interrelationships among problem solving creativity and transfer of learning

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• Published: 11 January 2023

The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature

• Enwei Xu   ORCID: orcid.org/0000-0001-6424-8169 1 ,
• Wei Wang 1 &
• Qingxia Wang 1  

Humanities and Social Sciences Communications volume  10 , Article number:  16 ( 2023 ) Cite this article

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Collaborative problem-solving has been widely embraced in the classroom instruction of critical thinking, which is regarded as the core of curriculum reform based on key competencies in the field of education as well as a key competence for learners in the 21st century. However, the effectiveness of collaborative problem-solving in promoting students’ critical thinking remains uncertain. This current research presents the major findings of a meta-analysis of 36 pieces of the literature revealed in worldwide educational periodicals during the 21st century to identify the effectiveness of collaborative problem-solving in promoting students’ critical thinking and to determine, based on evidence, whether and to what extent collaborative problem solving can result in a rise or decrease in critical thinking. The findings show that (1) collaborative problem solving is an effective teaching approach to foster students’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]); (2) in respect to the dimensions of critical thinking, collaborative problem solving can significantly and successfully enhance students’ attitudinal tendencies (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI[0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI[0.58, 0.82]); and (3) the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have an impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. On the basis of these results, recommendations are made for further study and instruction to better support students’ critical thinking in the context of collaborative problem-solving.

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Introduction

Although critical thinking has a long history in research, the concept of critical thinking, which is regarded as an essential competence for learners in the 21st century, has recently attracted more attention from researchers and teaching practitioners (National Research Council, 2012 ). Critical thinking should be the core of curriculum reform based on key competencies in the field of education (Peng and Deng, 2017 ) because students with critical thinking can not only understand the meaning of knowledge but also effectively solve practical problems in real life even after knowledge is forgotten (Kek and Huijser, 2011 ). The definition of critical thinking is not universal (Ennis, 1989 ; Castle, 2009 ; Niu et al., 2013 ). In general, the definition of critical thinking is a self-aware and self-regulated thought process (Facione, 1990 ; Niu et al., 2013 ). It refers to the cognitive skills needed to interpret, analyze, synthesize, reason, and evaluate information as well as the attitudinal tendency to apply these abilities (Halpern, 2001 ). The view that critical thinking can be taught and learned through curriculum teaching has been widely supported by many researchers (e.g., Kuncel, 2011 ; Leng and Lu, 2020 ), leading to educators’ efforts to foster it among students. In the field of teaching practice, there are three types of courses for teaching critical thinking (Ennis, 1989 ). The first is an independent curriculum in which critical thinking is taught and cultivated without involving the knowledge of specific disciplines; the second is an integrated curriculum in which critical thinking is integrated into the teaching of other disciplines as a clear teaching goal; and the third is a mixed curriculum in which critical thinking is taught in parallel to the teaching of other disciplines for mixed teaching training. Furthermore, numerous measuring tools have been developed by researchers and educators to measure critical thinking in the context of teaching practice. These include standardized measurement tools, such as WGCTA, CCTST, CCTT, and CCTDI, which have been verified by repeated experiments and are considered effective and reliable by international scholars (Facione and Facione, 1992 ). In short, descriptions of critical thinking, including its two dimensions of attitudinal tendency and cognitive skills, different types of teaching courses, and standardized measurement tools provide a complex normative framework for understanding, teaching, and evaluating critical thinking.

Cultivating critical thinking in curriculum teaching can start with a problem, and one of the most popular critical thinking instructional approaches is problem-based learning (Liu et al., 2020 ). Duch et al. ( 2001 ) noted that problem-based learning in group collaboration is progressive active learning, which can improve students’ critical thinking and problem-solving skills. Collaborative problem-solving is the organic integration of collaborative learning and problem-based learning, which takes learners as the center of the learning process and uses problems with poor structure in real-world situations as the starting point for the learning process (Liang et al., 2017 ). Students learn the knowledge needed to solve problems in a collaborative group, reach a consensus on problems in the field, and form solutions through social cooperation methods, such as dialogue, interpretation, questioning, debate, negotiation, and reflection, thus promoting the development of learners’ domain knowledge and critical thinking (Cindy, 2004 ; Liang et al., 2017 ).

Collaborative problem-solving has been widely used in the teaching practice of critical thinking, and several studies have attempted to conduct a systematic review and meta-analysis of the empirical literature on critical thinking from various perspectives. However, little attention has been paid to the impact of collaborative problem-solving on critical thinking. Therefore, the best approach for developing and enhancing critical thinking throughout collaborative problem-solving is to examine how to implement critical thinking instruction; however, this issue is still unexplored, which means that many teachers are incapable of better instructing critical thinking (Leng and Lu, 2020 ; Niu et al., 2013 ). For example, Huber ( 2016 ) provided the meta-analysis findings of 71 publications on gaining critical thinking over various time frames in college with the aim of determining whether critical thinking was truly teachable. These authors found that learners significantly improve their critical thinking while in college and that critical thinking differs with factors such as teaching strategies, intervention duration, subject area, and teaching type. The usefulness of collaborative problem-solving in fostering students’ critical thinking, however, was not determined by this study, nor did it reveal whether there existed significant variations among the different elements. A meta-analysis of 31 pieces of educational literature was conducted by Liu et al. ( 2020 ) to assess the impact of problem-solving on college students’ critical thinking. These authors found that problem-solving could promote the development of critical thinking among college students and proposed establishing a reasonable group structure for problem-solving in a follow-up study to improve students’ critical thinking. Additionally, previous empirical studies have reached inconclusive and even contradictory conclusions about whether and to what extent collaborative problem-solving increases or decreases critical thinking levels. As an illustration, Yang et al. ( 2008 ) carried out an experiment on the integrated curriculum teaching of college students based on a web bulletin board with the goal of fostering participants’ critical thinking in the context of collaborative problem-solving. These authors’ research revealed that through sharing, debating, examining, and reflecting on various experiences and ideas, collaborative problem-solving can considerably enhance students’ critical thinking in real-life problem situations. In contrast, collaborative problem-solving had a positive impact on learners’ interaction and could improve learning interest and motivation but could not significantly improve students’ critical thinking when compared to traditional classroom teaching, according to research by Naber and Wyatt ( 2014 ) and Sendag and Odabasi ( 2009 ) on undergraduate and high school students, respectively.

The above studies show that there is inconsistency regarding the effectiveness of collaborative problem-solving in promoting students’ critical thinking. Therefore, it is essential to conduct a thorough and trustworthy review to detect and decide whether and to what degree collaborative problem-solving can result in a rise or decrease in critical thinking. Meta-analysis is a quantitative analysis approach that is utilized to examine quantitative data from various separate studies that are all focused on the same research topic. This approach characterizes the effectiveness of its impact by averaging the effect sizes of numerous qualitative studies in an effort to reduce the uncertainty brought on by independent research and produce more conclusive findings (Lipsey and Wilson, 2001 ).

This paper used a meta-analytic approach and carried out a meta-analysis to examine the effectiveness of collaborative problem-solving in promoting students’ critical thinking in order to make a contribution to both research and practice. The following research questions were addressed by this meta-analysis:

What is the overall effect size of collaborative problem-solving in promoting students’ critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills)?

How are the disparities between the study conclusions impacted by various moderating variables if the impacts of various experimental designs in the included studies are heterogeneous?

This research followed the strict procedures (e.g., database searching, identification, screening, eligibility, merging, duplicate removal, and analysis of included studies) of Cooper’s ( 2010 ) proposed meta-analysis approach for examining quantitative data from various separate studies that are all focused on the same research topic. The relevant empirical research that appeared in worldwide educational periodicals within the 21st century was subjected to this meta-analysis using Rev-Man 5.4. The consistency of the data extracted separately by two researchers was tested using Cohen’s kappa coefficient, and a publication bias test and a heterogeneity test were run on the sample data to ascertain the quality of this meta-analysis.

Data sources and search strategies

There were three stages to the data collection process for this meta-analysis, as shown in Fig. 1 , which shows the number of articles included and eliminated during the selection process based on the statement and study eligibility criteria.

This flowchart shows the number of records identified, included and excluded in the article.

First, the databases used to systematically search for relevant articles were the journal papers of the Web of Science Core Collection and the Chinese Core source journal, as well as the Chinese Social Science Citation Index (CSSCI) source journal papers included in CNKI. These databases were selected because they are credible platforms that are sources of scholarly and peer-reviewed information with advanced search tools and contain literature relevant to the subject of our topic from reliable researchers and experts. The search string with the Boolean operator used in the Web of Science was “TS = (((“critical thinking” or “ct” and “pretest” or “posttest”) or (“critical thinking” or “ct” and “control group” or “quasi experiment” or “experiment”)) and (“collaboration” or “collaborative learning” or “CSCL”) and (“problem solving” or “problem-based learning” or “PBL”))”. The research area was “Education Educational Research”, and the search period was “January 1, 2000, to December 30, 2021”. A total of 412 papers were obtained. The search string with the Boolean operator used in the CNKI was “SU = (‘critical thinking’*‘collaboration’ + ‘critical thinking’*‘collaborative learning’ + ‘critical thinking’*‘CSCL’ + ‘critical thinking’*‘problem solving’ + ‘critical thinking’*‘problem-based learning’ + ‘critical thinking’*‘PBL’ + ‘critical thinking’*‘problem oriented’) AND FT = (‘experiment’ + ‘quasi experiment’ + ‘pretest’ + ‘posttest’ + ‘empirical study’)” (translated into Chinese when searching). A total of 56 studies were found throughout the search period of “January 2000 to December 2021”. From the databases, all duplicates and retractions were eliminated before exporting the references into Endnote, a program for managing bibliographic references. In all, 466 studies were found.

Second, the studies that matched the inclusion and exclusion criteria for the meta-analysis were chosen by two researchers after they had reviewed the abstracts and titles of the gathered articles, yielding a total of 126 studies.

Third, two researchers thoroughly reviewed each included article’s whole text in accordance with the inclusion and exclusion criteria. Meanwhile, a snowball search was performed using the references and citations of the included articles to ensure complete coverage of the articles. Ultimately, 36 articles were kept.

Two researchers worked together to carry out this entire process, and a consensus rate of almost 94.7% was reached after discussion and negotiation to clarify any emerging differences.

Eligibility criteria

Since not all the retrieved studies matched the criteria for this meta-analysis, eligibility criteria for both inclusion and exclusion were developed as follows:

The publication language of the included studies was limited to English and Chinese, and the full text could be obtained. Articles that did not meet the publication language and articles not published between 2000 and 2021 were excluded.

The research design of the included studies must be empirical and quantitative studies that can assess the effect of collaborative problem-solving on the development of critical thinking. Articles that could not identify the causal mechanisms by which collaborative problem-solving affects critical thinking, such as review articles and theoretical articles, were excluded.

The research method of the included studies must feature a randomized control experiment or a quasi-experiment, or a natural experiment, which have a higher degree of internal validity with strong experimental designs and can all plausibly provide evidence that critical thinking and collaborative problem-solving are causally related. Articles with non-experimental research methods, such as purely correlational or observational studies, were excluded.

The participants of the included studies were only students in school, including K-12 students and college students. Articles in which the participants were non-school students, such as social workers or adult learners, were excluded.

The research results of the included studies must mention definite signs that may be utilized to gauge critical thinking’s impact (e.g., sample size, mean value, or standard deviation). Articles that lacked specific measurement indicators for critical thinking and could not calculate the effect size were excluded.

Data coding design

In order to perform a meta-analysis, it is necessary to collect the most important information from the articles, codify that information’s properties, and convert descriptive data into quantitative data. Therefore, this study designed a data coding template (see Table 1 ). Ultimately, 16 coding fields were retained.

The designed data-coding template consisted of three pieces of information. Basic information about the papers was included in the descriptive information: the publishing year, author, serial number, and title of the paper.

The variable information for the experimental design had three variables: the independent variable (instruction method), the dependent variable (critical thinking), and the moderating variable (learning stage, teaching type, intervention duration, learning scaffold, group size, measuring tool, and subject area). Depending on the topic of this study, the intervention strategy, as the independent variable, was coded into collaborative and non-collaborative problem-solving. The dependent variable, critical thinking, was coded as a cognitive skill and an attitudinal tendency. And seven moderating variables were created by grouping and combining the experimental design variables discovered within the 36 studies (see Table 1 ), where learning stages were encoded as higher education, high school, middle school, and primary school or lower; teaching types were encoded as mixed courses, integrated courses, and independent courses; intervention durations were encoded as 0–1 weeks, 1–4 weeks, 4–12 weeks, and more than 12 weeks; group sizes were encoded as 2–3 persons, 4–6 persons, 7–10 persons, and more than 10 persons; learning scaffolds were encoded as teacher-supported learning scaffold, technique-supported learning scaffold, and resource-supported learning scaffold; measuring tools were encoded as standardized measurement tools (e.g., WGCTA, CCTT, CCTST, and CCTDI) and self-adapting measurement tools (e.g., modified or made by researchers); and subject areas were encoded according to the specific subjects used in the 36 included studies.

The data information contained three metrics for measuring critical thinking: sample size, average value, and standard deviation. It is vital to remember that studies with various experimental designs frequently adopt various formulas to determine the effect size. And this paper used Morris’ proposed standardized mean difference (SMD) calculation formula ( 2008 , p. 369; see Supplementary Table S3 ).

Procedure for extracting and coding data

According to the data coding template (see Table 1 ), the 36 papers’ information was retrieved by two researchers, who then entered them into Excel (see Supplementary Table S1 ). The results of each study were extracted separately in the data extraction procedure if an article contained numerous studies on critical thinking, or if a study assessed different critical thinking dimensions. For instance, Tiwari et al. ( 2010 ) used four time points, which were viewed as numerous different studies, to examine the outcomes of critical thinking, and Chen ( 2013 ) included the two outcome variables of attitudinal tendency and cognitive skills, which were regarded as two studies. After discussion and negotiation during data extraction, the two researchers’ consistency test coefficients were roughly 93.27%. Supplementary Table S2 details the key characteristics of the 36 included articles with 79 effect quantities, including descriptive information (e.g., the publishing year, author, serial number, and title of the paper), variable information (e.g., independent variables, dependent variables, and moderating variables), and data information (e.g., mean values, standard deviations, and sample size). Following that, testing for publication bias and heterogeneity was done on the sample data using the Rev-Man 5.4 software, and then the test results were used to conduct a meta-analysis.

Publication bias test

When the sample of studies included in a meta-analysis does not accurately reflect the general status of research on the relevant subject, publication bias is said to be exhibited in this research. The reliability and accuracy of the meta-analysis may be impacted by publication bias. Due to this, the meta-analysis needs to check the sample data for publication bias (Stewart et al., 2006 ). A popular method to check for publication bias is the funnel plot; and it is unlikely that there will be publishing bias when the data are equally dispersed on either side of the average effect size and targeted within the higher region. The data are equally dispersed within the higher portion of the efficient zone, consistent with the funnel plot connected with this analysis (see Fig. 2 ), indicating that publication bias is unlikely in this situation.

This funnel plot shows the result of publication bias of 79 effect quantities across 36 studies.

Heterogeneity test

To select the appropriate effect models for the meta-analysis, one might use the results of a heterogeneity test on the data effect sizes. In a meta-analysis, it is common practice to gauge the degree of data heterogeneity using the I 2 value, and I 2  ≥ 50% is typically understood to denote medium-high heterogeneity, which calls for the adoption of a random effect model; if not, a fixed effect model ought to be applied (Lipsey and Wilson, 2001 ). The findings of the heterogeneity test in this paper (see Table 2 ) revealed that I 2 was 86% and displayed significant heterogeneity ( P  < 0.01). To ensure accuracy and reliability, the overall effect size ought to be calculated utilizing the random effect model.

The analysis of the overall effect size

This meta-analysis utilized a random effect model to examine 79 effect quantities from 36 studies after eliminating heterogeneity. In accordance with Cohen’s criterion (Cohen, 1992 ), it is abundantly clear from the analysis results, which are shown in the forest plot of the overall effect (see Fig. 3 ), that the cumulative impact size of cooperative problem-solving is 0.82, which is statistically significant ( z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]), and can encourage learners to practice critical thinking.

This forest plot shows the analysis result of the overall effect size across 36 studies.

In addition, this study examined two distinct dimensions of critical thinking to better understand the precise contributions that collaborative problem-solving makes to the growth of critical thinking. The findings (see Table 3 ) indicate that collaborative problem-solving improves cognitive skills (ES = 0.70) and attitudinal tendency (ES = 1.17), with significant intergroup differences (chi 2  = 7.95, P  < 0.01). Although collaborative problem-solving improves both dimensions of critical thinking, it is essential to point out that the improvements in students’ attitudinal tendency are much more pronounced and have a significant comprehensive effect (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]), whereas gains in learners’ cognitive skill are slightly improved and are just above average. (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

The analysis of moderator effect size

The whole forest plot’s 79 effect quantities underwent a two-tailed test, which revealed significant heterogeneity ( I 2  = 86%, z  = 12.78, P  < 0.01), indicating differences between various effect sizes that may have been influenced by moderating factors other than sampling error. Therefore, exploring possible moderating factors that might produce considerable heterogeneity was done using subgroup analysis, such as the learning stage, learning scaffold, teaching type, group size, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, in order to further explore the key factors that influence critical thinking. The findings (see Table 4 ) indicate that various moderating factors have advantageous effects on critical thinking. In this situation, the subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), learning scaffold (chi 2  = 9.03, P  < 0.01), and teaching type (chi 2  = 7.20, P  < 0.05) are all significant moderators that can be applied to support the cultivation of critical thinking. However, since the learning stage and the measuring tools did not significantly differ among intergroup (chi 2  = 3.15, P  = 0.21 > 0.05, and chi 2  = 0.08, P  = 0.78 > 0.05), we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving. These are the precise outcomes, as follows:

Various learning stages influenced critical thinking positively, without significant intergroup differences (chi 2  = 3.15, P  = 0.21 > 0.05). High school was first on the list of effect sizes (ES = 1.36, P  < 0.01), then higher education (ES = 0.78, P  < 0.01), and middle school (ES = 0.73, P  < 0.01). These results show that, despite the learning stage’s beneficial influence on cultivating learners’ critical thinking, we are unable to explain why it is essential for cultivating critical thinking in the context of collaborative problem-solving.

Different teaching types had varying degrees of positive impact on critical thinking, with significant intergroup differences (chi 2  = 7.20, P  < 0.05). The effect size was ranked as follows: mixed courses (ES = 1.34, P  < 0.01), integrated courses (ES = 0.81, P  < 0.01), and independent courses (ES = 0.27, P  < 0.01). These results indicate that the most effective approach to cultivate critical thinking utilizing collaborative problem solving is through the teaching type of mixed courses.

Various intervention durations significantly improved critical thinking, and there were significant intergroup differences (chi 2  = 12.18, P  < 0.01). The effect sizes related to this variable showed a tendency to increase with longer intervention durations. The improvement in critical thinking reached a significant level (ES = 0.85, P  < 0.01) after more than 12 weeks of training. These findings indicate that the intervention duration and critical thinking’s impact are positively correlated, with a longer intervention duration having a greater effect.

Different learning scaffolds influenced critical thinking positively, with significant intergroup differences (chi 2  = 9.03, P  < 0.01). The resource-supported learning scaffold (ES = 0.69, P  < 0.01) acquired a medium-to-higher level of impact, the technique-supported learning scaffold (ES = 0.63, P  < 0.01) also attained a medium-to-higher level of impact, and the teacher-supported learning scaffold (ES = 0.92, P  < 0.01) displayed a high level of significant impact. These results show that the learning scaffold with teacher support has the greatest impact on cultivating critical thinking.

Various group sizes influenced critical thinking positively, and the intergroup differences were statistically significant (chi 2  = 8.77, P  < 0.05). Critical thinking showed a general declining trend with increasing group size. The overall effect size of 2–3 people in this situation was the biggest (ES = 0.99, P  < 0.01), and when the group size was greater than 7 people, the improvement in critical thinking was at the lower-middle level (ES < 0.5, P  < 0.01). These results show that the impact on critical thinking is positively connected with group size, and as group size grows, so does the overall impact.

Various measuring tools influenced critical thinking positively, with significant intergroup differences (chi 2  = 0.08, P  = 0.78 > 0.05). In this situation, the self-adapting measurement tools obtained an upper-medium level of effect (ES = 0.78), whereas the complete effect size of the standardized measurement tools was the largest, achieving a significant level of effect (ES = 0.84, P  < 0.01). These results show that, despite the beneficial influence of the measuring tool on cultivating critical thinking, we are unable to explain why it is crucial in fostering the growth of critical thinking by utilizing the approach of collaborative problem-solving.

Different subject areas had a greater impact on critical thinking, and the intergroup differences were statistically significant (chi 2  = 13.36, P  < 0.05). Mathematics had the greatest overall impact, achieving a significant level of effect (ES = 1.68, P  < 0.01), followed by science (ES = 1.25, P  < 0.01) and medical science (ES = 0.87, P  < 0.01), both of which also achieved a significant level of effect. Programming technology was the least effective (ES = 0.39, P  < 0.01), only having a medium-low degree of effect compared to education (ES = 0.72, P  < 0.01) and other fields (such as language, art, and social sciences) (ES = 0.58, P  < 0.01). These results suggest that scientific fields (e.g., mathematics, science) may be the most effective subject areas for cultivating critical thinking utilizing the approach of collaborative problem-solving.

The effectiveness of collaborative problem solving with regard to teaching critical thinking

According to this meta-analysis, using collaborative problem-solving as an intervention strategy in critical thinking teaching has a considerable amount of impact on cultivating learners’ critical thinking as a whole and has a favorable promotional effect on the two dimensions of critical thinking. According to certain studies, collaborative problem solving, the most frequently used critical thinking teaching strategy in curriculum instruction can considerably enhance students’ critical thinking (e.g., Liang et al., 2017 ; Liu et al., 2020 ; Cindy, 2004 ). This meta-analysis provides convergent data support for the above research views. Thus, the findings of this meta-analysis not only effectively address the first research query regarding the overall effect of cultivating critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills) utilizing the approach of collaborative problem-solving, but also enhance our confidence in cultivating critical thinking by using collaborative problem-solving intervention approach in the context of classroom teaching.

Furthermore, the associated improvements in attitudinal tendency are much stronger, but the corresponding improvements in cognitive skill are only marginally better. According to certain studies, cognitive skill differs from the attitudinal tendency in classroom instruction; the cultivation and development of the former as a key ability is a process of gradual accumulation, while the latter as an attitude is affected by the context of the teaching situation (e.g., a novel and exciting teaching approach, challenging and rewarding tasks) (Halpern, 2001 ; Wei and Hong, 2022 ). Collaborative problem-solving as a teaching approach is exciting and interesting, as well as rewarding and challenging; because it takes the learners as the focus and examines problems with poor structure in real situations, and it can inspire students to fully realize their potential for problem-solving, which will significantly improve their attitudinal tendency toward solving problems (Liu et al., 2020 ). Similar to how collaborative problem-solving influences attitudinal tendency, attitudinal tendency impacts cognitive skill when attempting to solve a problem (Liu et al., 2020 ; Zhang et al., 2022 ), and stronger attitudinal tendencies are associated with improved learning achievement and cognitive ability in students (Sison, 2008 ; Zhang et al., 2022 ). It can be seen that the two specific dimensions of critical thinking as well as critical thinking as a whole are affected by collaborative problem-solving, and this study illuminates the nuanced links between cognitive skills and attitudinal tendencies with regard to these two dimensions of critical thinking. To fully develop students’ capacity for critical thinking, future empirical research should pay closer attention to cognitive skills.

The moderating effects of collaborative problem solving with regard to teaching critical thinking

In order to further explore the key factors that influence critical thinking, exploring possible moderating effects that might produce considerable heterogeneity was done using subgroup analysis. The findings show that the moderating factors, such as the teaching type, learning stage, group size, learning scaffold, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, could all support the cultivation of collaborative problem-solving in critical thinking. Among them, the effect size differences between the learning stage and measuring tool are not significant, which does not explain why these two factors are crucial in supporting the cultivation of critical thinking utilizing the approach of collaborative problem-solving.

In terms of the learning stage, various learning stages influenced critical thinking positively without significant intergroup differences, indicating that we are unable to explain why it is crucial in fostering the growth of critical thinking.

Although high education accounts for 70.89% of all empirical studies performed by researchers, high school may be the appropriate learning stage to foster students’ critical thinking by utilizing the approach of collaborative problem-solving since it has the largest overall effect size. This phenomenon may be related to student’s cognitive development, which needs to be further studied in follow-up research.

With regard to teaching type, mixed course teaching may be the best teaching method to cultivate students’ critical thinking. Relevant studies have shown that in the actual teaching process if students are trained in thinking methods alone, the methods they learn are isolated and divorced from subject knowledge, which is not conducive to their transfer of thinking methods; therefore, if students’ thinking is trained only in subject teaching without systematic method training, it is challenging to apply to real-world circumstances (Ruggiero, 2012 ; Hu and Liu, 2015 ). Teaching critical thinking as mixed course teaching in parallel to other subject teachings can achieve the best effect on learners’ critical thinking, and explicit critical thinking instruction is more effective than less explicit critical thinking instruction (Bensley and Spero, 2014 ).

In terms of the intervention duration, with longer intervention times, the overall effect size shows an upward tendency. Thus, the intervention duration and critical thinking’s impact are positively correlated. Critical thinking, as a key competency for students in the 21st century, is difficult to get a meaningful improvement in a brief intervention duration. Instead, it could be developed over a lengthy period of time through consistent teaching and the progressive accumulation of knowledge (Halpern, 2001 ; Hu and Liu, 2015 ). Therefore, future empirical studies ought to take these restrictions into account throughout a longer period of critical thinking instruction.

With regard to group size, a group size of 2–3 persons has the highest effect size, and the comprehensive effect size decreases with increasing group size in general. This outcome is in line with some research findings; as an example, a group composed of two to four members is most appropriate for collaborative learning (Schellens and Valcke, 2006 ). However, the meta-analysis results also indicate that once the group size exceeds 7 people, small groups cannot produce better interaction and performance than large groups. This may be because the learning scaffolds of technique support, resource support, and teacher support improve the frequency and effectiveness of interaction among group members, and a collaborative group with more members may increase the diversity of views, which is helpful to cultivate critical thinking utilizing the approach of collaborative problem-solving.

With regard to the learning scaffold, the three different kinds of learning scaffolds can all enhance critical thinking. Among them, the teacher-supported learning scaffold has the largest overall effect size, demonstrating the interdependence of effective learning scaffolds and collaborative problem-solving. This outcome is in line with some research findings; as an example, a successful strategy is to encourage learners to collaborate, come up with solutions, and develop critical thinking skills by using learning scaffolds (Reiser, 2004 ; Xu et al., 2022 ); learning scaffolds can lower task complexity and unpleasant feelings while also enticing students to engage in learning activities (Wood et al., 2006 ); learning scaffolds are designed to assist students in using learning approaches more successfully to adapt the collaborative problem-solving process, and the teacher-supported learning scaffolds have the greatest influence on critical thinking in this process because they are more targeted, informative, and timely (Xu et al., 2022 ).

With respect to the measuring tool, despite the fact that standardized measurement tools (such as the WGCTA, CCTT, and CCTST) have been acknowledged as trustworthy and effective by worldwide experts, only 54.43% of the research included in this meta-analysis adopted them for assessment, and the results indicated no intergroup differences. These results suggest that not all teaching circumstances are appropriate for measuring critical thinking using standardized measurement tools. “The measuring tools for measuring thinking ability have limits in assessing learners in educational situations and should be adapted appropriately to accurately assess the changes in learners’ critical thinking.”, according to Simpson and Courtney ( 2002 , p. 91). As a result, in order to more fully and precisely gauge how learners’ critical thinking has evolved, we must properly modify standardized measuring tools based on collaborative problem-solving learning contexts.

With regard to the subject area, the comprehensive effect size of science departments (e.g., mathematics, science, medical science) is larger than that of language arts and social sciences. Some recent international education reforms have noted that critical thinking is a basic part of scientific literacy. Students with scientific literacy can prove the rationality of their judgment according to accurate evidence and reasonable standards when they face challenges or poorly structured problems (Kyndt et al., 2013 ), which makes critical thinking crucial for developing scientific understanding and applying this understanding to practical problem solving for problems related to science, technology, and society (Yore et al., 2007 ).

Suggestions for critical thinking teaching

Other than those stated in the discussion above, the following suggestions are offered for critical thinking instruction utilizing the approach of collaborative problem-solving.

First, teachers should put a special emphasis on the two core elements, which are collaboration and problem-solving, to design real problems based on collaborative situations. This meta-analysis provides evidence to support the view that collaborative problem-solving has a strong synergistic effect on promoting students’ critical thinking. Asking questions about real situations and allowing learners to take part in critical discussions on real problems during class instruction are key ways to teach critical thinking rather than simply reading speculative articles without practice (Mulnix, 2012 ). Furthermore, the improvement of students’ critical thinking is realized through cognitive conflict with other learners in the problem situation (Yang et al., 2008 ). Consequently, it is essential for teachers to put a special emphasis on the two core elements, which are collaboration and problem-solving, and design real problems and encourage students to discuss, negotiate, and argue based on collaborative problem-solving situations.

Second, teachers should design and implement mixed courses to cultivate learners’ critical thinking, utilizing the approach of collaborative problem-solving. Critical thinking can be taught through curriculum instruction (Kuncel, 2011 ; Leng and Lu, 2020 ), with the goal of cultivating learners’ critical thinking for flexible transfer and application in real problem-solving situations. This meta-analysis shows that mixed course teaching has a highly substantial impact on the cultivation and promotion of learners’ critical thinking. Therefore, teachers should design and implement mixed course teaching with real collaborative problem-solving situations in combination with the knowledge content of specific disciplines in conventional teaching, teach methods and strategies of critical thinking based on poorly structured problems to help students master critical thinking, and provide practical activities in which students can interact with each other to develop knowledge construction and critical thinking utilizing the approach of collaborative problem-solving.

Third, teachers should be more trained in critical thinking, particularly preservice teachers, and they also should be conscious of the ways in which teachers’ support for learning scaffolds can promote critical thinking. The learning scaffold supported by teachers had the greatest impact on learners’ critical thinking, in addition to being more directive, targeted, and timely (Wood et al., 2006 ). Critical thinking can only be effectively taught when teachers recognize the significance of critical thinking for students’ growth and use the proper approaches while designing instructional activities (Forawi, 2016 ). Therefore, with the intention of enabling teachers to create learning scaffolds to cultivate learners’ critical thinking utilizing the approach of collaborative problem solving, it is essential to concentrate on the teacher-supported learning scaffolds and enhance the instruction for teaching critical thinking to teachers, especially preservice teachers.

Implications and limitations

There are certain limitations in this meta-analysis, but future research can correct them. First, the search languages were restricted to English and Chinese, so it is possible that pertinent studies that were written in other languages were overlooked, resulting in an inadequate number of articles for review. Second, these data provided by the included studies are partially missing, such as whether teachers were trained in the theory and practice of critical thinking, the average age and gender of learners, and the differences in critical thinking among learners of various ages and genders. Third, as is typical for review articles, more studies were released while this meta-analysis was being done; therefore, it had a time limit. With the development of relevant research, future studies focusing on these issues are highly relevant and needed.

Conclusions

The subject of the magnitude of collaborative problem-solving’s impact on fostering students’ critical thinking, which received scant attention from other studies, was successfully addressed by this study. The question of the effectiveness of collaborative problem-solving in promoting students’ critical thinking was addressed in this study, which addressed a topic that had gotten little attention in earlier research. The following conclusions can be made:

Regarding the results obtained, collaborative problem solving is an effective teaching approach to foster learners’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]). With respect to the dimensions of critical thinking, collaborative problem-solving can significantly and effectively improve students’ attitudinal tendency, and the comprehensive effect is significant (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

As demonstrated by both the results and the discussion, there are varying degrees of beneficial effects on students’ critical thinking from all seven moderating factors, which were found across 36 studies. In this context, the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have a positive impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. Since the learning stage (chi 2  = 3.15, P  = 0.21 > 0.05) and measuring tools (chi 2  = 0.08, P  = 0.78 > 0.05) did not demonstrate any significant intergroup differences, we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving.

Data availability

All data generated or analyzed during this study are included within the article and its supplementary information files, and the supplementary information files are available in the Dataverse repository: https://doi.org/10.7910/DVN/IPFJO6 .

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Acknowledgements

This research was supported by the graduate scientific research and innovation project of Xinjiang Uygur Autonomous Region named “Research on in-depth learning of high school information technology courses for the cultivation of computing thinking” (No. XJ2022G190) and the independent innovation fund project for doctoral students of the College of Educational Science of Xinjiang Normal University named “Research on project-based teaching of high school information technology courses from the perspective of discipline core literacy” (No. XJNUJKYA2003).

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Xu, E., Wang, W. & Wang, Q. The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature. Humanit Soc Sci Commun 10 , 16 (2023). https://doi.org/10.1057/s41599-023-01508-1

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Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University

Transfer of learning, the application of learning to different contexts over time, is important to all learning for development. As 21st century skills specifically aim to be “generic,” there is an assumption that they can be transferred from context to context. We investigate the process of transfer in problem solving, with specific focus on mathematical problem solving tasks. Problem solving is highly valued in 21st century workplaces, where mathematical skills are also considered to be foundational in STEM and of paramount importance. This study examines the transfer of first semester mathematics learning to problem solving in second semester physics at university. We report on: (1) university students’ (n = 10) “think-aloud” accounts of the process of transfer; and (2) students’ (n = 10) and academics’ (n = 8) perspectives on transfer processes and problem solving. Think-aloud accounts show students’ recursive use of interpretation, integration, planning and execution thinking processes and highlight the meta-cognitive strategies used in transfer. Academics’ and students’ perspectives on transfer show disparities. Understanding these perspectives is important to current initiatives to integrate and optimise 21st century learning within universities. We argue that renewed attention on the concept of transfer is needed if the generic aims of 21st century skills are to be understood and promoted.

1. Introduction

This article focuses on the centrality and potential of transfer of learning for 21st Century (21C). In particular we explore mathematics transfer, which is widely recognised as central to human development, educational systems and economies, and has recently been the focus of many research projects, policy and public campaigns internationally ( Australian Industry Group 2013 ; Office of the Chief Scientist 2013 ; National Research Council 2013 ; The Royal Society Science Policy Centre 2014 ; U.S. Congress Joint Economic Committee 2012 ; U.S. Department of Education 2016 ). However, given its importance, relatively little recent research exists that examines how mathematics learning is applied in other contexts; and even less research that operationalises and explores transfer of learning within authentic educational systems ( Nakakoji and Wilson 2018 ). We argue that considering the transfer of mathematics learning is critical to the development of 21C skills; and that attention to the role of transfer more broadly, within holistic conceptions of 21C learning, is needed to progress understanding in the field.

While looking at transfer between mathematics and physics, we also consider the relationship between mathematics and science more broadly within the context of university. The importance of this relationship to industry and society is undisputed; and learning in these disciplinary areas is seen as key to 21Century education. Yet, within universities, optimal productivity at the nexus between mathematics and science is often assumed and rarely examined. Most universities, for example, offer mathematics “service courses” to a wide range of degree programs and assume that learning in mathematics courses is effectively applied across degree curriculum. However, sparse research is published to support this assumption and less still that can inform practice in interdisciplinary teaching and learning ( Nakakoji and Wilson 2018 ).

This paper makes two contributions. First, we examine students’ think-aloud accounts of the processes they use in transfer of learning tasks. Second, we examine both student and academic staff perspectives on the mathematics/science relationship, including their views on factors that promote or hinder transfer.

We take a process approach to examine transfer of learning, in relation to 21C problem solving skills. The process-oriented approach ( Sternberg 2000 ) aims to “isolate a finite set of competent that can be combined in various ways to perform any cognitive task” ( Meichenbaum 1980, p. 271 ). We adopt a think-aloud protocol to collect data and document the cognitive processes used in transfer of mathematics learning to science problem-solving tasks. This approach, in tandem with post-task interviews, has strong practical relevance to education because it can identify barriers and stumbling blocks to student learning.

This small-scale exploratory research is nonetheless significant given the historical relationship between mathematics and science and the relative lack of research attention given to this important interdisciplinary relationship. The study makes a preliminary contribution to applied understanding of what can be done to promote the transfer of learning for 21C skills at this important disciplinary nexus.

1.1. Transfer and 21st Century Learning

Rapid technological advancement has changed the skills and knowledge used in workplaces. The change requires employees to process various types of information, analyse big data, interact with and communicate with people, and apply their prior knowledge and experience to a new situation to solve the complex problem in different contexts. Industries demand graduates with generic skills, such as higher-order critical thinking and problem solving, and also strong metacognitive and communication skills. According to the Organization for Economic Co-operation and Development (OECD), 21st century skills refer to “skills and competencies young people will be required to have in order to be effective workers and citizens in the knowledge society of the 21st century” ( Ananiadou and Claro 2009, p. 8 ). Problem solving is consistently identified as central to these requirements, and routinely listed as a desirable graduate attribute for employability and as an integral component in 21C learning.

Transfer is viewed as critical for future education and central to the application of 21st century skills ( OECD 2018 ). Logic dictates that transfer is needed for the application of these generic and “transferable” skills—although this has not, as yet, been as widely acknowledged in the academic literature as might be expected. With predictions of rapidly evolving environments in our future, the ability to apply prior learning to new contexts is essential. Transfer of mathematical learning is the ability of students to apply mathematical skills, knowledge, and reasoning to other disciplines, and this is likely to be particularly important to 21st century skills. Demonstration of this ability is a central issue in mathematics and science education ( Tariq 2013 ; King et al. 2015 ).

1.2. The Relationship between Mathematics and Science

National reports suggest many countries are concerned about participation, standards, and capacity building in mathematics and science for STEM related industry and labor markets ( Australian Industry Group 2013 ; National Research Council 2013 ; Office of the Chief Scientist 2013 ; The Royal Society 2014 ; U.S. Department of Education 2016 ). This has increased research focus on education in these fields. However, learning in these fields is not discrete, they are intertwined, and we need to know more about how learning in mathematics and science is transferred and shared. The study presented here is part of a larger project exploring learning at the mathematics/science nexus at one Australian university ( Nakakoji and Wilson 2014 , 2018 ; Nakakoji et al. 2014 ).

Interdisciplinary relationships between mathematics and science are critically important as mathematics is applied in diverse disciplines and these relationships lead to advancements across disciplines ( National Research Council 2013 ). Both mathematics and science educators need to consider the implications of this interdisciplinarity in terms of effective teaching and learning in schools and universities. For example, U.S. primary and secondary education standards require teachers to enhance the synergy between these disciplines by application of mathematics to science, e.g., mathematical modelling and statistics ( Stage et al. 2013 ). The close relationship is also evident in the fact that mathematical learning in high schools and university is a strong predictor of attainment in science; with mathematics scores explaining between 43 and 87 percent of the variance in a range of science subjects ( Nakakoji and Wilson 2014 ; Sadler and Tai 2007 ). Furthermore, there is currently a flux of research into STEM and internationally many universities conduct educational research in these disciplines in order to improve their teaching ( King and Sen 2013 ). As part of this, it is widely recognised that effective communication and collaboration between academics in STEM disciplines is important ( Anderson et al. 2011 ; Blumberg et al. 2005 ; Orton and Roper 2000 ).

The strong correlations found between mathematics and science learning ( Nakakoji and Wilson 2014 ; Sadler and Tai 2007 ) can be attributed to both the underlying shared general and specific intelligence factors ( Nisbett et al. 2012 ) and the transfer of specific learning and skills between these disciplines ( Roberts et al. 2007 ; Nakakoji and Wilson 2018 ), including, for example, the common and shared use of problem solving schemata and other cognitive strategies. The g factor or general ability is the underlying foundation to diverse cognitive abilities that directly or indirectly affects all learning, including in mathematics and science. According to Cattell-Horn-Carroll theory, g is the strongest factor analytical construct in the hierarchical model of intelligence (see for example, Taub et al. 2008 ). Furthermore, the g factor has been shown to be highly correlated with international assessments of educational attainment, such as PISA and TIMSS, and IQ tests ( Rindermann 2007 ). When examining two different educational attainments it is unsurprising to see high levels of correlation, due the fact that both will draw on the g factor.

The relationship between mathematics and general ability has been examined empirically and is particularly strong. The g factor was correlated with 25 secondary school subjects in the UK; and mathematics had the strongest association with g (r = 0.77), explaining approximately sixty percent of the variance in general ability ( Deary et al. 2007 ). This suggests that among educational attainments, mathematics is particularly linked to g and that this might also explain how mathematics would be a strong predictor of other educational attainments.

Complicating this picture of correlated educational attainments is the unique relationship between mathematics and science. As these are highly cognate disciplines, they may share additional factors. These may contribute to g, and they may also be specific intelligence factors which explain additional achievement variance beyond g . Taub and colleagues (2008) have demonstrated how cognitive ability factors, including fluid reasoning and processing speed, are highly associated with mathematical attainment. Science and mathematics problem-solving assessments may share requirements for fluid reasoning and processing speed (which contribute to g). Science and mathematics may also both draw on specific factors, like gq (quantitative knowledge), so we might expect higher levels of correlation between them than between other, less cognate disciplines. We might also expect these factors to be drawn on in transfer of learning tasks ( Richmond et al. 2011 ) and in problem solving tasks ( Decker and Roberts 2015 ).

1.3. University Mathematics Service Courses and Science Courses

In many countries, including Australia, universities and higher education institutions adopt a “service course” model. In this model, first year mathematics courses are provided by mathematics departments to students from diverse STEM disciplines, such as biology, chemistry, engineering, and physics. These courses cover calculus, differential equations, and linear algebra; and can be offered at different levels (fundamental, intermediate, advanced) according to these students’ prior learning in high school mathematics and the requirements for their degree programs (see for example Nakakoji et al. 2014 ). As sciences are viewed as mathematically cognate disciplines, many institutions have mandatory requirements for science students to study first year mathematics service courses where mathematical skills, knowledge, and reasoning are developed so that they can be applied in other disciplinary learning. This approach often goes unchallenged, and yet it is built upon a range of assumptions, including: (i) students engage with service courses and effective learning occurs; (ii) skills and understanding in mathematics service courses are transferred to other disciplines; (iii) transfer of skills and understanding is assessed in other disciplinary areas, and (iv) the mathematics learnt in service courses is useful in building 21C skills for professions and working life. Interrogating these assumptions within one Australian university, we examined the correlation between mathematics and science learning ( Nakakoji and Wilson 2014 ; Nakakoji et al. 2014 ) and looked for evidence of transfer of learning between them. Using extant university exam assessments, we were able to demonstrate transfer of learning in some science courses, but not others, dependent on the requirements for mathematical reasoning and calculation evident within the course assessments ( Nakakoji and Wilson 2018 ). Specifically, we found assessment of mathematical learning and transfer of learning was evident in only engineering and physics course assessments; and mathematics was not assessed in biology and biochemistry courses in a way that enabled testing and demonstration of transfer from the mathematics service courses.

The larger project employed mixed method design to explore transfer of mathematical learning in a range of different ways (see Nakakoji et al. 2014 ), and findings demonstrated the importance and predictive power of mathematics to the subsequent learning in university science ( Nakakoji and Wilson 2014 , 2018 ). In Nakakoji and Wilson ( 2014 ), multiple regression analysis confirmed strong relationships between mathematics and science attainment. For example, 84% of the variance in second semester biology was explained by first semester biology and mathematics marks; with mathematics uniquely explaining 3.8 percent. In addition, in Nakakoji and Wilson ( 2018 ), transfer was quantitatively measured, using a previously established Transfer Index ( Roberts et al. 2007 ). Our research applied this index to pre-existing university tests and exam data; demonstrating, for the first time, that these can be used to assess transfer. We were able to identify transfer of mathematics learning to physics and engineering; however, in biology and biochemistry, the papers provided no opportunity to assess transfer. Path analyses showed significant direct-transfer effect in the advanced physics course, i.e., the final marks in the course increased by a 0.67 standard deviation when transfer was increased by one standard deviation. This analysis relied on data from marks in exams and demonstrated the importance of mathematics to science learning; however, it did not examine the processes of transfer.

In this paper, we report on students’ think-aloud accounts of transfer and the views of students and academics from across this range of science courses. We focus specifically on a problem-solving physics task requiring mathematical curriculum from the service courses. This sort of challenging task requires complex higher-order thinking and utilization of problem-solving schemata.

The study presented here examines the processes involved in transfer, using a think-aloud method ( Van Someren et al. 1994 ) to see how students articulate their thinking while attempting a task requiring transfer (task details are provided later). Think-aloud is a method that can be used to explore and examine the cognitive processes of thinking by asking participants to constantly verbalise what they are thinking while doing assigned tasks ( Van Someren et al. 1994 ). Think-aloud reports provide rich information on cognition during complex problem solving tasks ( Lohman 2000 ). This method was originally used in psychology (ibid.), and is currently applied in education, psychology, and learning science research on metacognitive skills ( Bannert and Mengelkamp 2008 ), collaborative problem solving ( Siddiq and Scherer 2017 ), the process of schematic representation ( Anwar et al. 2018 ), web-based learning ( Young 2005 ), and science text and diagrams ( Cromley et al. 2010 ). However, in relation to mathematics education, there are few studies using think-aloud (e.g., Brennan et al. 2010 ; Ke 2008 ; Monaghan 2005 ) and these papers focus on primary school mathematics. We were unable to find prior research using the think-aloud method in the context of university mathematics education.

Analysis of the think-aloud accounts is framed by mathematical problem-solving theory ( Mayer 1992 ; Okamoto 2008 ; Seo 2010 ) and also Bloom’s taxonomy ( Bloom et al. 1956 ). We also use qualitative data from interviews to further explore student and academic perspectives on problem solving, transfer, and the broader relationship between mathematics and science learning. We address the following research questions:

• What are the processes of transfer (if any) evident in students’ “think-aloud” accounts while solving physics exam questions requiring knowledge and understanding from their mathematics service courses?
• What are the challenges in transfer of learning reported by students and by academics?
• What teaching and learning factors do students and academics believe to enhance transfer?

2. Literature Review

We present a brief review of literature relating to transfer of learning, including important theoretical explanations of how transfer might occur and what factors are involved in it. The review focuses specifically on the research examining transfer in mathematical and science contexts, although a vast corpus of literature exists on transfer more generally.

2.1. Transfer of Learning

Many researchers in education and psychology have investigated transfer since the late nineteenth century. In a broad sense, as learning involves the application of prior learning to new similar or different contexts, transfer is related to all learning. Accordingly, transfer can be seen as the central or primary goal in education ( Bransford et al. 1999 ; De Corte 1995 , 2003 ; Engle 2012 ; Mestre 2003 ; Siler and Willow 2014 ). The acknowledgment that 21C learning will require adaption to rapidly evolving environments means that transfer of learning is key to efficient and effective education in the future. Teaching and learning that is able to optimise transfer of learning is an important goal. In their account of 21C learning, Saavedra and Opfer point out:

Students must apply the skills and knowledge they gain in one discipline to another and what they learn in school to other areas of their lives. A common theme is that ordinary instruction doesn’t prepare learners well to transfer what they learn, but explicit attention to the challenges of transfer can cultivate it. ( Saavedra and Opfer 2012, p. 10 )

As transfer can be related to all learning, the list of factors associated with it can be extensive and we review here only some indicative starting points. Billing ( 2007 ) identifies nine major factors to facilitate transfer based on his survey of several hundred papers. They are (i) motivation, (ii) metacognitive strategies and skills, (iii) learning in context, (iv) principles, rules, and schemata acquisition, (v) similarity and analogy, (vi) varied examples and contexts, (vii) reduced cognitive load, (viii) active learning, and (ix) learning by discovery. In addition, transfer can be promoted by the necessity of prior knowledge (see Mestre 2003 ) and learning with deep understanding ( Barnett and Ceci 2002 ; Brown and Kane 1988 ; Chi et al. 1989 , 1994 ).

2.2. Cognitive Explanations of Transfer

Undoubtedly, transfer involves higher order thinking ( Anderson et al. 2001 ; Bloom et al. 1956 ), as does mathematical problem solving ( Mayer and Wittrock 1996 ). Consequently, to analyse transfer in the problem-solving task in this study, we use both a framework for mathematical problem solving ( Seo 2010 ) and Bloom’s seminal cognitive taxonomy which outlines lower and higher-order thinking ( Bloom et al. 1956 ).

2.2.1. Higher Order Thinking and Bloom’s Taxonomy

The taxonomy of educational objectives developed by Bloom and colleagues ( Anderson et al. 2001 ; Bloom et al. 1956 ; Krathwohl et al. 1964 ) has been influential in providing the foundations to understand transfer of learning ( Krathwohl 2002 ). Bloom’s taxonomy (1956), which is one of the most seminal books in education ( Anderson 2002 ), presents a framework that is used for various purposes, including assessment, curriculum development, instruction, and learning theory ( Seaman 2011 ). Importantly, Bloom’s taxonomy is a useful tool to understand the different levels of cognitive processes related to higher order thinking. The taxonomy recognises three major domains in cognitive functioning: the cognitive domain ( Bloom et al. 1956 ), affective domain ( Krathwohl et al. 1964 ), and psychomotor domain. The first domain is relevant to this study as it is helpful to classify lower and higher order thinking. The cognitive domain entails “the recall or recognition of knowledge and the development of intellectual abilities and skills” ( Krathwohl 2002, p. 7 ). The taxonomy categorises and ranks increasing levels of higher order thinking: (i) knowledge, (ii) comprehension, (iii) application, (iv) analysis, (v) synthesis, and (xi) evaluation. The taxonomy hierarchy is cumulative and easy to understand, with many examples listed by Bloom and colleagues (1956), a wealth of supporting empirical studies (e.g., Kropp et al. 1966 ; Miller et al. 1979 ), and well established validity ( Seaman 2011 ). A summary, with explanations and some mathematical examples, is shown in Table 1 . Research suggests that learners need to use all these cognitive processes, including the higher order thinking, for effective transfer of learning.

Summary of Bloom’s original taxonomy (1956) with mathematics application examples.

Source: ( Bloom et al. 1956 ).

In mathematics education, Bloom’s taxonomy has been widely used for teaching and assessment for over a half century ( Thompson 2008 ). However, it has faced criticism with claims that the taxonomy fails to distinguish between the different levels of mathematical reasoning ( Suzuki 1997 ). Another issue is its apparent inaccuracy in predicting which of the cognitive processes students use to solve problems in mathematics tests ( Gierl 1997 ). To address these weaknesses when analysing transfer of mathematical learning, we use Bloom’s cognitive taxonomy alongside more specific theory of mathematical problem-solving.

2.2.2. Mathematical Problem-Solving Theories and Transfer

We use mathematical problem-solving theory to analyse students’ accounts of the transfer task to address the criticisms of Bloom’s taxonomy, and also because the nature of the exam questions used in the task is built around a problem-solving approach.

Problem-solving is defined as “cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver” ( Mayer and Wittrock 1996, p. 47 ). Sub-processes of problem-solving cover representation, planning, and executing ( Mayer and Wittrock 1996 ), all of which can involve metacognition which is “cognition on cognition” or “thinking about thinking.” Metacognition enhances student performance via conscious and deliberative problem-solving strategies in planning, monitoring, and evaluating ( Ali et al. 2018 ). In order for successful transfer to occur, a learner selects appropriate previous skills and knowledge, applies them into new problems, and monitors the appropriate general and specific cognitive processes to solve the problems ( Mayer and Wittrock 1996 ).

Literature on mathematical problem-solving falls into three categories ( Okamoto 2008 ): mere calculation problems (e.g., Brown and Burton 1978 ), algebraic word problems (e.g., Kintsch 1986 ), and geometric problems (e.g., Owen and Sweller 1985 ). Metacognition is thought to be more deeply involved with word problems and geometric problems than calculation problems ( Okamoto 2008 ). Some literature provides additional detail on four levels of cognitive processes in this mathematical problem solving (e.g., Mayer 1992 ; Okamoto 2008 ) and a specific model of cognitive sequencing and flow, see Figure 1 , which is translated from Seo ( 2010 ).

Four levels of procedures of mathematical problem solving. ( Source: Seo 2010, p. 230 translated by Yoshitaka Nakakoji).

In this study, we use Seo’s model as a theoretical framework, alongside Bloom’s taxonomy, to analyse the processes of transfer. Whilst Seo’s model comes from the Japanese literature on mathematical learning, it shows some remarkable similarities to the OECD PISA problem solving assessment framework ( DeBortoli and Macaskill 2014 ). The model covers basic procedural steps, and it was anticipated that this could frame students’ descriptions in their think-aloud accounts. This was considered appropriate for a first pass application of think-aloud to transfer, some of the more complex models of problem solving (e.g., Ichikawa et al. 2009 ) may be appropriate for follow-up research.

In particular, a problem schema is seen as important in mathematical learning ( Okamoto 2008 ; Silver 1987 ) and is defined as patterned knowledge about structures of problems and ways of solving problems ( Seo 2010 ). The schema referred to in Seo’s model is “a cluster of knowledge representing a particular generic procedure, object, percept, event, sequence of events, or social situation” ( Thorndyke 1984, p. 167 ), which is featured by five characteristics: abstraction, instantiation, prediction, induction, and hierarchical organization ( Reed 1993 ). This schema is understood to be especially useful in solving word problems ( Seo 2010 ) and geometric problems ( Okamoto 2008 ). Students with insufficient problem schema may experience mathematical learning difficulties. Many primary and secondary students, although good at calculations, have difficulties with word problems ( Seo 2010 ). This is because they fail to understand the meaning of problems and form the representation of the whole problems ( Seo 2010 ). This is related to the translation and integration procedures in Figure 1 , which involve use of a problem schema. According to schema theory, transfer of learning is heavily subject to whether appropriate anticipatory schemata are activated ( Salomon and Perkins 1989 ). For example, vertical transfer (transfer of basic knowledge to higher level understanding) needs to activate procedural schemata which have been developed previously ( Royer 1979 ).

2.3. Socio-Cultural Explanations of Transfer

Socio-cultural theories of transfer emphasise the importance of social and cultural learning interactions and contexts. In particular, situated learning (for example, see Greeno 2011 ; Greeno et al. 1993 ; Lave 1988 ; Lave and Wenger 1991 ) and the actor-oriented approach to transfer ( Karakok 2009 ; Lobato 2006 , 2008a , 2008b ) are relevant to transfer of mathematical learning. In this study, these perspectives are useful for understanding the context of transfer in mathematics and science education at university. These theories highlight the importance of individuals’ personally constructed learning and the interplay between their understanding of mathematics and the process of solving transfer tasks.

Lave ( 1988 ) first pointed out a paucity of transfer studies in natural settings and academic disregard of the need for problem-solving in daily life situations. Lave’s research on situated learning makes it clear that learners, their thinking, and learning activities are not independent from their contexts, thus “cognition and performance are context-specific, in a fundamental sense” ( Evans 1998, p. 270 ). Greeno et al. ( 1993 ) expanded and articulated the situatived perspective on transfer, emphasising not only the importance of the situation where learning occurs, but also the learner’s ability to interact with other people and the various materials available for learning ( Marton 2006 ). In relation to our study, this perspective acknowledges that while mathematical learning occurs in formal course settings, including university lectures and tutorials with interactions between lecturers, tutors, and peers, but we cannot view this learning as complete. In addition, learning can be constructed when students study by themselves, or with others, at home, in a library, or elsewhere, and by utilising physical and online materials.

In experimental studies, researchers have highlighted difficulties in observing transfer (see for example, Detterman 1993 ; Hatano and Greeno 1999 ; Lobato 2006 ). The sociocultural actor-oriented approach attempts to overcome this difficulty, and some other weaknesses in understanding transfer, by shifting to a learner-centred perspective. This approach specifically examines transfer processes by looking at how learners relate to learning experiences within novel situations. Thus, this approach enables researchers to consider the occurrence of transfer even if students provide incorrect or non-standard performance in tasks, while this situation would be treated as failure of transfer in experimental studies where transfer is measured as a dichotomised absolute. Adopting an actor-oriented approach, Karakok ( 2009 ) identified transfer of students’ understanding of the concepts in linear algebra to quantum physics contexts. In this paper, although we utilise cognitive theoretical frameworks for analysis of individuals’ account of transfer processes, socio-cultural perspectives, acknowledging individuals own constructions of their learning, and the importance of context, frame the study more fully.

3.1. Research Design

There are two data collection strategies employed. First, student interviews look at both the processes of transfer tasks, using a think-aloud method, and a post-task interview data is also gathered on students’ perceptions about the relationship between mathematics and science and the issues related to transfer. Second, interviews with academic teaching staff explored the relationship between mathematics and science; and factors promoting and hindering transfer.

3.2. Student Think-Aloud Study

We examined first-year university students’ transfer processes by giving them physics exam questions and getting them to think-aloud while they completed this task. Post-task interviews were conducted with questions exploring student perspectives of mathematics and science learning and transfer between these.

3.2.1. Sample

Ten students in STEM degrees at an Australian university agreed to participate in this study. We purposively selected student cohorts studying first semester mathematics service courses and second semester physics courses and recruited volunteers from classes. Relevant background information was collected: age (mode = 21, range 19–32), gender (female 30%), and degree (all were Bachelor of Science, three in advanced courses and two in degrees combined with arts or education).

3.2.2. Data Collection

Individual interviews were conducted to collect the following information. First, a short half-page questionnaire was used to collect the background information of students. Second, a cognitive interview was conducted using the think-aloud method to examine the learning processes of transfer; this is explained in detail in the following sections. A third element was a post-task interview used to identify the strategies used and difficulties faced by students in the transfer task.

3.2.3. Think-Aloud Tasks

Students were asked to speak aloud about what they were thinking, while solving two physics questions extracted from first year second semester past exams (see Figure 2 ). These physics questions required mathematical skill, knowledge, and reasoning learned in mathematics service courses. The first question involved calculation of a partial derivative. The second question could be mathematically answered by solving Schrödinger equation; however, the application of mathematics to solve the second question was not obvious and an intuitive approach to physics could be employed as an alternative. To extract more insight into the students’ thinking, a follow-up interview was also used to retrospectively explore how the question was attempted.

Think-aloud physics Questions 1 and 2 used for examination of transfer processes.

3.2.4. Data Analysis

In order to look at the processes of transfer, the six categories in Bloom’s taxonomy (see Table 1 ) and the four levels of processes of mathematical problem-solving (based on Seo 2010 , see Figure 1 ) were used as descriptive categories for the various processes students described in their think-aloud accounts. Drawing on both the think-aloud transcripts and the working out evident on the transfer task answer sheets, each process was also coded by the language used (words and phrases), mathematical expressions, formulae, or graphs (see some examples in Table 2 ). In addition, qualitative thematic analysis was used to identify themes generated from the post-task interview data.

Examples of words, phrases, and mathematical expression associated with coding four processes in solving transfer questions 1 and 2.

3.3. Interviews with Academics

An interview survey was conducted to ascertain academic practitioners’ views on teaching and learning. The academics were experts in their own research fields, and experienced in higher education teaching and learning. Almost all of them held post-graduate qualifications in higher education teaching and learning and they were able to provide informed and articulate comment on learning issues and challenges. However, it needs to be acknowledged and emphasized that whilst they provided a range of authentic practitioner perspectives, none were expert on transfer of learning. Academic teaching staff had been involved in, and interviewed, previous teaching and learning research, but not research on transfer of mathematical learning. This study addresses that gap with a preliminary, small sample.

3.3.1. Sample

The sample included eight senior teaching academics across four disciplines (5 mathematics, 1 IT, 1 physics, and 1 bioscience; 4 males and 4 females) who were invited to participate in this study and were required to meet the three conditions: (i) knowledge and experience with the issues under investigation; (ii) capacity and willingness to participate; (iii) sufficient time to participate in the interviews.

3.3.2. Data Collection and Analysis Methods

There were two rounds of interviews asking experts fifteen open-ended questions in total. In the first round, sample questions covered “What mathematical knowledge and skills taught in first year mathematics do you think are most relevant to study in biology, biochemistry, engineering and physics?” and “What factors enhance or hinder students’ application of mathematical skills and knowledge in biology, biochemistry, engineering and physics?” On the basis of analysis of responses, a second round of questions were made for clarification. For their convenience, participants were invited to answer these questions by e-mail. This email interview technique has been used in other studies on teaching and learning mathematics ( Petocz et al. 2006 ). The interview data transcribed was analysed with using thematic analysis, according to the principles outlined by Braun and Clarke ( 2006 ).

4. Results and Discussion

We present our findings and discussion around the four research questions.

4.1. Research Question 1: What Are the Processes of Transfer (If Any) Evident in Students “Think-Aloud” Accounts While Solving Physics Exam Questions Requiring Knowledge and Understanding from Their Mathematics Service Courses?

For each student, their progression through the elements of Seo’s theory (2010) interpretation, integration, planning, and execution was coded and presented in a temporal model (see Figure 3 a for question 1(a) and in Figure 3 b for question 1(b). The coding of the processes was based on categorisation of phrases and synonyms relating to each of Seo’s four processes. For example, the students’ think-aloud account was categorised as interpretation if they reported “I’m reading the question and I’m thinking about the relationship between frequency and wavelength” (Student 1) or planning if they stated “we just need to replace in the formula where the wavelength is used, … and … to multiply it by dλ df” (Student 2).

( a ) Think-aloud reports on transfer processes for question 1. ( b ) Think-aloud reports on transfer processes for question 2.

To interpret Figure 3 a,b, the reader can view each students’ progression, and/or recursive movement, from left to right; this reflects their reported thinking over time as they attempted to answer the physics question. Comments provide additional important information and the transfer column highlights whether any transfer of mathematics learning to the physics task was evident. All students followed the interpretation, integration, planning, and execution processes highlighted in Seo’s model. But, there was one small exception, student 4, who missed the integration process. Further, some students (e.g., 5) made errors at various points (see dotted line boxes) and others still (e.g., 1) took recursive steps to modify their approach to the problem when they made initial errors.

The extent to which transfer was demonstrated varied among students and this may be related to question difficulty. In the sub-question 1(a), five students out of ten got correct answers, therefore demonstrating transfer, while in the sub-question 1(b) only three students managed this. The small proportion demonstrating transfer in the latter question was not surprising given the increased complexity of the calculation. Two students in the former and three students in the latter also were able to demonstrate understanding related to transfer to some extent; socio-cultural theories of transfer suggest this can be considered partial transfer.

This variation in how transfer was demonstrated among students is consistent with socio-cultural studies on transfer, in particular the actor-oriented approach to transfer (see for example, Karakok 2009 ), which highlights diversity in how transfer of learning is constructed within individuals learning. Furthermore, analysis of the think-aloud accounts made it evident that metacognition was also an important part in all four of the transfer processes. For example, students reviewed their planning and calculation, some of them more than once, and this reflects their conscious monitoring of their own thinking.

There were students who couldn’t solve the questions. For example, some students had the right approach, but couldn’t fully demonstrate transfer (see cases 6 and 7 in Figure 3 b). It is important to consider what made the questions difficult for these students. There were five main issues identified from the analysis of the transfer processes: (i) lack of mathematical knowledge, related to the first and foundational category of Bloom’s taxonomy—knowledge; (ii) difficulties in understanding the question, i.e., issues in translation and integration processes, related to the second category of Bloom’s Taxonomy—comprehension (iii) issues in recalling prior learning, which are also related to the first category of Bloom’s taxonomy as well as planning and execution processes; (iv) a lack of procedural knowledge to solve the question—or poor use of problem schema; (v) a lack of practice and/or a technical error in calculation, related to the execution process. In the post-task interviews, students mentioned difficulties in understanding the problem, for example:

“in the first question with the Planck’s formula, it’s difficult to understand the question to begin with. So reading through the question is a lot to figure out. And you also have to recognise lots of different symbols, maths symbols which I’m sure if I didn’t know what they were I would be very lost, even more than I was.”

In regard to problem schema, another student also stated that “I understand what it’s asking me to do. I just don’t know how to do it.” These issues, mentioned by students, are consistent with the difficulties in mathematical learning outlined by Seo ( 2010 ). These difficulties are orientated to the nature of the question task as questions 1(a) and (b) relied mostly on calculation. However, as we will explain, these contrasted substantially with difficulties in Question 2, which had much greater demands on reasoning abilities.

Question 2 (see Figure 2 ), required thinking about probability distributions and exponential decay, however, no students were able to solve this question using mathematical formula in physics or mathematical reasoning in an explicit way. This led to difficulty in conducting the analysis and transfer could not be observed. For question 2, the students preferred recalling their basic knowledge of physics and tended to utilise an intuitive approach to solve the physics questio—rather than employing the mathematical methods taught in their service course in the previous semester. Although it is important for students to understand the concepts in physics in their disciplinary ways, the understanding of mathematical expressions behind the physical world is equally important and of perhaps greater utility in terms of development of their generic 21C skills.

The students’ inability to approach Question 2 with mathematical understanding may be related to a disparity between mathematical service courses and physics courses, wherein the content taught in mathematics classes is assumed and not reinforced within the physics classes. In other words, conceptual understanding in physics may be overemphasised or presented without in-depth exploration and reinforcement of the relevant mathematical aspects. This issue was touched on by one academic who suggested in interview: “I suppose there is so much to test conceptually in the sciences in an examination that educators do not want students to spend time on mathematical working.”

It is impossible to verify if this was the case. We can only report that for Question 2, the analysis of transfer of mathematical learning was not possible because mathematical reasoning was not evident in the students’ think-aloud accounts.

4.2. Research Question 2: What Are the Challenges in Transfer of Learning Reported by Students and by Academics?

Interviews explored students’ and academics’ perspectives on what would hinder and enhance transfer of learning. This was done through a series of open-ended questions which generated a large amount of data. After coding students and academics responses separately and producing highly synthesised themes, we compare and contrast these in Table 3 .

Potential factors promoting or hindering transfer.

The factors reported here were consistent with a range of educational research studies and theories. For example, Hattie’s ( 2009 ) syntheses of meta-analyses in education also highlights teacher clarity and feedback as among the most influential teaching factors. Unsurprisingly, mathematical anxiety was mentioned by both students and academics. Academics are aware of their students’ mathematics anxiety: “I imagine this is to do with a perceived lack of mathematical understanding, and fear of mathematics, among students (and society in general)” (Academic 2). Both students’ and academics’ comments on mathematics anxiety made it clear that this was an issue central to teaching and learning. We wondered if students’ fear of mathematics and academics’ sensitivity to this fear was related to the low level of science academics’ inclusion of mathematics in science assessments. We reviewed a range of science exam papers in the university and found there were no questions aligning with the tertiary mathematics taught in mathematics service courses in either biology or bio-chemistry, despite the fact that students were required to undertake those university mathematics courses. In physics, just a few questions were aligned with the university mathematics service courses, with high-school level mathematics (advanced and extension courses) more evident in the first-year physics papers. If mathematical learning is not assessed in the science disciplinary context, why are mathematics courses compulsory for science students at university? Unfortunately, in the interviews academics provided no direct replies to this question. What was evident is that first, both students and academics acknowledge the importance of mathematics, but academics think that understanding concepts in their disciplines is more important than mathematical application. Second, scientist academics think that mathematical skills and understanding should be assessed in mathematics service courses, not science courses. Academics also acknowledged that it is very problematic and difficult to provide mathematics courses to accommodate learning needs for students from diverse backgrounds and disciplines, such as science and engineering.

Finally, “translation” emerged as an important theme for academics (see Table 3 ), and although students did not mention it in interview, it did emerge as an issue for students in the think-aloud task. Translation was seen as important as science problems with mathematical content can be expressed in the form of word problems. One academic commented: “the most difficult problem for many students is not with the maths but converting from words to maths and back” (Academic 3). Our analysis of the transfer processes showed translation was an important strategy for successful transfer of learning and problem-solving (see student 1 in Figure 3 a).

4.3. Research Question 3: What Teaching and Learning Factors Do Students and Academics Believe to Enhance Transfer?

Interviews also explored students and academic perspectives on what would enhance transfer of learning, shown in Table 3 . Overall, the reported factors are consistent with a range of educational theories; for example, transfer theory suggests that higher order thinking, like application of understanding in real world problems and rehearsal (repeated practice) are enablers of transfer. Furthermore, students’ self-beliefs and confidence are important in learning in general ( Malmberg et al. 2013 ); as is prior learning ( Martin et al. 2013 ). Thus early experiences in mathematical learning are likely to be a key to successful transfer of mathematical understanding in university.

There was one noticeable gap between student and academic perspectives. While academics expect students to: “try to see wider connections and the historical development of mathematics in science, rather than only focus on a narrow disciplinary context” (Academic 4); the student perspective suggests that in relation to science learning “the basics of maths that I’ve done [are] very useful, but the stuff I’m doing right now is not quite so” (Student 10). In interviews, academics identified the relevance of mathematics and interdisciplinary learning as important; by contrast, students did not mention these as important to promoting transfer ( Table 3 ). This demonstrates an apparent gap in how the utility of mathematics is perceived by students and academics. Socio-cultural theories of transfer suggest that students need to value learning in order to transfer it effectively. If academics are not able to communicate the value and potential of the mathematics learning, this presents a barrier to transfer.

There are factors identified in research as important to transfer, which were not apparent in these academics’ and students’ responses. First, although socio-cultural theories emphasise the importance of learning contexts, none of the interview respondents mentioned contextual factors like interaction with lecturers, tutors and/or peers in lectures, tutorials, or other situations and material available for learning.

Also unexpectedly, neither students nor academics mentioned metacognition in the interviews. This is despite the fact that research literature views metacognitive knowledge and activities, such as monitoring and control, as essential to mathematical learning and transfer ( Billing 2007 ; Mayer and Wittrock 1996 ; Okamoto 2008 ; Seo 2010 ). It was evident in think-aloud that students employed metacognitive strategies; for example, when monitoring calculation a student stated “Did I do something wrong?” (Student 3). However, they may not have been aware of this strategy and did not discuss this in the post-task interviews. As education systems shift to focus on transferability of generic skills, they are likely to promote a stronger focus on metacognitive skills.

5. Conclusions

Although it is ubiquitous to all conceptions of learning, “transfer of learning” is in keener focus within the modern conception of 21C skills due to their aims to provide generic skills and competencies. Such skills need to be carried and applied, or transferred, to a wide range of contexts. It is anticipated that the demands of future learning, within rapidly changing environments, will require increasing competence in transfer of learning. Our experience exploring “transfer of learning” between mathematics and science at one university has highlighted a range of issues and possibilities.

Firstly, through this study, we were able to demonstrate how a process-oriented approach ( Sternberg 2000 ) to learning could be applied in an authentic educational context, documenting student thinking during a mathematical problem-solving transfer task. Although small in scale, the findings demonstrate how a larger study using this approach could be used to provide diagnostic information to strengthen teaching and learning. Using Seo’s problem-solving theory and Bloom’s taxonomy, students’ thinking pathways and stumbling blocks in this process could be analysed by teacher academics. Common difficulties can be identified and teaching and learning designed to rectify them. While diagnostic assessments for school and adult literacy and numeracy have been available for decades, there remains potential to develop similar methods for practitioners teaching for 21C skills, including transfer of learning. There are already a range of assessments of problem-solving skills that might be adapted to this purpose.

Our findings provide the opportunity to reflect on models for the teaching of mathematics in universities. We argue that transfer of learning between mathematics and science is a neglected area of enormous potential. Given societal demands for increasing STEM capacity, it is critical and inevitable that we reflect on the following questions: What are the key challenges in mathematics learning for sciences? How should mathematics be taught to university science students? Transfer of learning, and associated cognitive and socio-cultural theories, provide a conceptual framework through which these questions can be considered and explored.

The close relationship between mathematics and intellgience factors, the demands for numeracy and data skills from industry, and the potential of mathematics in problem-solving all suggest that reforms in mathematics education are needed and should be discssed as part of the 21C skills agenda. There are currently different models for higher education mathematics across countries and institutions. For some countries, such as Australia, mathematics service courses are provided, particularly for first year undergraduate students. These aim to enable students to apply mathematics to diverse contexts, but might make it difficult for students to see the connection with their own discipline. In other countries, such as Japan, it is more common that each discipline teaches mathematics, and courses are embedded into each discipline’s degree programs. This may help students to apply mathematics to their own disciplinary context, but may restrict the applicability of mathematics to more diverse contexts. Investigations into the effectiveness of each approach would be useful, as would broader contemplation of how mathematics might be positioned within interdisciplinary learning and strategy for developing 21C skills.

Finally, this project has led us to believe that greater attention could, and should, be paid to the concept of “transfer of learning” in order to promote 21C skills. While there is a long literature on transfer, there are enormous gaps in relation to how transfer might be measured, evaluated, and promoted within schools and universities. Now is the time to bring what we know of transfer into the realm of educational practice; and where understanding is lacking we must develop a program of research. In particular, interdisciplinary learning, which has been positioned as a key goal for the future (as technology and industry developments are expected to morph and transform traditional disciplinary boundaries) has not been thoroughly explored at the level of institutional practice.

While 21C skills lauded as critical for the new century include problem-solving skills, and recent educational rhetoric continues to exhort the value of integrated interdisciplinary learning, particularly in STEM, there remains a need for research examining if, and how, pursuit of these goals is evident in current educational practice.

Author Contributions

Conceptualisation, Y.N.; methodology, Y.N. and R.W.; formal analysis, Y.N; writing—original draft, Y.N.; writing—review & editing, R.W. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

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On the Relationship Between Problem-Solving Skills and Professional Practice

• First Online: 01 January 2010

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• Kevin W. Eva 2  

Part of the book series: Innovation and Change in Professional Education ((ICPE,volume 5))

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It is often thought that one can create expert practitioners by enabling novices, training to work within the same field, to think like the experts. That is, by understanding the expert’s thought processes and problem-solving tendencies we can more efficiently help students learn to reason like experts, thus becoming expert themselves. In the early days of research into problem solving it did indeed appear as though general heuristics could be identified that could be inculcated into our educational programs for the purpose of nurturing good problem-solving skills on the part of our students. In the 1970s, however, as this view was revolutionising the educational systems of health care professionals, research began to suggest that, in fact, there are few differences in the ways that experts and novices approach medical problems. More fundamentally, the mediating impact of knowledge came to be seen as playing a larger role in problem solving than anything that could be labelled a problem-solving skill or clinical reasoning strategy. This chapter will review the history of this literature, culminating in both (a) a description of a current model of clinical reasoning in which flexibility of approach is much more central than specificity of strategy, and (b) suggestions for how the concept of problem solving might be reformulated as a means towards the end of clinical expertise rather than as its defining quality.

• Problem-Solving Skills
• Professional Development
• Professional Practice

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To fully appreciate the problems, the reader is encouraged to pause to try solving each in turn. The answers will be provided later in the text.

Create a cube by using eight matchsticks to form two squares and joining the corners of each square with the 4 remaining matchsticks. The top, bottom, and four sides of the cube form six squares.

Again, the reader is encouraged to pause and try to solve these problems before the answers are revealed later in the chapter.

In fact the “hint” condition may be the most analogous to the way in which the problems were utilised in this chapter given that the context in which this section is presented may provide a sufficient hint.

Ark, T. K., Brooks, L. R., & Eva, K. W. (2006). Giving learners the best of both worlds: Do clinical teachers need to guard against teaching pattern recognition to novices? Academic Medicine , 81 , 405–409.

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Eva, K.W. (2009). On the Relationship Between Problem-Solving Skills and Professional Practice. In: Kanes, C. (eds) Elaborating Professionalism. Innovation and Change in Professional Education, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2605-7_2

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Home > Libraries > LIBRARIESPUBLISHING > PUPOAJ > IJPBL > Vol. 10 (2016) > Iss. 2

Detangling the Interrelationships Between Self-Regulation and Ill-Structured Problem Solving in Problem-Based Learning

Xun Ge , University of Oklahoma Follow Victor Law , University of New Mexico Follow Kun Huang , Mississippi State University Follow

One of the goals for problem-based learning (PBL) is to promote self-regulation. Although self-regulation has been studied extensively, its interrelationships with ill-structured problem solving have been unclear. In order to clarify the interrelationships, this article proposes a conceptual framework illustrating the iterative processes among problem-solving stages (i.e., problem representation and solution generation) and self-regulation phases (i.e., planning, execution, and reflection). The dynamics of the interrelationships are further illustrated with three ill-structured problem-solving examples in different domains (i.e., information problem solving, historical inquiry, and science inquiry). The proposed framework contributes to research and practice by providing a new lens to examine self-regulation in ill-structured problem solving and offering guidelines to design effective tools and strategies to scaffold and assess PBL.

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Ge, X. , Law, V. , & Huang, K. (2016). Detangling the Interrelationships Between Self-Regulation and Ill-Structured Problem Solving in Problem-Based Learning. Interdisciplinary Journal of Problem-Based Learning, 10 (2). Available at: https://doi.org/10.7771/1541-5015.1622

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An investigation of the interrelationships between motivation, engagement, and complex problem solving in game-based learning

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Digital game-based learning, especially massively multiplayer online games, has been touted for its potential to promote student motivation and complex problem-solving competency development. However, current evidence is limited to anecdotal studies. The purpose of this empirical investigation is to examine the complex interplay between learners' motivation, engagement, and complex problem-solving outcomes during game-based learning. A theoretical model is offered that explicates the dynamic interrelationships among learners' problem representation, motivation (i.e., interest, competence, autonomy, relatedness, self-determination, and self-efficacy), and engagement. Findings of this study suggest that learners' motivation determine their engagement during gameplay, which in turn determines their development of complex problem-solving competencies. Findings also suggest that learner's motivation, engagement, and problem-solving performance are greatly impacted by the nature and the design of game tasks. The implications of this study are discussed in detail for designing effective game-based learning environments to facilitate learner engagement and complex problem-solving competencies.

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• Dynamic modeling as a cognitive regulation scaffold for developing complex problem-solving skills in an educational massively multiplayer online game environment Eseryel, D. ; Ge, X. ; Ifenthaler, Dirk ; Law, V. ( 2011 ) Following a design-based research framework, this article reports two empirical studies with an educational MMOG, called McLarin's Adventures, on facilitating 9th-grade students' complex problem-solving skill acquisition ...
• Theoretical Considerations for Game-Based e-Learning Analytics Gibson, David ; Jakl, P. ( 2015 ) In an interactive digital-game, traces of a learner’s progress, problem-solving attempts, self-expressions and social communications can entail highly detailed and time-sensitive computer-based documentation of the context, ...
• Theoretical considerations for game-based e-learning analytics Gibson, David ; Jakl, P. ( 2015 ) In an interactive digital game or gamified e-learning experience, mapping a learner’s progress, problem-solving attempts, self-expressions and social communications can entail highly detailed and time-sensitive computer-based ...

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An Investigation of the Interrelationships between Motivation, Engagement, and Complex Problem Solving in Game-based Learning

Related Papers

In D. Ifenthaler, D. Eseryel, & X. Ge (Eds.). Assessment in game-based learning: Foundations, innovations, and perspectives (pp. 257-285). New York: Springer.

Deniz Eseryel

Complex problem solving and motivation are often argued as the most important benefits of massively multiplayer role-playing online games. However, little empirical research exists to support these assertions. Current research and educational game design theory are insufficient to explain the relationship between complex problem solving, motivation, and games; nor do they support the design of educational games intended to promote motivation and complex problem-solving skills. For the past few years, we have been engaged with design-based research (DBR) to address this gap in the literature. In this chapter, we present the findings of this study in a framework for designing and assessing educational MMORPGs for facilitating learners’ motivation and complex problem-solving skill acquisition. This game design and assessment framework bridges three levels of interactivity that were identified in a series of DBR studies as being crucial for effective educational game design: (1) interface interactivity, (2) narrative interactivity, and (3) social interactivity. In this chapter, we present Interactivity3 design and assessment framework and discuss the findings of a study that shows the validity of this framework for designing and assessing educational MMORPGs.

In D. Ifenthaler, Kinshuk, P. Isaias, D. G. Sampson, & J. M. Spector (Eds.), Multiple perspectives on problem solving and learning in the digital age (pp. 159-178)

The central thesis of this chapter is that emerging technologies such as digital games compel educators, educational researchers, and instructional designers to conceptualize learning, instruction, and assessment in fundamentally different ways. New technologies, including massively multi-player digital games offer new opportunities for learning and instruction; however, there is as yet insufficient evidence to support sustained impact on learning and instruction, apart from the case of military training based on large simulated war games. Technologically sophisticated design and assessment frameworks are likely to facilitate progress in this area, and that is our focus in this chapter. Specifically, we provide an integrated framework for assessing complex problem solving in digital game-based learning in the context of a longitudinal design-based research study.

Journal of Educational Computing Research

Aroutis Foster , Mamta Shah

There is a need for game-based learning frameworks that provide a lens for understanding learning experiences afforded in digital games. These frameworks should aim to facilitate game analyses, identification of learning opportunities, and support for learner experiences. This article uses the inquiry, communication, construction, and expression (ICCE) framework to examine a mathematics game (Dimension M) to support learners. The study was conducted using mixed-methods with interviews, observations, and pre-post assessments, in addition to analyzing learner experiences using the ICCE framework. Results showed that the twenty 9th graders’ in the game-based learning course had statistically significant mathematics gains, but not for motivation. Interpretive results highlight how ICCE as enacted in the game design supported learners’ experiences. The ICCE framework may be a valuable tool for aiding teachers to assess the efficacy of games for learning and for students to benefit from the possible designed experiences within games.

Journal of Educational Computing Research, 45(3), 265-287

Following a design-based research framework, this article reports two empirical studies with an educational MMOG, called McLarin’s Adventures, on facilitating 9th-grade students’ complex problem-solving skill acquisi- tion in interdisciplinary STEM education. The article discusses the nature of complex and ill-structured problem solving and, accordingly, how the game-based learning environment can facilitate complex problem-solving skill acquisition. The findings of the first study point to the importance of supporting cognitive regulation of students for successful complex problem- solving skill acquisition in digital game-based learning. The findings of the follow-up study show that when scaffolded by dynamic modeling, students made significant improvement in their complex problem-solving outcomes. Implications drawn from the findings of these two studies are discussed related to: (1) educational game design strategies to effectively facilitate complex problem-solving skill development; and (2) stealth or embedded assessment of progress in complex problem solving during digital game-based learning.

Journal of Research on Technology in Education

This paper reports results from a yearlong project at a high school that used the Play Curricular activity Reflection Discussion (PCaRD) model for integrating games in classrooms. PCaRD was implemented using three games for supporting teachers and students in an elective course. Qualitative data sources such as interviews and field notes were primarily used to understand the process of students’ content knowledge and motivation to learn Mathematics, Physics, and Social Studies supported by quantitative assessments for measuring achievement gains and motivational changes. Interpretive analysis indicated that PCaRD aided student learning, motivation to learn, and identification with the content. We found mixed quantitative results for student knowledge gain with only statistical significant gains for mathematics. We also found that PCaRD provided teachers with an adaptive structure for integrating games in an existing and new curriculum. PCaRD has implications for research, teaching, and design of games for learning.

Technology Enhanced Language Learning (TELL) is the buzzword of new learning strategies in the classroom. Game-based applications used in the classroom can investigate the interest generated and performance in learning. They can be used to apply insights and develop the pedagogy. Although there are many games-enhanced and game-based perspectives (Chik, 2014), there is a need for more research on the former. The experiment and survey was conducted during a National Seminar held at the Malabar Christian College, Calicut, S. India with over one hundred participants from more than 16 colleges all over India. A simple game format using web resources was administered to the participants with a questionnaire before and after the game to measure their interest. The results highlighted the fact that game-based pedagogy stimulates learners and engages their attention. The experiment is hoped to have far-reaching consequences in the educational world

This thesis reviews and utilizes concepts from cognitive psychology, developmental psychology and game design to bring forth a number of design principles for educational games that may improve students’ motivation to learn. The main contribution of this thesis is a novel approach to serious game design, namely envisioning play and learning as a restructuring practice. This change of perspective, from a formal game design approach (focused on rules and regulations) towards a more activity centered approach (focused on process and style), may help designers to leverage the motivational potential of games, in order to make education more engaging to students. The main research question of this thesis is: How to design autonomy-supportive learning games and how can these games improve students’ motivation to learn? After the introduction, section 2 describes developments in education. Whereas, ‘traditional’ education focused on the transfer of content and the training of rather specific skills, social constructivist thought in Dutch education brought forward a focus on meta-cognitive skills, such as problem-solving, empathic understanding and entrepreneurship. As a result, Dutch educational system attempts to make students increasingly responsible for their own learning process. One way of doing this is by creating autonomy-supportive learning environments. In these, students have the opportunity to explore, experiment and struggle with the learning content. This manner of learning appears rather playful. Therefore this section concludes that autonomy- supportive learning may proof a valuable approach for serious game designers. Section 3 stresses the correspondence between autonomy-supportive learning and gameplay. It shows how games have become increasingly autonomy-supportive. For example, players can find multiple solutions to a problem, they can play in accordance to their favored playing styles, and players are increasingly able to self-express themselves through social negotiations with others. Additionally, section 3 introduces the term: restructuring. Restructuring suggests the rearrangement and manipulation of existing structures to create something new. It is suggests that play can be characterized as a restructuring practice, and that this may help designers to integrate the learning into the gameplay. Section 4 suggests that both education and the game industry present their audiences with autonomy-supportive environments. In addition, it suggests that learning and playing can be characterized as a restructuring practice. For example, learners rearrange, manipulate and change existing knowledge actors and structures to construct new knowledge. In comparison, players rearrange, manipulate and change exiting objects, rules, goals and experiences to create something new too. Play and learning are both considered restructuring practice though social negotiations in a socio-cultural network of human and inhuman actors. Serious game designers can search for the restructureable elements in the learning content and transform them into playful activities. In short, designers could determine what can be changed without changing the learning content and translate this to game mechanics and dynamics. Searching for restructureable elements is considered the main design guideline to integrate the learning into the gameplay. Section 5 suggests ten designs steps to integrate the learning into the gameplay. Consecutively, Section 6 illustrates the ten steps of embedding the learning content in the gameplay with the development of Combinatorics (a game about permutations), followed by Section 7, which combines all insights from development psychology and game design in the Applied Game Design Model. This model describes the ten steps of ‘getting the learning into the game’. The Applied Game Design Model describes the initial concept design of an educational game. Section 8 contributes to this design with various ways to leverage the motivational potential of games. The section starts with explaining the reasons to use Self-Determination Theory as theoretical framework for motivation and consequently suggests various design decisions to satisfy needs for competence, autonomy and relatedness. These design tools are illustrated with the further development of Combinatorics in section 9. Section 10 examined the motivational impact of Combinatorics. It describes a comparative study between the experienced regulatory style that was reported by players of an autonomy-supportive version and a restrictive (Drill & Practice) version of Combinatorics. Findings suggest that autonomy-supportive games can positively influence motivation towards learning. However, the restrictive version may positively influence motivation to learn as well. It becomes clear that different design decisions lead to different changes in motivation. Future research could study these differences in more detail and over a longer period of time, trying to get a better understanding of restructuring practices and their impact on motivation. Section 11 discusses the main contributions and positioning of this thesis, followed by the final conclusions in section 12, which revisits the concepts of cognitive psychology, developmental psychology, the Applied Game Design Model and the validation study to suggest a number of design principles for educational games that my improve students’ motivation to learn.

Educational Technology Research & Development

The important but little understood problem that motivated this study was the lack of research on valid assessment methods to determine progress in higher-order learning in situations involving complex and ill-structured problems. Without a valid assessment method, little progress can occur in instructional design research with regard to designing effective learning environments to facilitate acquisition of expertise in complex, ill-structured knowledge domains. In this paper, we first present a method based on causal representations for assessing progress of learning in complex, ill-structured problem solving and discuss its theoretical framework. Then, we present an experimental study investigating its validity against adapted protocol analysis (Ericsson & Simon, 1984). This study explored the impact of a massively multiplayer online educational game, which was designed to support an interdisciplinary STEM education on ninth-grade students' complex, ill-structured problem solving skill acquisition. We identify conceptual similarities and differences between the two methods, present our comparative study and its results, and then discuss implications for diagnostics and applications. We conclude by determining how the two approaches could be used in conjunction for further research on complex and ill-structured problem solving.

Ignatius T Endarto

In this 21st century, technology has revolutionized almost all aspects of life, including language learning. However, the trend of testing has substantially remained unchanged. Since the ultimate goal of language testing is to judge and gather information about learners' proficiency, one might archetypically describe it as either having students sit at their tables with paper and pencils trying to answer a number of questions individually and in a very formal manner, or asking them to perform something like a presentation or role play in front of the examiner. Those kinds of tests tend to bring a nerve-racking atmosphere which might hinder students in demonstrating their actual competence. Relating to the issue, this study dwells upon the use of web-based platforms in the gamification of language testing. Gamification is the adoption of game elements for non-game purposes. By promoting gamified testing via web-based platforms, this research seeks to make language assessments more fun and motivating, and of course less terrifying for learners.

Diego Ponte

In this paper, we investigate the impact of flow (operationalized as heightened challenge and skill), engagement, and immersion on learning in game-based learning environments. The data was gathered through a survey from players (N ¼ 173) of two learning games (Quantum Spectre: N ¼ 134 and Spumone: N ¼ 40). The results show that engagement in the game has a clear positive effect on learning, however, we did not find a significant effect between immersion in the game and learning. Challenge of the game had a positive effect on learning both directly and via the increased engagement. Being skilled in the game did not affect learning directly but by increasing engagement in the game. Both the challenge of the game and being skilled in the game had a positive effect on both being engaged and immersed in the game. The challenge in the game was an especially strong predictor of learning outcomes. For the design of educational games, the results suggest that the challenge of the game should be able to keep up with the learners growing abilities and learning in order to endorse continued learning in game-based learning environments.

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