \(\gamma >1\) and reduces to subset selection when \(\gamma =0\) . The bridge estimator reduces to the LASSO estimator when \(\gamma =1\) and to the ridge estimator when \(\gamma =2\) . Specifically,

where \(\Vert . \Vert _1\) is the \(\ell _1\) -norm, and

The bridge estimator also enjoys several other useful and interesting properties (see [ 22 , 23 ] for more details). We summarize these salient properties with emphasis on the special cases of the LASSO ( \(\gamma =1\) ) and the ridge estimators ( \(\gamma =2\) ).

The asymptotic properties of bridge estimators have been studied in detail by [ 22 ]. In particular, where \(p<n\) , with p increasing to infinity as n grows, and under appropriate regularity conditions, bridge estimators enjoy the oracle property for \(0<\gamma <1\) . This implies that neither the LASSO nor the ridge estimator possesses the oracle property [ 16 , 17 ]. If \(p>>n\) and no assumptions are imposed on the covariate matrix, then the regression parameters are generally non-identifiable. However, if a suitable structure is assumed for the covariate matrix, then bridge estimators achieve consistent variable selection and estimation [ 22 ].

Although the LASSO estimator performs automatic variable selection, it is a biased and inconsistent estimator [ 24 , 25 ]. Moreover, it is unstable with high-dimensional data because it

cannot select a larger number of predictors p than the sample size n if \(p>>n\) ;

arbitrarily selects one member of a set of pairwise highly correlated predictors and ignores the other.

The ridge estimator performs well for many predictors each of which has a small effect but cannot shrink the coefficients to become exactly zero. Moreover, the ridge estimator

prevents coefficients of linear regression models with many correlated variables from being poorly determined and exhibiting high variance;

shrinks coefficients of correlated predictors equally towards zero and towards each other;

retains all predictor variables in the model leading to complex and less interpretable models.

In addition, RR has close connections with marker-based best linear unbiased prediction (BLUP) and genomic best linear unbiased prediction (GBLUP) [ 26 ], which we clarify in what follows. The ridge estimator is given by

where, if \(\lambda\) is estimated by cross-validation as suggested above, then the ridge estimator may be denoted by RR-CV. Another way of looking at the ridge estimator is to assume in ( 1 ) that \(\varvec{\beta }\sim N({\textbf {0}},{\textbf {I}}\sigma ^2_{\beta })\) is a random vector of unknown marker effects and that \(\varvec{\varepsilon }\sim N({\textbf {0}},{\textbf {I}}\sigma ^2_{e})\) is an unknown random error term, where \(\sigma ^2_{\beta }\) and \(\sigma ^2_{e}\) are the unknown marker-effect and error variances, respectively. Model ( 1 ), written in matrix form as

is now a linear mixed model and hence, the variances can be estimated via the restricted maximum likelihood (REML) method. Observing that \({\textbf{y}}\sim N({\varvec{0}},{\textbf{K}}\sigma ^2_{\beta }+{\textbf{I}}\sigma ^2_{\varepsilon })\) , where \({\textbf{K}}={\textbf {X}}'{} {\textbf {X}}\) is the kinship or genomic relationship matrix, the BLUP solution for the marker effects under model ( 5 ) is given by ([ 27 ]; p.270)

Now defining \({\textbf {H}}={\textbf {I}} \frac{\sigma ^2_{\varepsilon }}{\sigma ^2_{\beta }}\) to simplify the notation and pre-multiplying \(\widehat{\varvec{\beta }}_{BLUP}\) with \(({\textbf {X}}'{} {\textbf {X}}+{\textbf {H}})^{-1}{} {\textbf {X}}'({\textbf {K}}+{\textbf {H}}){\textbf {K}}^{-1}{} {\textbf {X}}\) we obtain

Finally, observing that \(({\textbf {X}}'{} {\textbf {X}}+{\textbf {H}})^{-1}{} {\textbf {X}}'({\textbf {K}}+{\textbf {H}}){\textbf {K}}^{-1}{} {\textbf {X}}={\textbf {X}}'{} {\textbf {K}}^{-1}{} {\textbf {X}}\) (see Appendix ) and that \({\textbf {X}}'{} {\textbf {K}}^{-1}{} {\textbf {X}}{} {\textbf {X}}'={\textbf {X}}'\) we find that

establishing the equivalence of BLUP and RR [ 28 , 29 ] and that one can actually estimate the ridge parameter \(\lambda\) by \(\widehat{\lambda }=\frac{\widehat{\sigma }^2_{e}}{\widehat{\sigma }^2_{\beta }}\) . Because we use REML to estimate the two variance components in \(\widehat{\varvec{\beta }}_{BLUP}\) , we refer to this RR appproach as RR-REML. Our basic regression model ( 5 ) can be written as

where, \({\textbf {g}}={\textbf {X}}\varvec{\beta }\) . Making the same assumptions as for RR-REML, i.e., assuming that \(\varvec{\beta }\sim N({\textbf {0}},{\textbf {I}}\sigma ^2_{\beta })\) and \(\varvec{\varepsilon }\sim N({\textbf {0}},{\textbf {I}}\sigma ^2_{e})\) , we have that \({\textbf {g}}\sim N({\textbf {0}},{\textbf {K}}\sigma ^2_{\beta })\) . The BLUP of \({\textbf {g}}\) , also known as genomic estimated breeding values (GEBV) or gBLUP, under this model is ([ 27 ]; p.270)

Now pre-multiplying \(\widehat{{\textbf {g}}}_{BLUP}\) with \({\textbf {X}}({\textbf {X}}'{} {\textbf {X}}+{\textbf {H}})^{-1}{} {\textbf {X}}'({\textbf {K}}+{\textbf {H}}){\textbf {K}}^{-1}\) we obtain

Finally, observing that \({\textbf {X}}({\textbf {X}}'{} {\textbf {X}}+{\textbf {H}})^{-1}{} {\textbf {X}}'({\textbf {K}}+{\textbf {H}}){\textbf {K}}^{-1}={\textbf {I}}\) (see Appendix ), we find that \(\widehat{{\textbf {g}}}_{BLUP}={\textbf {X}}\widehat{\varvec{\beta }}_{BLUP}\) establishing the equivalence of RR-REML and gBLUP [ 30 , 31 ].

Due to the nature of the \(\ell _1\) penalty, particularly for high values of \(\lambda\) , the LASSO estimator will shrink many coefficients to exactly zero, something that never happens with the ridge estimator.

Elastic net estimator

The elastic net estimator blends two bridge-type estimators, the LASSO and the ridge, to produce a composite estimator that reduces to the LASSO when \(\lambda _2=0\) and to the ridge when \(\lambda _1=0\) . Specifically, the elastic net estimator is specified by

with \(k=1+\lambda _2\) if the predictors are standardized (as we assume) or \(k=1+\lambda _2/n\) otherwise. Even when \(\lambda _1,\lambda _2\ne 0\) , the elastic net estimator behaves much like the LASSO but with the added advantage of being robust to extreme correlations among predictors. Moreover, the elastic net estimator is able to select more than n predictors when \(p>>n\) . Model sparsity occurs as a consequence of the \(\ell _1\) penalty term. Mazumder et al. [ 32 ] proposed an estimation procedure based on sparse principal components analysis (PCA), which produces an even more sparse model than the original formulation of the elastic net estimator [ 20 ]. Because it blends two bridge-type estimators, neither of which enjoys the oracle property, the ENET also lacks the oracle property.

Other competitive regularization methods that are asymptotically oracle efficient ( \(p<n\) with p increasing to infinity with n ), which do not fall into the category of bridge-type estimators, are the smoothly clipped absolute deviations (SCAD [ 17 , 33 ]) and the minimax concave penalty (MCP [ 25 , 34 ]) methods. Details of the penalty functions and other important properties of both methods can be found elsewhere [ 9 , 35 ].

Adaptive regularized regression methods

The adaptive regularization methods are extensions of the regularized regression methods that allow the resulting estimators to achieve the oracle property under certain regularity conditions. Table 4 (lower half) summarizes the adaptive methods considered here.

Adaptive bridge-type estimators

Adaptive bridge estimators extend the bridge estimators by introducing weights in the penalty term. More precisely,

where \(\{{w}_j\}_{j=1}^p\) are adaptive data-driven weights. As with the bridge-type estimator, the adaptive bridge estimator simplifies to the adaptive LASSO ( a LASSO) estimator when \(\gamma =1\) and to the adaptive ridge estimator when \(\gamma =2\) . Chen et al. [ 36 ] studied the properties of adaptive bridge estimators for the particular case when \(p<n\) (with p increasing to infinity with n ), \(0<\gamma <2\) and \({w}_j=(\vert \widehat{\beta }_j^{init}\vert )^{-1}\) with \(\widehat{\varvec{\beta }}^{init}=\widehat{\varvec{\beta }}_{ols}\) . They showed that for \(0<\gamma <1\) , and under additional model assumptions, adaptive bridge estimators enjoy the oracle property. For \(p>>n\) , \(\widehat{\varvec{\beta }}_{ols}\) cannot be computed and thus other initial estimates, such as \(\widehat{\varvec{\beta }}_{ridge}\) , have to be used. Theoretical properties of the adaptive bridge estimator for \(p>>n\) do not seem to have been well studied thus far.

The adaptive LASSO estimator was proposed by [ 37 ] to remedy the problem of the lack of the oracle property of the LASSO estimator [ 16 , 17 ]. The penalty for the adaptive LASSO is given by (adaptive bridge estimator with \(\gamma =1\) )

where the adaptive data-driven weights \(\{{w}_j\}_{j=1}^p\) can be computed as \({w}_j=(\vert \widehat{\beta }_j^{init}\vert )^{-\nu }\) with \(\widehat{\varvec{\beta }}^{init}\) an initial root- n consistent estimate of \(\varvec{\beta }\) obtained through least squares (or ridge regression if multicollinearity is important) and \(\nu\) is a positive constant. Consequently,

with \(\nu\) chosen appropriately, performs as well as the oracle, i.e., the adaptive LASSO achieves the oracle property. Nevertheless, this estimator still inherits the LASSO’s instability with high dimensional data. The values of \(\lambda\) and \(\nu\) can be simultaneously selected from a grid of values, with values of \(\nu\) selected from \(\{0.5,1,2\}\) , using two-dimensional cross-validation [ 37 ].

Grandvalet [ 38 ] shows that the adaptive ridge estimator (adaptive bridge estimator with \(\gamma =2\) ) is equivalent to the LASSO in the sense that both produce the same estimate and thus the adaptive ridge is not considered further.

Adaptive elastic-net

The adaptive elastic-net ( a ENET) combines the ridge and a LASSO penalties to achieve the oracle property [ 39 ] while at the same time alleviating the instability of the a LASSO with high dimensional data. The method first computes \(\widehat{\varvec{\beta }}_{enet}\) as described above for the elastic net estimator, then constructs the adaptive weights as \(\widehat{w}_j=(|\widehat{\beta }_{j,enet}|)^{-\nu }\) , where \(\nu\) is a positive constant, and then solves

where \(k=1+\lambda _2\) if the predictors are standardized (as we assume) or \(k=1+\lambda _2/n\) otherwise. In particular, when \(\lambda _2=0\) the adaptive elastic-net reduces to the a LASSO estimator. This is also the case when the design matrix is orthogonal regardless of the value of \(\lambda _2\) [ 20 , 37 , 39 ].

Other adaptive regularization methods are the multi-step adaptive ENET ( ma ENET), the adaptive smoothly clipped absolute deviations ( a SCAD) and the adaptive minimax concave penalty ( a MCP) methods. Details of the penalty functions and noteworthy properties of the latter three methods are summarized elsewhere [ 6 , 40 ].

Regularized group regression methods

Regularized regression methods that select individual predictors do not exploit information on potential grouping structure among markers, such as that arising from the association of markers with particular Quantitative Trait Loci (QTL) on a chromosome or haplotype blocks, to enhance the accuracy of genomic prediction. The nearby SNP markers in such groups are linked, producing highly correlated predictors. If such grouping structure is present but is ignored by using models that select individual predictors only, then such models may be inefficient or even inappropriate, reducing the accuracy of genomic prediction [ 9 ]. Regularized group regression methods are regularized regression methods with penalty functions that enable the selection of the important groups of covariates and include group bridge ( g bridge), group LASSO ( g LASSO), group SCAD ( g SCAD) and group MCP ( g MCP) methods (see [ 9 , 41 , 42 , 43 , 44 , 45 , 46 ] for detailed reviews). Some grouping methods such as the group bridge, sparse group LASSO ( sg LASSO) and group MCP, besides allowing for group selection, also select the important members of each group [ 43 ] and are therefore said to perform bi-level selection, i.e., group-wise and within-group variable selection. Bi-level selection is appropriate if predictors are not distinct but have a common underlying grouping structure.

Estimators and penalty functions for the regularized grouped methods can be formulated as follows. Consider subsets \(A_1,\ldots ,A_L\) of \(\{1,\dots ,p\}\) ( L being the total number of covariate groups), representing known covariate groupings of design vectors, which may or may not overlap. Let \(\varvec{\beta }_{A_l}=(\beta _k , k \in A_l)\) be the regression coefficients in the l -th group and \(p_l\) the cardinality of the l -th group (i.e., the number of unique elements in \(A_l\) ). Regularized group regression methods estimate \(\varvec{\beta }=(\varvec{\beta }_{A_1},...,\varvec{\beta }_{A_L})'\) by minimizing

where \({\textbf{X}}_{.l}\) is a matrix with columns corresponding to the predictors in group l .

Because \(\sum \limits _{i=1}^n\Big (y_i-\sum \limits _{l=1}^L {\textbf{X}}_{il}\varvec{\beta }_{A_l}\Big )^2\) in ( 10 ) is equivalent to RSS some authors use the RSS formulation directly. It is assumed that all the covariates belong to at least one of the groups. Table 5 summarizes the methods described in this section.

Group bridge-type estimators

Group bridge-type estimators use in ( 10 ) the penalty term \(p_{\lambda }(\varvec{\beta })=\lambda \sum \limits _{l=1}^L c_l\Vert \varvec{\beta }_{A_l}\Vert _1^{\gamma }\) with \(c_l\) constants that adjust for the different sizes of the groups. The group bridge-type estimators are thus obtained as

A simple and usual choice for the \(c_l\) constants consists in considering each \(c_l\propto p_l^{1-\gamma }\) . When \(0<\gamma <1\) group bridge can be used simultaneously for group and individual variable selection. Also, note that under these assumptions, the group bridge estimator correctly selects groups with nonzero coefficients with probability converging to one under reasonable regularity conditions, i.e., it enjoys the oracle group selection property (see [ 47 ] for details). When the group sizes are all equal to one, i.e., \(p_l=1 \ \forall \ 1\le l \le L\) , then group bridge estimators reduce to the bridge estimators.

Group LASSO and sparse group LASSO

Group LASSO regression uses in ( 10 ) the penalty function \(\texttt{p}_{\lambda }(\varvec{\beta })=\lambda \sum \limits _{l=1}^L\sqrt{p_l}||\varvec{\beta }_{A_l}||_2\) . The group LASSO estimator is thus given by

Unlike the group bridge estimator ( \(0<\gamma <1\) ), g LASSO is designed for group selection, but does not select individual variables within the groups. Indeed, its formulation is more akin to that of the adaptive ridge estimator [ 47 ]. As with the group-bridge estimator, when the group sizes are all equal to one, i.e., \(p_l=1 \ \forall \ 1\le l \le L\) , the g LASSO estimator reduces to the LASSO estimator.

Because the g LASSO does not yield sparsity within a group (it either discards or retains a whole group of covariates) the sparse group lasso ( sg LASSO), which blends the LASSO and the g LASSO penalties, was proposed [ 48 , 49 ]. Specifically, the sg LASSO estimator is given by

where \(\alpha \in [0,1]\) provides a convex combination of the lasso and group lasso penalties ( \(\alpha =0\) gives the g LASSO fit, \(\alpha =1\) gives the LASSO fit). The g LASSO is superior to the standard LASSO under the strong group sparsity and certain other conditions, including a group sparse eigenvalue condition [ 50 ]. Because the sgLASSO lacks the oracle property, the adaptive sparse group LASSO was recently proposed to remedy this drawback [ 51 ].

Note that there are two types of sparsity, i.e., (i) “groupwise sparsity”, which refers to the number of groups with at least one nonzero coefficient, and (ii) “within group sparsity” that refers to the number of nonzero coefficients within each nonzero group. The “overall sparsity” usually refers to the total number of non-zero coefficients regardless of grouping.

Other group regularization methods are the hierarchical group LASSO ( h LASSO), the group smoothly clipped absolute deviations ( g SCAD) and the group minimax concave penalty ( g MCP) methods. Details of the penalty functions and salient properties of these methods can be found in [ 9 , 52 , 53 , 54 , 55 ].

Bayesian regularized estimators

The two Bayesian methods we consider are based on the Bayesian basic linear regression model [ 10 ]. They assume a continuous response \({\textbf{y}}=(y_1, \ldots , y_n)\) so that the regression equation can be represented as \(y_i = \eta _i + \varepsilon _i\) , where \(\eta _i\) is a linear predictor (the expected value of \(y_i\) given predictors) and \(\varepsilon _i\) are independent normal model residuals with mean zero and variance \(w_i^2\sigma ^2_{\varepsilon }\) , with \(w_i\) representing user defined weights and \(\sigma ^2_{\varepsilon }\) is a residual variance parameter. The model structure for the linear predictor \(\varvec{\eta }\) is constructed as follows

with an intercept \(\mu\) (equivalent to \(\beta _0\) in equation ( 1 )), design \(n\times p\) matrix \({\textbf{X}}\) for predictor vectors \({\textbf{X}}_j = (x_{ij})\) and fixed effects vectors \(\varvec{\beta }_j\) associated with the the predictors \({\textbf{X}}_j\) .

The likelihood function of the data has the following conditional distribution:

with the general parameter vector \(\varvec{\theta }\) representing the vector of all unknowns, such as the intercept, all the regression coefficients and random effects, the residual variance as well as parameters and hyper-parameters subject to inference in the hierarchical Bayesian model.

The prior distribution factorises as follows:

In the basic form of the model the following prior settings are typically chosen:

The intercept is assigned a flat prior \(p(\mu ) = \frac{1}{\sqrt{2 \cdot \pi } \sigma _M} e^{-\frac{\mu ^2}{2 \cdot \sigma _M^2}}\) with prior hyper-parameter \(\sigma _M^2\) chosen to be very large to make the prior flat.

The residual variance is assigned a scaled-inverse \(\chi ^2\) density \(p(\sigma ^2) = \chi ^{-2}(S_{\varepsilon }|\text {df}_{\varepsilon })\) with degrees of freedom parameter \(\text {df}_{\varepsilon }\) (> 0) and scale parameter \(\text {S}_{\varepsilon }\) (> 0).

The priors for the regression coefficients \(\beta _{jk}\) can be chosen in different ways, for example, as flat priors similar to the intercept, which is considered an uninformative choice. Choosing informative priors not only provides a chance to introduce information on the coefficients known from previous runs of the study, but also allows performing penalized or regularized regression, such as Ridge regression or the LASSO through the choice of suitable priors.

Those coefficients utilizing flat priors are called “fixed” effects, as the estimation of the posterior is based only on information contained in the data itself, encoded by the likelihood. This is the reference model for regularised Bayesian models.

Choosing a Gaussian prior, according to [ 18 ], yields Ridge regression shrinkage estimation. Similar to [ 10 ] we call this approach the Bayesian ridge regression. Choosing double-exponential priors corresponds to the Bayesian LASSO model [ 10 ].

Ensemble methods

Ensemble methods build multiple models using a given learning algorithm and then combine their predictions to produce an optimal estimate. The two most commonly used algorithms are bagging (or bragging) and boosting . Whereas bagging is a stagewise procedure that combines the predictions of multiple models (e.g., classification or regression trees) to yield an average prediction, boosting is a stagewise process in which each stage attempts to improve the predictions at the previous stage by up-weighting poorly predicted values. Below, we briefly discuss two popular ensemble methods, namely, random forests, an extension of bagging, and gradient boosting algorithms. Note that, although variable scaling (centering or standardizing) might accelerate convergence of the learning algorithms, the ensemble methods do not require it. Indeed, the collection of partition rules used with the ensemble methods should not change with scaling.

Random forests (RF)

The random forests algorithm is an ensemble algorithm that uses an ensemble of unpruned decision (classification or regression) trees, each grown using a bootstrap sample of the training data, and randomly selected (without replacement) subsets of the predictor variables (features) as candidates for splitting tree nodes. The randomness introduced by bootstrapping and selecting a random subset of the predictors reduces the variance of the random forest estimator, often at the cost of a slight increase in bias. The RF regression prediction for a new observation \(y_i\) , say \(\widehat{y}_i^B\) , is made by averaging the output of the ensemble of B trees \(\{T(y_i,\Psi _b)\}_{b=1,...,B}\) as [ 56 ]

where \(\Psi _b\) characterizes the b -th RF tree in terms of split variables, cut points at each node, and terminal node values. Recommendations on how to select the number of trees to grow, the number of covariates to be randomly chosen at each tree node and the minimum size of terminal nodes of trees, below which no split is attempted, are provided by [ 57 , 58 ]. We refer to [ 56 , 57 , 58 ] for further details on the RF regression.

Stochastic gradient boosting (SGB)

Boosting enhances the predictive performance of base learners such as classification or regression trees, each of which performs only slightly better than random guessing, to become arbitrarily strong [ 56 ]. As with RF, boosting algorithms can also handle interactions, nonlinear relationships, automatically select variables and are robust to outliers, missing data and numerous correlated and irrelevant variables. In regression, boosting is an additive expansion of the form

where \(\beta _1,\dots ,\beta _M\) are the expansion coefficients and the basis functions \(h({\textbf {X}};\gamma )\) , base learners, are functions of the multivariate argument \({\textbf {X}}\) , characterized by a set of parameters \(\gamma =(\gamma _1,\dots ,\gamma _M)\) . Typically these models are fit by minimizing a loss function L (e.g., the squared-error loss) averaged over the training data

We used regression trees as basis functions in which the parameters \(\gamma _m\) are the splitting variables, split points at the internal nodes, and the predictions at the terminal nodes. Boosting regression trees involves generating a sequence of trees, each grown on the residuals of the previous tree. Prediction is accomplished by weighting the ensemble outputs of all the regression trees. We refer to [ 49 , 56 , 59 ] for further details on SGB (see, e.g., [ 59 ] for the interpretation of boosting in terms of regression for a continuous, normally distributed response variable).

Instance-based methods

For the instance-based methods, scaling before applying the method is crucially important. Scaling the variables (features) prior to model fitting prevents possible numerical difficulties in the intermediate calculations and helps avoid domination of numeric variables with smaller by those with greater magnitude and range.

Support vector machines

Support vector machines (SVM) is a popular supervised learning technique for classification and regression of a quantitative response y on a set of predictors, in which case the method is called support vector regression or SVR [ 60 ]. In particular, SVR uses the model

with \({\textbf {x}}_i=(x_{i1},\dots ,x_{ip})'\) and where the approximating function \(f({\textbf {x}}_i)\) is a linear combination of basis functions \(h({\textbf {x}}_i)^T\) , which can be linear (or nonlinear) transformations of \({\textbf {x}}_i\) . The goal of SVR is to find a function f such that \(f({\textbf {x}}_i)\) deviates from \(y_i\) by a value no greater than \(\varepsilon\) for each training point \({\textbf {x}}_i\) , and at the same time is as flat as possible. This so-called \(\varepsilon\) -insensitive SVR, or simply \(\varepsilon\) -SVR, thus fits a model ( 14 ) using only those residuals which are smaller in absolute value than \(\varepsilon\) and a linear loss function for larger residuals. The choice of the loss function (e.g., linear, quadratic, Huber) usually considers the noise distribution pertaining to the data samples, level of sparsity and computational complexity.

If Eq. ( 14 ) is the usual linear regression model, i.e., \(y_i=f({\textbf {x}}_i)=\beta _0+{\textbf {x}}_i^T\varvec{\beta }\) , one considers the following minimization problem

where \(\lambda\) is the regularization parameter (cost) that controls the trade-off between flatness and error tolerance, \(\Vert .\Vert\) refers to the norm under a Hilbert space (i.e., \(\Vert \textbf{x} \Vert = \sqrt{\langle \textbf{x}{,} \textbf{x}\rangle }\) with \(\textbf{x}\) a \(p\ge 1\) dimensional vector) and

is an \(\varepsilon\) -insensitive linear loss. Given the minimizers of ( 15 ) \(\hat{\beta }_0\) and \(\hat{\varvec{\beta }}\) , the solution function has the form

where \(\hat{\alpha }^*_i, \ \hat{\alpha }_i\) are positive weights given to each observation (i.e., to the column vector \({\textbf{x}}_i\) ) estimated from the data. Typically only a subset of \((\hat{\alpha }_i^*-\hat{\alpha }_i)\) are non-zero with the observations associated to these so called support vectors , and thus the name of the method, SVM. More details on SVM can be found in [ 56 ].

Deep learning methods

Deep learning (DL) algorithms are implemented through neural networks, which encompass an assortment of architectures (e.g., convolutional, recurrent and densely connected neural networks) and depend on many parameters and hyperparameters whose careful optimization is crucial to enhancing predictive accuracy and minimizing overfitting (see [ 8 , 61 , 62 , 63 , 64 , 65 ] for further insights into DL architectures and other particulars and the supplementary materials https://github.com/miguelperezenciso/DLpipeline of [ 8 ] for a list of the main DL hyperparameters, their role and related optimization issues). It can be very challenging to achieve great improvements in predictive accuracy in genomic prediction studies with DL because hyperparameter optimization can be extremely demanding and also because DL requires very large training datasets which might not always be available [ 1 , 2 , 3 , 4 ].

After selecting a DL architecture there is usually a large set of parameters to be set in order to minimize some fitting criterion such as least squares or some measure of entropy from some training data (network training). Therefore, an optimization method must also be selected. The three top ranked optimizers for neural networks are mini-batch gradient descent, gradient descent with momentum and adaptive moment estimation (ADAM; [ 66 ]). Among the three, the mini-batch gradient descent and Adam are usually preferred, because they perform well most of the time. In terms of convergence speed, ADAM is often clearly the winner and thus a natural choice [ 67 ].

Next, we offer a few more details on the feed-forward and convolutional neural networks, which, besides being some of the most popular DL architectures, are well suited for regression problems. These models can be represented graphically as a set of inputs linked to the outputs through one or more hidden layer. Figure 1 a represents such a model (either FFNN or CNN) with a single hidden layer.

figure 1

Graphical representation of a a feed-forward neural network (FFNN) with one hidden layer; and b a convolution of a filter \((v_1,v_2,v_3)\) , with stride=2, on the Input Channel \((x_1,x_2,\dots )\) . The result is in the Output Channel \((y_1,y_2,\dots )\)

Further details on neural networks in general and FFNN and CNN in particular can be found in [ 1 , 2 , 3 , 4 , 8 , 56 ]. Note that, to avoid potential numerical difficulties, it is recommended that both the target (response variable; here assumed to be continuous and normally distributed), and the features (covariates) are standardized prior to training the network [ 8 ].

Feed-forward neural network (FFNN)

A feed-forward neural network (FFNN), also known in the literature as a multi-layer perceptron (MLP), is a neural network that does not assume a specific structure in the input features (i.e., in the covariates). This neural network consists of an input layer, an output layer and multiple hidden layers between the input and output layers.

The model for a FFNN with one hidden layer expressed as a multiple linear regression model ( 1 ) is given by

where the \(y_i\) (output) and \(x_{ij}\) (input) are defined as in model ( 1 ), \(\alpha\) is the output bias, h runs over the units of the hidden layer, \(\alpha _h\) refers to the bias of the h -th unit of the hidden layer, \(w_{jh}\) refer to the weights between the inputs and the hidden layer, \(w_h\) refer to the weights between the hidden layer and the output, \(\phi\) is the activation function of the hidden layer. The model parameters \(\alpha\) , \(\alpha _h\) , \(w_h\) and \(w_{jh}\) are unknown network parameters that need to be estimated in the network training process.

Convolutional neural network (CNN)

A convolution neural network (CNN) is a neural network that contains one or more convolution layers, which are defined by a set of filters. Although a CNN generally refers to a 2-dimensional neural network, which is used for image analysis, in this study we consider a 1-dimensional (1D) CNN. Here, the input to the 1D convolution layer is a vector \({\textbf{x}}=(x_1,\dots ,x_p)\) equal to one row of the \(n\times p\) marker matrix \(\textbf{X}\) . The 1D convolution filter is defined by a vector \({\textbf{v}}=(v_1,\dots ,v_d)\) where \(d<p\) . The convolution of a filter \({\textbf{v}}\) with \({\textbf{x}}\) , which is called a channel , is a vector \({\textbf{y}}=(y_1,y_2,\dots )\) satisfying

where s , i.e., the stride length, is the shift displacement of the filter across the input data. An activation function is applied after each convolution to produce an output. Figure 1 b depicts a 1D convolution of a filter \((v_1,v_2,v_3)\) on the input vector \((x_1,x_2,\dots ,x_9,\dots )\) , considering a stride of length \(s=2\) , which results in the output channel \((y_1,y_2,\dots )\) . Filter values \(v_1,\dots , v_d\) are model parameters that are estimated in the neural network training process.

Performance assessment

For the simulated dataset, we assessed predictive performance using predictive accuracy (PA), the Pearson correlation between the predicted (PGBVs) and the simulated true (TGBVs) breeding values. For all the three KWS empirical data sets, predictive performance was expressed as predictive ability (PA), the Pearson correlation between the PGBVs and the observed (adjusted means estimated from phenotypic analysis) genomic breeding values (OGBVs), also calculated using cross validation. The simulated true breeding values are specified in the simulation model and therefore are known exactly. In contrast, for empirical data, the true breeding values are unknown and are approximated by the observed breeding values estimated as adjusted means during phenotypic analysis. The higher the PA, the better is the relative predictive performance of a method. We additionally assessed the predictive performance of the methods using the out-of-sample mean squared prediction error (MSPE) and the mean absolute prediction error (MAPE). Specifically,

where the \(y_i\) and \(\bar{y}\) are, respectively, the TGBVs and mean TGBVs for the single simulated dataset, but the OGBVs and mean OGBVs for the empirical datasets, and the \(\hat{y}_i\) and \(\bar{\hat{y}}_i\) are, respectively, the PGBVs and mean PGBVs. 10-fold CV is used to assess the PA for each method for the simulated datasets in contrast to the 5-fold CV used with the three empirical maize datasets. Although we report both the prediction errors and the PA, breeders are primarily interested in the final ordering of the genotypes, which the PA captures better than the prediction errors.

For the cross validation, we aimed to have at least 150 individuals per fold. Accordingly, each phenotypic dataset was randomly split into k approximately equal parts. The breeding values for each of the k folds were predicted by training the model on the \(k-1\) remaining folds and a CV error (CVE) computed for each of the k folds. The method with the smallest CVE was selected to predict the breeding values for the unphenotyped genotypes for the simulated dataset, and the phenotyped genotypes in the validation sets for each of the three empirical maize datasets.

All the methods are implemented in the R software and are available in various R packages [ 10 , 32 , 40 , 43 , 48 , 54 , 58 , 68 , 69 , 70 , 71 , 72 , 73 ]. Table S1 (Additional file 5 , Section 3) lists the R packages we used to analyse the synthetic and real datasets. For the deep learning methods, and because of fine tuning requirements, we used the Python software and packages Numpy, Pandas and Tensorflow [ 74 , 75 ]. All R and Python codes referring to the simulated data are provided in Additional files 2 & 3 .

Noteworthy details of model fitting are available in the supplementary materials (Additional file 5 , Section 2).

Although we did not fully quantify the computational costs of the different methods, the computational burden increased strikingly from the simple regularized through the adaptive to the grouped methods. A similar trend was also apparent from the ensemble, through the instance-based to the deep learning methods. Computational time may be reduced greatly by parallelizing the estimation or optimization algorithms, but this strategy may not always be available and can be challenging to implement for some methods.

The relative performances of the various methods on the simulated data varied with the target trait and with whether performance was assessed in terms of predictive accuracy or prediction error. Performance also varied in terms of computational cost with some methods requiring considerably more time than others. Results of genomic prediction accuracy for the simulated data are displayed in Figs. 2 , 3 and 4 and Tables S2-S5 (Additional file 5 , Section 3). Tables S6 & S7 (Additional file 5 , Section 3) report the calibration details for the fitted feed-forward and convolutional neural networks.

figure 2

Prediction accuracy (PA) of the regularized, adaptive regularized and Bayesian regularized methods, computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. The choice of \(\lambda\) , where applicable, was based on the 10-fold CV. The mean squared and absolute prediction errors are also provided. See Table S 2 for details

figure 3

Prediction accuracy (PA) of the group regularized methods (mean and range values of PA across the different groupings), computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. Choice of \(\lambda\) was based on the 10-fold CV. Display refers to the mean, max and min values of PA across all the 10 grouping schemes. The mean squared and absolute prediction errors are also provided. See Table S 3 for details

figure 4

Prediction accuracy (PA) of the ensemble, instance-based and deep learning methods, computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. See Tables S 4 -S 5 for details

Table 6 displays the range of the observed predictive accuracies across all the classes of the regularized methods for traits \(T_1-T_3\) . Neither the adaptive, group, nor Bayesian regularized methods seem to improve upon the results of their regularized counterparts, although group regularized methods do provide some slight improvement upon the results of the adaptive regularized methods. Even though all the regularized regression methods had comparable overall performance, the best compromise between high PA ( \(\ge 0.77\) for \(T_1\) , 0.82 for \(T_2\) and 0.81 for \(T_3\) ) and small prediction errors was achieved by the LASSO, ENET, sENET and SCAD (Fig.  2 and Table S 2 ; first half). Within the class of adaptive regularized methods, the best compromise was achieved by aLASSO and aENET (Fig.  2 and Table S 2 ; second half; PA \(\ge 0.72\) for \(T_1\) , 0.78 for \(T_2\) and 0.80 for for \(T_3\) ). For the group regularized methods, a good compromise was achieved by the gLASSO and gSCAD (Fig.  2 and Table S 2 ; mean PA values \(\ge 0.76\) for \(T_1\) , 0.82 for \(T_2\) and 0.81 for \(T_3\) ). Whereas the worst performing group regularized methods in terms of the estimated PAs were the cMCP and gel for \(T_1\) (PA \(<0.7\) ), sgLASSO and gel for \(T_2\) (PA \(<0.8\) ) and hLASSO and gel for \(T_3\) (PA \(<0.8\) ), the worst performing methods in terms of prediction errors were the gel ( \(T_1\) & \(T_2\) only) and sgLASSO ( \(T_3\) only). Of all the group regularized methods, the most time consuming were the sgLASSO and hLASSO, with sgLASSO requiring several more months to compute results for trait \(T_1\) than for traits \(T_2\) or \(T_3\) . In the comparisons between the two Bayesian regularized methods, Lasso Bayes consistently outperformed the Ridge Bayes method across all the three traits, demonstrating superior predictive accuracy and generally smaller prediction errors.

The ensemble, instance-based and deep learning methods did not improve upon the results of the regularized, the group or the Bayesian regularized methods (Fig.  4 and Tables S 4 & S 5 ). Among the ensemble and instance-based groups of methods, RF provided the best compromise between high PA and small prediction errors. For the deep learning methods, the FFNN provided consistently higher PA values than CNN across all the three traits from the simulated data.

Predictive performance varied not only among the methods but also with the target quantitative traits. Specifically, trait \(T_3\) had the highest predictive accuracies for the adaptive methods, whereas trait \(T_2\) was generally top ranked across all the remaining methods.

The ridge regression methods plus the overall best performing methods (high PA values and low prediction errors) for each class of methods based on the analysis of the simulated dataset, were applied to each of the three KWS empirical maize datasets. The specific methods fitted to the KWS maize datasets comprised RR-CV, RR-REML, sENET, aENET (enet penalty), gLASSO, RF, FFNN and lBayes.

Results are displayed in Fig.  5 and Table S8 (Additional file  5 , Section 3). Across the three real maize datasets, the highest predicitive abilities were obtained for the 2010 dataset. The estimated predictive abilities (PA) are under 0.7 for the 2010 dataset but under 0.6 for the 2011 dataset and generally under 0.6 for the 2012 dataset (RR-REML and lBayes excluded with estimated PAs of 0.616 and 0.624, respectively), regardless of the method used. The lBayes and RR-REML (2011 & 2012 datasets) and RF, RR-REML and lBayes (2010 dataset) are evidently the best performing methods (higher PA values and lower prediction errors). On the other hand, aENET \(^e\) (2010 & 2011 datasets) and RF (2012 dataset) are the worst performing methods (lower PA and higher prediction errors). Interestingly, the RF performed both the best (2010 dataset) and the worst (2012 dataset), clearly emphasizing that the methods are strongly data dependent.

figure 5

Predictive ability (PA; mean and range values computed across the 5-fold validation datasets and 10 replicates) of the regularized and adaptive regularized methods, computed as the Pearson correlation coefficient between the observed breeding values (OBVs) and the predicted breeding values (PBVs), for the KWS datasets. The choice of \(\lambda\) , where applicable, was based on 4-fold CV. See Table S 8 for details

We have investigated the predictive performance of several state-of-the art machine learning methods in genomic prediction via the use of one simulated and three real datasets. All the methods showed reasonably high predictive performance for most practical selection decisions. But the relative predictive performance of the methods was both data and target trait dependent, complicating and precluding omnibus comparative evaluations of the genomic prediction methods, thus ruling out selection of one procedure for routine use in genomic prediction. These results broaden the findings of earlier studies (e.g. [ 9 ]) to encompass a wider range of groups of methods. If reproducibility of results, low computational cost and time are important considerations, then using the regularized regression methods comes highly recommended because they consistently produced, with relatively lower computational cost and computing time, reasonably accurate and competitive predictions relative to the other groups of methods for the simulated and the three real datasets. Even among the regularized regression methods, increasing model complexity from simple through the adaptive to grouped or even the Bayesian regularized methods, generally only increased computing time without clearly improving predictive performance.

The ensemble, instance-based and deep-learning ML methods need the tuning of numerous hyperparameters thus requiring considerable computing time to adequately explore the entire hyperparameter space. This will not always be possible in most applications because of limiting time and computational resources leading to potentially less than optimal results and may well partly explain why these methods did not clearly outperform the classical ML methods. Indeed, the computational costs of the ensemble, instance-based and deep learning methods can quickly become prohibitive, if all the parameters are tuned by searching over the often large grid of values. This will typically require not only proficiency in programming and algorithm parallelization and optimization, but excellent computing resources. These constraints, plus the growing size of phenotypic and genomic data, make it difficult to identify methods for routine use in genomic prediction and call for greater focus on and investment in enhancing the computational efficiencies of algorithms and computing resources.

We have considered only well tested and established off-the-shelf machine learning methods and one simulated and three real datasets. We are extending this work to cover the following four objectives. (1) Comparing the performance of methods that use advanced techniques for feature selection or dimensionality reduction on multiple synthetic datasets simulated using different configurations or scenarios. (2) Exploring how the methods generalize based on different training/test splits across simulations/real-world datasets, individuals/samples, or chromosomes. (3) Evaluating the sensitivity of the different methods to hyperparameter selection. (4) Assessing the training and testing complexity for the different methods.

Machine learning methods are well suited for efficiently handling high dimensional data. Particularly, supervised machine learning methods have been successfully used in genomic prediction or genome-enabled selection. However, their comparative predictive accuracy is still poorly understood, yet this is a critical issue in plant and animal breeding studies given that increasing methodological complexity can substantially increase computational complexity or cost. Here, we showed that predictive performance is both data and target trait dependent thus ruling out selection of one method for routine use in genomic prediction. We also showed that for this reason, relatively low computational complexity and competitive predictive performance, the classical linear mixed model approach and regularized regression methods remain strong contenders for genomic prediction.

Availability of data and materials

The simulated animal data from the QTLMAS workshop 2012 is provided in the supplementary materials together with the annotated R and Python codes used to analyse these data. The KWS data is proprietary data and cannot be shared publicly for confidentiality reasons. These can only be shared upon reasonable request and with KWS' express consent. This notwithstanding, we provide a synthetic dataset that mimics the KWS data, which can be used with our codes to illustrate the implementation of the ML methods.

Abbreviations

Adaptive moment estimation

Best linear unbiased prediction

Cross-validation

Deep learning

Elastic net

Feed-forward neural network

  • Genomic prediction
  • Genomic selection

Least absolute shrinkage and selection operator

Mean absolute prediction error

Minimax concave penalty

Machine learning

Multi-layer perceptron

Mean squared prediction error

Ordinary least squares

Predictive accuracy/ability

Principal component analysis

Predicted genomic breeding value

Quantitative trait loci

Restricted maximum likelihood

Random forests

Ridge regression

Residual sum of squares

Smoothly clipped absolute deviation

Stochastic gradient boosting

Single nucleotide polymorphism

True genomic breeding value

Support vector machine

Support vector regression

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Acknowledgements

We thank KWS for providing the maize datasets. We thank the Centre for Mathematical Analysis, Geometry, and Dynamical Systems, from Instituto Superior Técnico (IST) of the University of Lisbon, for granting access to their computing resources to run the Deep Learning Models.

Open Access funding enabled and organized by Projekt DEAL. This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications). The German Federal Ministry of Education and Research (BMBF) funded this research within the AgroClustEr “Synbreed - Synergistic plant and animal breeding” (Grant ID: 0315526). JOO was additionally supported by the German Research Foundation (DFG, Grant # 257734638). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Joseph O. Ogutu & Hans-Peter Piepho

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Contributions

VML, JOO and HPP conceived the project. RAPR wrote the Python code, selected and trained the deep learning models. AP selected and programmed the Bayesian models and wrote the corresponding theory. VML and JOO wrote the R code, performed the simulations and all the other analyses. VML wrote the initial draft of the manuscript. JOO, RAPR and HPP contributed to writing and revising the manuscript. All authors read and approved the final version of the manuscript.

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Correspondence to Vanda M. Lourenço or Joseph O. Ogutu .

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Supplementary Information

Additional file 1..

Simulated (animal breeding) dataset. Includes four txt files: one for the grouping schemes, one for the QTLMAS prediction data, one for the QTLMAS training data, and one for the validation trait values.

Additional file 2.

R codes used to fit the ML algorithms to the simulated (animal breeding) dataset. Includes six R files: one for the simple regularized methods, one for the adaptive regularized methods, one for the group regularized methods, one for the Bayesian regularized methods, one for the ensemble methods, and one for the instance-based methods.

Additional file 3.

Python codes used to fit the deep learning (FFNN & CNN) algorithms to the simulated (animal breeding) dataset. Includes six py and three pnz files: three of the py files refer to the FFNN fits and the other three to the CNN fits; each of the three pnz files include six npy files referring to the training of the FFNNs for traits 1, 2 & 3, respectively.

Additional file 4.

Includes SAS code for (i) the phenotypic data analysis (S1 Text.doc); (ii) SNP grouping schemes (S2 Text.doc); and (iii) the 5-fold data split (S3 Text.doc & S4 Text.doc) for the KWS \(2010-2012\) data sets.

Additional file 5.

Includes the RR-BLUP model used to estimate variance components for the KWS real maize data (Section 1), the Noteworthy details of model fitting (Section 2) plus the additional Tables of results (Section 3). Table S1. List of R and Python packages used in this paper. Table S2. Prediction accuracy (PA) of the regularized, adaptive regularized and Bayesian regularized methods, computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. The choice of \(\lambda\) , where applicable, was based on the 10-fold CV. The mean squared and absolute prediction errors are also provided. Table S3. Prediction accuracy (PA) of the group regularized methods (mean and range values of PA across the different groupings), computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. Choice of \(\lambda\) was based on the 10-fold CV. Display refers to the mean, max and min values of PA across all the 10 grouping schemes. The mean squared and absolute prediction errors are also provided. Table S4. Prediction accuracy (PA) of the ensemble and instance-based methods, computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. Table S5. Prediction accuracy (PA) of the deep learning methods, computed as the Pearson correlation coefficient between the true breeding values (TBVs) and the predicted breeding values (PBVs), for the simulated dataset, where \(T_1-T_3\) refer to three quantitative milk traits. Table S6. Best FFNN model calibration parameters selected for each of the three quantitative milk traits \(T_1-T_3\) . Table S7. Best CNN model calibration parameters (Number of epochs/Learning rate) selected for each of the three quantitative milk traits \(T_1-T_3\) . Table S8. Predictive ability (PA; mean and range values computed across the 5-fold validation datasets and 10 replicates) of the regularized, adaptive regularized, group regularized, Bayesian regularized, ensemble, instance-based and deep learning methods, computed as the Pearson correlation coefficient between the observed breeding values (OBVs) and the predicted breeding values (PBVs), for the KWS datasets. The choice of \(\lambda\) , where applicable, was based on 4-fold CV.

Observation for \(\widehat{\varvec{\beta }}_{BLUP}\) derivation:

Observation for \(\widehat{\textbf{g}}_{blup}\) derivation:, rights and permissions.

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Lourenço, V., Ogutu, J., Rodrigues, R. et al. Genomic prediction using machine learning: a comparison of the performance of regularized regression, ensemble, instance-based and deep learning methods on synthetic and empirical data. BMC Genomics 25 , 152 (2024). https://doi.org/10.1186/s12864-023-09933-x

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Accepted : 20 December 2023

Published : 07 February 2024

DOI : https://doi.org/10.1186/s12864-023-09933-x

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  • Breeding value
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