ZDM (2019) 51:979–990 https://doi.org/10.1007/s11858-019-01075-3 ORIGINAL ARTICLE The use and potential of Fermi problems in the STEM disciplines to support the development of twenty‑first century competencies Jonas Bergman Ärlebäck1 · Lluís Albarracín2 Accepted: 19 July 2019 / Published online: 31 July 2019 © The Author(s) 2019 Abstract This paper presents the results of a literature review focusing on the use and potential of Fermi problems (FPs) in the STEM disciplines. We identify different usages and roles given to FPs in each of the STEM disciplines and highlight their potential to promote modeling of real phenomena and situations as well as learning about the estimation of quantities. Modeling is found to be at the heart of integrating the different STEM disciplines and estimation is a key facilitator that enables students to tackle complex situations and content in all disciplines. The results of the review are discussed in terms of the potential and possibilities of FPs to support the development of twenty-first century competencies in the STEM disciplines. Keywords Fermi problems · STEM Education · Mathematical modeling · Estimation 1 Introduction and background the protectionist view of the individual disciplines as illus- trated in the case of mathematics by Shaughnessy (2013). He STEM Education, which brings together the disciplines of argued that mathematics “will become silent if not given sig- Science, Technology, Engineering and Mathematics into a nificant attention” (p. 324) when disciplines are integrated single integrated field of study (Kelley and Knowles 2016), in STEM education programs. It is still an open question aims at providing students with competencies to meet for STEM educators how the disciplines can be effectively employers’ growing demands in these areas. Both integra- integrated while at the same time ensuring the integrity of tive and holistic approaches have been used to conceptual- each individual discipline (English 2017). ize the interdisciplinary nature of STEM (Fitzallen 2015). Although little is known about how this integration can However, the integration of the different disciplines has be achieved (Honey et al. 2014), interdisciplinary and/or proven challenging. During the last two decades STEM integrated approaches to STEM education are nevertheless education primarily has focused on improving Science widely stressed and argued for (The Common Core State and Mathematics with little emphasis on integration and Standards for Mathematics (http://www.cores​t anda​rds. only limited attention given to Technology or Engineering org/Math/); the Next Generation Science Standards (http:// (Bybee 2010). For example, the learning opportunities for www.nextge​ nscie​ nce.org/); STEM Task Force Report 2014). enhancing the understanding and developing of mathemati- For example, Kennedy and Odell (2014) have argued that cal skills within a STEM context was illustrated by Magiera “STEM education requires programs to include rigorous cur- (2013), but the way in which mathematics can contribute riculum, instruction, and assessment, integrate technology to the development of an understanding of the concepts of and engineering into the science and mathematics curricu- the other STEM disciplines has not been given much atten- lum, and also promotes scientific inquiry and the engineer- tion (Fitzallen 2015). Perhaps this is not so surprising given ing design process” (p. 246). However, to be able to imple- ment a “rigorous curriculum”, it is essential to know more * Jonas Bergman Ärlebäck precisely what should be implemented. Although there are jonas.bergman.arleback@liu.se numerous attempts in the literature trying to establish a set of skills and specify the knowledge needed to be developed 1 Department of Mathematics, Linköping Univeristy, in order to be a competent professional in the STEM field, Linköping, Sweden no consensus has yet been reached (Binkley et al. 2012). 2 Universitat Autònoma de Barcelona, Cerdanyola del Vallès, One such attempt is by Binkley et al. (2012) who analyzed Spain 13 Vol.:(0123456789) 980 J. B. Ärlebäck, L. Albarracín twelve twenty-first century skills frameworks from a number Given this recommendation from STEM education of different countries and identified ten skills grouped into research to focus on real world problems, and motivated four broad categories: by recent research carried out on modeling in the field of Ways of thinking mathematics education showing that Fermi problems (FPs) can be useful in introducing mathematical modeling at 1. Creativity and innovation. educational levels ranging from primary to tertiary (Albar- 2. Critical thinking, problem solving, decision making. racín and Gorgorió 2014; Ärlebäck 2009; Czocher 2016; 3. Learning to learn. Peter-Koop 2009), we focus our review on what FPs have to offer in relation to STEM education and twenty-first century Ways of working competences. 4. Communication. 1.1 Fermi problems, estimation and mathematical 5. Collaboration (teamwork). modeling Tools for working FPs were originally used by the physicist Enrico Fermi to show the power of deductive thinking and to prepare his 6. Information literacy (includes research on sources, evi- students for experimental laboratory work. Efthimiou and dence, biases, etc.). Llewellyn (2006) characterized FPs as open questions offer- 7. ICT (Information and Communication Technology) lit- ing little or no specific information for the problem solv- eracy. ers to direct them in the solution process. Examples of FPs include: estimating the number of cars on a 3 km tailback Living in the World on a motorway (Peter-Koop 2009); estimating to within one order of magnitude the number of hemoglobin molecules 8. Citizenship—local and global. that your body makes every second (White 2004); and esti- 9. Life and career. mating how many shopping malls are in the United States 10. Personal and social responsibility—including cultural (Anderson and Sherman 2010). The most prominent char- awareness and competence. acteristic that defines FPs is the way in which their solu-   (p. 36, italics in original). tions are achieved. By focusing on a question asking for an estimate of a specific quantity, the idea is to tackle the Jang (2016), on the other hand, used data from STEM problem by making assumptions based on the knowledge workplaces and identified the competencies called for as already available to the problem solver and following sim- explicated in STEM job listings. Jang’s analysis resulted in ple chains of reasoning. The procedure proposed by Fermi a list of 18 skills, seven categories of knowledge and 27 was to decompose the original problem into simpler sub- work activities important in the STEM professions. Given problems and to reach a solution to the original question by the breadth and diversity of the competencies identified, making reasonable estimates or educated guesses after first Jang (2016) concluded that students should be motivated to considering the individual sub-problems (Carlson 1997). In solve interdisciplinary sets of complex problems collabora- the literature this way of working is known as the Fermi tively using critical thinking and knowledge from all STEM (estimates) method. Thus, in our review, we consider that disciplines. Drawing on multiple studies in different STEM FPs are those questions that ask students to estimate quanti- disciplines highlighting the possibilities of learning from ties in the real world by following the Fermi method. real problems, Jang (2016) claimed that “problem solving Given the emphasis on and the role of estimates and edu- that addresses ill-defined problems and demands the evalua- cated guesses in solving a FP, the research on FP can be seen tion of multiple solution paths should be more encouraged in as connected to the long tradition on estimation in the teach- STEM education programs” (p. 296). This is consistent with ing and learning of mathematics. Generally, estimation is a Fitzallen (2015) who cited multiple authors advocating the process that gives a rough solution to a problem in counting integration of the STEM disciplines through engaging stu- or measurement (Siegel et al. 1982). Wagner and Davies dents in real-world problem solving, and Clark and Sengupta (2010) connected estimation skills with number sense, (2013) have argued that modeling “can productively engage which must be practiced with students until they reach a students in inquiry-based activities that support learning of critical level allowing them not only to manipulate quantities complex scientific concepts as well as the core argumenta- meaningfully, but also to derive meaning from them so that tion and modeling practices at the heart of scientific inquiry” they can be useful in making decisions. In mathematics, the (p. 85). focus on using estimates often occurs as students develop an understanding of, and methods for, counting and measuring 13 The use and potential of Fermi problems in the STEM disciplines to support the development of… 981 (Reys 1984). But when FPs are used in other disciplines, all relevant journal articles, conference papers, book chap- the understanding of quantities and their relationships are ters, theses and research reports. To this end we used vari- on equal footing with concepts and skills of counting and ous databases, particularly Google Scholar, Scopus, Web of measuring. This equal footing is directly related to seeing Science, Jstor and Crossref, and search phrases involving FPs as modeling problems. Fermi problem/s, Fermi question/s and Back-of-envelope and Mathematical modeling, as understood in this paper, is their counterparts in German, Portuguese, French, Spanish a general problem-solving process encompassing the math- and Swedish. The searching and screening for inclusion was ematising of a real-world phenomenon and elaborating a complicated due to the wide and diverse scientific produc- mathematical model to describe, understand or predict the tion of Enrico Fermi; a large number of physics phenom- behavior or properties of the phenomenon studied. We fol- ena, institutions and instruments are associated with his low Lesh and Harel (2003), who define mathematical models name such as for example Fermi surfaces, the Fermi-Ulam as conceptual systems that describe other systems and take model, the Fermi paradox, Fermi large area telescope, and models to be constructed of a set of concepts to describe or Fermilab. explain the mathematical objects relevant to the phenom- Our search resulted in a set of documents that was further enon studied. Such models include the procedures used to expanded using forward- and backward searches (cf. Tem- create useful constructions, manipulations, or predictions plier and Paré 2015), by looking through the references of for achieving clearly recognized goals. From this perspec- the documents found and using the Google Scholar tool that tive, we concur with Robinson (2008), who considered FPs lists the publications citing a specific source. This allowed to be miniature-modeling problems in the sense that they us to identify a set of 117 relevant publications. Given our are smaller, well-defined and delimited contextualized prob- goal to be as exhaustive as possible and that the traditions lems, not real world problems given in all their complex- and norms of writing in different disciplines follow differ- ity. Both full scale modeling problems and FPs confront the ent standards, we did not discriminate between research- or problem solver with the need to conduct a detailed analysis practitioner-oriented publications when screening the docu- of the situation presented in the statement of the problem, to ments for inclusion and assessing their quality. For exam- decompose the original problem into simpler sub-problems, ple, FPs appeared in American university physics classes and to engage in validating the results in the context of the (and in science and engineering classes more generally) in original problem. In this paper, we consider modeling being the middle of the twentieth century, predating the modern connected to real world problems whereas generic problem development of didactical fields of research. Hence, there solving refers to general reasoning skills and metacognition. exists a long tradition and shared knowledge about the learn- ing promoted by FPs among students in physics and other 1.2 Research questions science and engineering disciplines. However, the first aca- demic work we found investigating the didactic potential of To investigate the role FPs can have as a facilitator and inte- FPs was by Moore (1987). Thus, the knowledge about the grator in terms of teaching and learning in the STEM dis- potential of FPs as didactical tools in the STEM disciplines ciplines and developing twenty-first century competencies, is expressed in two ways: first, in articles focusing on the we review the literature to answer the following research reporting of knowledge obtained through experience and questions: reflection on accumulated teaching practices, and, second and more recently, in traditional research studies. • How, and based on what arguments, have FPs been used During the screening of articles, we found multiple in the different STEM disciplines? instances of one or more authors having a number of publi- • What are the research findings on the use of FPs in the cations addressing similar questions and presenting basically STEM disciplines? the same data sets and analysis in different publications. A typical example of such an instance is when a conference paper later was developed into a full journal article. In these 2 Method and analysis cases, only the more extended publication presenting the research was included in the review. We also excluded pub- We have conducted a systematic literature review inspired lications from the original 117 when the document consisted by the six steps outlined by Templier and Paré (2015): for- of a mere list of problems without any discussion or argu- mulating research questions and objectives; searching the mentation on the use of the problems. Most of the documents literature; screening for inclusion; assessing the quality of found are written in English, but we also found documents primary studies; extracting data; and analyzing data. Given in Spanish, Portuguese, Swedish, French, Dutch, German, our research questions and objectives, we set out to do an Japanese and Turkish. We managed to translate all papers exhaustive search of the literature to ensure the inclusion of written in languages we could not read using online tools, 13 982 J. B. Ärlebäck, L. Albarracín Table 1  Coding schema with themes and guiding questions Theme Guiding question A Type of publication Is the paper a research publication or a more argumentative publication? A Sender Who is talking? A Targeted audience Whom is the intended audience? A Main research field From which discipline is the publication? A Year Publication year B Aim of the research What is the aim of the research? B Research questions studied What are the research questions studied? C Definition/description of FPs How is the notion of FP defined/described? C Rationale for using FPs What is the motivation for using FPs in the paper? C Related content What are the contents related to the FPs? D Subjects’ age What age are the subjects? D Number of subjects How many subjects participated in the study? D Nationality of the subjects Where is the study conducted? What is the nationality of the subjects? D Nature of subjects’ work with FP Are the students working individually or in groups? E Nature of methods used Qualitative, quantitative or mixed? E Data collected What data are collected and how? E The actual FPs used What explicit problems are used? E Arguments for using FP What are the arguments for using FP? F Type of analysis applied How are the data analyzed? F Theoretical framework What are the theoretical frameworks used? G Results What are the results of the study? G Suggestions and implications What are the suggestions and implications put forward in the publication? except for two Japanese articles which appeared to be prac- reported on empirical studies focusing on teaching or learn- titioner-oriented papers. The screening process reduced the ing using FPs. Eighteen of the 91 documents argued for the 117 initial publications to a core of 91 documents explicitly use of FPs in connection with curriculum changes, compe- addressing the use of FPs in different educational settings. tence development or other education goals (such as chang- All 91 documents were read and analyzed using a coding ing beliefs or fostering critical thinking). Notably, 30 of the schema containing seven broader themes with accompany- 91 publications describing the possibilities and potential of ing guiding questions. The themes were: (A) Type of paper; using FPs were written based on the author’s own teaching (B) Focus and aim of the paper; (C) Definition of FP; (D) experiences. Setting of the research presented; (E) Methods applied in The development of publications over time is shown in the research presented; (F) Analysis; and (G) Results and Table 2 by period of publication and STEM discipline. In implications (see Table 1). During the first reading of the science we found 22 documents with 19 of those documents documents some of the data were straightforwardly recorded in physics. This is not surprising given the origin of FPs in (such as year of publication or classifying the publication as the work of Fermi. Mathematics had the largest number of being research- or practitioner-oriented) or resulted in vari- documents (48) which is not surprising given the develop- ous emerging lists (such as the theoretical frameworks or the ments in the field of mathematical modeling in mathematics actual FPs used in the publication). For some of the catego- ries the relevant section(s) in the documents were marked in order to be later collectively subjected to open coding. Table 2  The distribution of documents by time and STEM discipline This process of analysis allowed us to connect the results S TE M General Total of different empirical and practitioner-oriented studies to the Before 2000 3 4 2 0 9 theoretical foundations on which other studies were based. 2000–2004 7 1 2 0 10 The 91 documents analyzed were diverse in nature. Arti- 2005–2009 4 2 11 2 19 cles published in research journals in different fields used 2010–2014 1 3 18 3 25 different frameworks and methodologies, especially when After 2015 7 3 15 3 28 research articles in education were published in pure sci- 22 13 48 8 91 entific journals. We found that 43 of the 91 publications 13 The use and potential of Fermi problems in the STEM disciplines to support the development of… 983 Table 3  Main sources of documents reviewed on FPs can support students developing twenty-first century Journals Conferences STEM competencies. The actual arguments identified are elaborated in the Praxis der Naturwissenschaften—Physik (5) ICTMAa (4) following sections, but Table 4 summarizes the total num- The Physics Teacher (4) CERMEb (2) ber arguments for, and results on, FPs in each discipline. Physics Education (3) Mathematics education was found to be the discipline that Delta Phi B (2) provided the most arguments and results on FPs, and is, in Educational Studies in Mathematics (2) addition the discipline that had the largest proportion of EURASIA Journal of Mathematics Science and research studies. Technology Education (2) In order to organize the identified arguments and results, Mathematics Teaching in the Middle School (2) we classified them by the type of content to which they are Modelling in Science Education Learning (2) related as a result of open coding: the development of skills The Mathematics Enthusiast (2) for estimation; number sense within the different disciplines ZDM (2) and changes in beliefs based on numerical analysis of phe- a International Conference on the Teaching of Mathematical Model- nomena; aspects related to generic problem solving such ling and Applications as general reasoning or metacognition; and aspects related b Conferences of European Society for Research in Mathematics Edu- to the modeling of real phenomena. The number of argu- cation ments and results for each of these content areas is shown in Table 5. Most identified arguments are related to prob- Table 4  Numbers of arguments and results by discipline lem solving, whereas most results are related to modeling. S TE M General edu- Total The latter finding is possibly due to the increase of empiri- cation cal studies of mathematical modeling in recent years (Schu- kajlow et al. 2018). Arguments 17 11 30 2 60 Results 9 4 27 1 41 Total 26 15 57 3 101 3.1 Estimation The role and function of making and using estimates in dif- ferent contexts is a common theme found in the literature. education over the last 10 years. The numbers of documents For example, Moore (1987) argued that engineers should found in technology and engineering were scarce and hence be able to make adequate estimates based on their knowl- are reported together. The documents were found in 61 dif- edge given limited data, and he advocated for introducing ferent sources and Table 3 lists the sources with two or more FPs in programming courses as a way to engage students in contributions. working with ill-defined problems. Similarly, Bentley (1999) advocated that computer science students needed to master some common techniques in programming similar to the 3 Results Fermi estimates method in order to decide if their programs are adequate and effective for solving the problems they In this section, we present the results from the literature are trying to address. One such example of an FP is about review with respect to our research questions. We first give deciding whether technical aspects (such as the speed of a an overview of the arguments and results identified in the 91 network) are adequate for the intended use, or whether it is publications analyzed. Then, we answer the second research worth investing a certain amount of time in improving an question organized in four sections: estimation; number algorithm given the impact on performance this can achieve. sense and beliefs; problem solving; and modeling. In the Working with first year engineering students, Shakerin result section we then discuss FPs as facilitators and integra- (2006) used a sequence of FPs focusing on the topic of tors among the different STEM disciplines, and how working Table 5  Number of arguments Estimation Number Sense and Problem Modeling Other Total and results per contents skills Beliefs solving Arguments 7 7 38 7 1 60 Results 0 3 13 23 2 41 Total 7 10 51 30 3 101 13 984 J. B. Ärlebäck, L. Albarracín energy. With the aim of developing students’ estimation meet every day. Albarracín and Gorgorió (2015) showed skills, the objective of the early tasks in the sequence was how secondary students developed their own methods to to create a set of reference points to be used in later tasks solve real situation FPs and became more confident in their as a basis for comparison. Students initially worked with calculations and results than in the data reported in newspa- basic estimates of quantities of length, area, volume, weight, pers. Along the same lines, Carlson (1997) argued that FPs density, energy and temperature. An important part of the and the Fermi method offer opportunities for presenting con- approach was to verify the estimated quantities experimen- temporary problems to students and proposed the study of tally. In one task students estimated the weight of a person’s environmental problems related to fuel consumption, such as limb, such as an arm or leg, by using the water displaced in a the consequences of changing large cars to smaller cars with bathtub. In other tasks, such as the estimation of the electri- lower fuel consumption. Carlson pointed out that working on cal energy consumed in a household, information measured FPs can quickly make students conscious of their abilities, and collected by others (e.g., meter readings) could be used skillful in giving quick answers based on simple calculations to verify the estimate. According to Shakerin, students can in various settings, and that engaging in problems connected experience the potential in the Fermi method and make con- with economic and social contexts enables the students to nections to the theories on which the meaning and relation- develop their opinions and beliefs in an informed manner. ships among quantities are based when they tackle more Although not experimentally verified, one of the most complex problem situations. Shakerin argued that this can be common claims for using FPs in the literature is that they done by attending to the units of the involved quantities, and promote understandings and meanings of quantities within that this sequencing of tasks supports the development of the disciplines. The quantitative study of Furjanic and Mül- the knowledge and skills needed to use estimation properly ler (2001) concluded that secondary students were better in engineering workplace activities. Using FPs in connec- at estimating everyday quantities than estimating the more tion with laboratory experiments and in situations where discipline specific quantities relevant for the physics courses it is not possible or feasible to make direct measurements they were taking. White (2004) argued that the quantitative to make decisions, resonates with the practice of engineer- approach promoted by FPs provides the basic component ing by encouraging students to use and integrate previous for understanding quantitative problems in biology. Phillips knowledge about physical laws, about units and dimension, and Milo (2009) stressed that for problems in biology that and their own estimates and references. cannot be addressed qualitatively, a quantitative approach The work of Moore (1987), Bentley (1999) and Shakerin can provide relevant information to better understand and (2006) illustrate educational usages of FPs in the decision- explain processes that are difficult to study. Thus, they pro- making and planning phases in engineering processes where posed that the Fermi method be used as a research tool in the estimation skills are crucial. Other authors working in dif- life sciences: “the time is ripe for the emergence of a similar ferent disciplines share this idea. Robinson (2008) described tradition [using Fermi estimates] in the biological setting order of magnitude calculations as an essential skill in because as we argue… such estimates can reveal gaps in our physics that can confirm the feasibility of a model or deter- understanding” (Phillips and Milo 2009, p. 21467). In order mine whether an onerous and time-consuming calculation to have reliable quantitative data in the field of biology read- is required. In economics, Anderson and Sherman (2010) ily available, Milo’s team created a free easy-access Wiki- observe that it is often necessary to make quick estimates pedia-like community effort to collect numerical data that when neither time nor resources are available for making have been published in peer-reviewed literature (see www. more time consuming or costly assessments, as in the first bionu​mbers​.org). The Fermi estimation method has been stages of product development when entrepreneurs evaluate used successfully on a large number of problems for which the commercial viability of new commodities. it is difficult to find a solution using traditional and common However, as illustrated in Table 5, the reviewed research methods in biology as illustrated by the work of Machtans has not focused on explicitly establishing evidence support- and Thogmartin (2014) on estimating bird populations from ing the argument that FPs promote students’ developing of very small data sets. These authors emphasized that Fermi their estimation skills. estimates often are more accurate than expected because the multiplication of several estimated factors will normally 3.2 Number sense and beliefs include both overestimated and underestimated quantities, potentially cancelling out some of the approximation error. Wagner and Davies (2010) examined how FPs promote con- Focusing on generating meaningful understanding of necting experiences of quantities and computation skills in quantities, Cordry (2010) used the Fermi method to address terms of number-, quantity- and critical sense. They argued the earth’s limited resources and its capacity to sustain the that it is essential that students as citizens be able to interpret world’s population from the point of view of thermody- and respond critically to the barrage of numbers that they namics. Cordry used FPs to introduce a unit of reference 13 The use and potential of Fermi problems in the STEM disciplines to support the development of… 985 in terms of the energy consumption a person needs for sur- key reference points, graphical sketching, and analyzing the vival. The suggested task raised multiple questions such physics of the situation), and their results suggested that this as: Will the world’s population continue to grow without the phase of problem solving in which students struggled the restraint? Are there sufficient sources of food and energy to most. Holubova’s analysis showed that students tended not provide a decent living-standard for all people? The students to validate their obtained results adequately and argued that are generally surprised by the results of their calculations, this crucial part of the problem-solving process needs to be especially when they learn that the present world popula- taught explicitly. tion’s energy needs exceeds the sources that they have just Some authors investigated the use of FPs complemented calculated are available. The activity was used at the college with different technologies in order to have a larger impact level in a unit on the conservation of energy and allowed the on the mathematical thinking of the students. Tangney and scientific content to be presented in a context relevant for Bray (2013), for example, proposed the use of mobile tools the students and to open up a discussion that goes beyond such as still or video cameras and stopwatches to capture the physics class. data on problems related to counting groups or flows of peo- The findings by Cordry (2010) can be complemented by ple in spaces such as schools or supermarkets. Keune and those of Morgan (2017), who studied the changes in college Henning (2003) pointed to the possibility of using spread- students’ perceptions regarding the prevalence of life in the sheets to provide a dynamic way for the students to test their galaxy after using estimates while working on the Drake solutions by slightly changing their estimated values for each equation. Using a pre-/post-test design, Morgan noted a shift sub-problem in order to see what might be a large effect. towards a more optimistic stance regarding the existence of At the college level, Vidal et al. (2017) discussed using extra-terrestrial life in the galaxy. Morgan argued that the Matlab to calculate estimated volumes of different shaped Fermi method provided a tool for making unknown aspects mountains. closer and more accessible to students and that FPs have a A main research result with respect to problem solving real potential to make an impact on students’ beliefs about is that the Fermi method can successfully be taught (Raviv the world. et al. 2016; Barahmeh et al. 2017). Barahmeh et al. (2017) demonstrated this aspect in the context of teaching 9th grade 3.3 Problem solving students physics, and illustrated how the students improved their skills in taking measurements, making measurement Many of the analyzed documents highlighted aspects of predictions, and making productive use of measurements. problem solving that can be worked on using FPs. Taggart Consistent with the claims of Robinson (2008), Barahmeh et al. (2007) stressed that FPs promote flexible thinking and et al. stressed the value of the Fermi method for students to creating one’s own method to deal with problems when no think freely and without predetermined formulas for solving more direct solution method is readily applicable or avail- the tasks, and for promoting discussions on different solu- able. Seiwald (2016) argued that FPs move away from a tions. A way of teaching FPs, argued for by Anderson and strong emphasis on routine tasks and focus on developing Sherman (2010), is based on creating a graphical representa- and deepening the understanding of important concepts. tion that structures and divides the main problem into differ- Seiwald claimed that FPs can support educational contexts ent sub-problems, which provides the students with a tool to in which different solutions to the same problem are pro- organize their work. Taken together, the studies mentioned moted and can support a culture of discussion and debate above exemplify the emphasis identified in the literature that in the classroom. Other researchers have tried to connect FPs promote the learning of generic problem solving skills FPs to various mathematical competencies, or in other ways in terms of general reasoning skills and metacognition. attempted to theorize the potential and possibilities of FPs in classrooms. For example, Sriraman and Knott (2009) 3.4 Modeling stressed that working with educated guesses in socially rel- evant contexts can be a form of fostering mathematical and According to Robinson (2008), in order to solve a FP, one critical thinking. Other studies have shown how teachers can has to: (1) synthesize a physical model; (2) examine the rele- use FPs to promote tolerance for the ambiguities arising in vant physical principles applicable; (3) determine constraints problem solving (Palmér et al. 2018). By studying students’ and boundary conditions; (4) decide how simple the model work on FPs in an engineering problem contest, Holubova can be while still maintaining realism; and (5) apply some (2017) showed that students can use many different activi- rough estimation. Using this approach, Robinson taught first ties in the analytical phase of solving a problem (selecting year college physics by proposing a problem and then allow- ing the students to generate their own methods and solutions, before engaging the students in discussing these among themselves in relation to a solution provided by the teacher. 13 986 J. B. Ärlebäck, L. Albarracín By studying the impact of the Fermi method on students’ solve a FP are complex even for these students, requiring examination marks Robinson showed that students can learn them to regulate their modeling processes by monitoring the Fermi method to solve FPs effectively, but Robinson how their immediate goals or subgoals are related to the found that this approach to develop more general problem- problem statement. In a follow-up study, Czocher (2018) solving competencies was challenging for some students. showed that college students do not construct their models In mathematics education, multiple studies have investi- independently of validation, but that validating is an integral gated the potential of FPs to introduce mathematical mod- part of the model construction process. eling with students of different age groups. In upper primary The articles collected in this section show that FPs can be school (ages 10–12), Peter-Koop (2009) used FPs to analyze used as modeling tasks with different goals depending on the students’ problem solving strategies, and concluded that (a) age of the students. For example, in primary education FPs students solve problems in several ways; (b) students develop can be used to introduce modelling, in secondary education new mathematical knowledge while solving the problem; they can support students to connect or refine knowledge and (c) students display problem-solving processes that are from different disciplines, and in higher education FPs can multi-cyclic in nature and relatable to the modeling cycle be used to introduce modeling as well as fostering self-mon- (cf. Blum and Leiss 2007). Other studies demonstrated the itoring and highlighting validation processes involved in the following: how primary school students engaged in model- modelling process. ling when solving FPs focusing on collecting and process- ing real data (Henze and Fritzlar 2010); the great variety of mathematical models that students generated when solving 4 Discussion and conclusions FPs (Albarracín and Gorgorió 2019); and the positive devel- opment of the modeling sub-competences such as simplify, The review of the literature on FPs in the STEM disciplines mathematize, interpret and explain (Haberzettl et al. 2018). summarized above shows a wide range of usages for FPs Ärlebäck (2009) analyzed the modeling activities of and the Fermi method. All the reviewed articles promote groups of secondary school mathematics students engaged the use of FPs for various reasons that in one way or another in solving a FP and highlighted the importance of social are connected to at least one of the two main characteristics relationships between the students, and the role of extra- of the Fermi method: modeling a situation by simplifying mathematical knowledge in the problem solving situation. and decomposing the initial problem into sub-problems; or The discussion within the groups was a key factor in gen- deriving and using appropriate estimates to make progress erating new mathematical knowledge connected to content on (sub-)problems. Based on the results from the previous from other disciplines (Ärlebäck and Frejd 2013). Also sections, we now discuss FPs as facilitators and integrators working with secondary students, Albarracín and Gorgorió among the different STEM disciplines, and how working on (2014) investigated students’ proposed solving strategies FPs can support students in developing twenty-first century on several FPs and observed that those FPs that allowed STEM competencies. an effective solution to the problems were connected to the construction of mathematical models. Similarly, by charac- 4.1 FPs as integrators between STEM disciplines terizing the outputs produced by secondary school students from a sequence of FPs, Ferrando et al. (2017) identified Creating models to describe, explain and make predictions the procedures and concepts that shaped the students’ mod- about different phenomena is fundamental in the sciences els. Ferrando et al. concluded that experienced students and engineering (Hestenes 2010). This role of modeling used a greater number of concepts to develop their models is consistent with the view in mathematics education that and that this resulted in more accurate and realistic models models are conceptual systems constructed to describe, when compared to the models developed by students without explain or predict some other systems for achieving clearly prior modeling experience. Albarracín and Gorgorió (2018) recognized goals (Lesh and Harel 2003). Our review of showed that sequences of FPs can encourage secondary stu- the literature reveals numerous articles highlighting the dents to construct mathematical models that successively potential of FPs to introduce and engage students in creat- better describe reality, allowing them to connect different ing models that describe phenomena or situations specific mathematical concepts in order to create a useful model in to science (Cordry 2010; Efthimiou and Llewellyn 2006; various situations. Their study also showed that students Robinson 2008), engineering (Shakerin 2006), or econom- changed the conceptual strategies that supported their mod- ics (Anderson and Sherman 2010). We have also shown that els based on procedural obstacles rather than on conceptual the literature on FPs promotes problem-solving competen- limitations. cies in contextualized situations in the teaching and learn- Lastly, Czocher (2016) used FPs with engineering stu- ing of mathematics (Albarracín and Gorgorió 2014, 2018; dents and confirmed that the modeling processes needed to Ärlebäck 2009; Czocher 2016, 2018; Ferrando et al. 2017; 13 The use and potential of Fermi problems in the STEM disciplines to support the development of… 987 Gallart et al. 2017; Keune and Henning 2003; Peter-Koop task at hand. This reduction of complexity allows students 2009). All these studies have in common that they confront to approach the solution in an alternative way by consider- students with accessible realistic situations and require an ing discipline specific quantities and relationships in order analysis using the Fermi method as a first attempt to cre- to simplify the situation to more manageable sub-problems ate a model, allowing the students to develop their meth- and by using estimates. In the Fermi method, working with ods and solutions by discussing these among themselves. and understanding individual quantities and their relation- Mathematization of realistic phenomena is a key process ships within different disciplines comes to the fore. It is in for creating conceptual models and developing meaning and this sense FPs work as facilitators for learning in the STEM relevant concepts within and across disciplines. Since real disciplines. Indeed, Vasterink (2011) argued that FPs can be phenomena are not neatly delimited and concerned only with a central part of the learning of estimation in the different a single knowledge domain, the integration of content and STEM disciplines. This is exemplified in science by Ches- disciplines through FPs comes naturally as a consequence nutt et al. (2018) who investigated the role of estimation on of well chosen and designed FPs and the use of the Fermi students’ developing understanding of size scale, and the method to approach the tasks. When studying a particular study of Resnick et al. (2017) on using scale as the basis phenomenon, different questions can and do arise, belonging for comparing size using relational reasoning. These last to different disciplines, which elicits the need for students two studies illustrate that FPs on interdisciplinary themes to identify, explore and understand relevant quantities from involving different scales (from the nanoscale to the macro the various disciplines. In other words, the type of modeling scale) can be a productive way to teach about concepts such required when working on a FP can act as an integrating as geological time and about estimating size and scale. activity for the STEM disciplines. In engineering, estimation plays a particularly important role in the preliminary stages of the engineering design pro- 4.2 FPs as a facilitator for learning in the STEM cess when decisions have to be made based on incomplete disciplines information or partly unavailable data, or when selections must be made from a multitude of options (Shakerin 2006). One of the features that define FPs is the need to deal with Estimation is often used to find answers to ill-defined prob- sub-problems based on estimates or educated guesses (Srira- lems, when detailed solutions not are required, or to check man and Knott 2009). Indeed, in solving FPs it is possible the validity of answers obtained experimentally. Success- for students to engage in making two different kind of esti- ful estimation in professional engineering practice is often mations: computational- and measurement estimation (cf. based on the knowledge of dimensions and units, basic laws Hogan and Brezinski 2003). What is essential however, is of physics, and the ability to define relationships and make that the process of assigning values to the quantities involved comparisons (Joshi et al. 2013). This use of estimation in in a problem is intimately connected to understanding the workplace environments has been proposed as one of the concepts and quantities being estimated (Resnick et al. twenty-first century STEM competencies in product design 2017). (Jang 2016). In schools, measurement estimation is often part of the mathematics curricula as a way of introducing and consoli- 4.3 Connection to twenty‑first century skills dating the learning of measurement (Sarama and Clements 2009). By tradition, measurement is learned in the disci- In relation to the development of twenty-first century compe- pline of mathematics where basic concepts and quantities tencies, this review shows that FPs promote the learning of are defined and students use different methods to measure several skills and competencies identified by Binkley et al. these quantities (e.g., length, area, volume, time). A key (2012) within and across multiple disciplines. When work- aspect for students to estimate quantities in an appropri- ing on ill-defined problems, FPs engage students in prob- ate way is the development of a well-defined set of mental lem solving, helping them learn how to deal with complex units of reference on which to base their estimates (Bright situations by developing their own strategies for tackling 1976). However, when dealing with any problem in a real unfamiliar problems (Moore 1987). situation the relevant quantities often belong to disciplines Researchers and practitioners have argued that FPs pro- other than mathematics, such as life sciences (heart rate) or mote communication and collaboration skills by engaging physics (energy). Vasterink (2011) argued that competence students in group work and class discussions, both within in estimation is universal to all the sciences and is widely single and across different disciplines (Efthimiou and applicable to and used in many other disciplines. In essence, Llewellyn 2006; Ärlebäck 2009). Although FPs generally estimation allows results to be obtained for quantities that are accessible to students, in a multidisciplinary context they otherwise can be almost impossible to measure. The Fermi naturally elicit non-trivial discussion and critical reflection method provides a tool for reducing the complexity of the on the selection and use of relevant concepts, constructs, 13 988 J. B. Ärlebäck, L. Albarracín and needed data (estimates). This way of working supports However, the few studies found suggest that technology the students in developing critical thinking skills when they might be a productive way for students to deal with com- evaluate the potential of each of the different strategies and plex cross-disciplinary FPs, which in turn could contribute solutions generated to solve the problem (Seiwald 2016). to students developing ICT literacy. Students’ critical thinking skills are also developed when Similarly to most of the papers we have reviewed, in this they see how strategies and solutions can be used in other paper we have focused on the potential of using FPs and not problems using the same basic strategies adjusted to other on limitations and challenges associated with working with contexts (Albarracín and Gorgorió 2018). The Fermi method FPs and the Fermi method. Although sparely mentioned in provides students with a tool to explore ways of thinking and the literature, we consider limitations and challenges impor- working without being tied to a concrete formula or routine tant to investigate in the future and in developing this area that must be learned and used, but rather students can work of research. freely, making their own decisions and developing as flexible Given both the strong connection between the learning of thinkers (Barahmeh et al. 2017; White 2004). measurement and the development of number sense, and that Some authors highlight that aspects of citizenship, infor- individuals in numerous diverse professions report applied mation literacy and personal and social responsibility can be measurement estimation skills to be a critical component for actualized for students through working with FPs situated success in their careers and as a tool to foster critical think- in cultural, social and environmental settings. Examples of ing, we concur with Sriraman and Knott (2009), who argued this aspect include FPs on the consumption of fossil fuels that the importance given to learning measurement estima- (Carlson 1997) and critical analysis of newspaper informa- tion and identifying key aspects of a problem (i.e., the Fermi tion (Albarracín and Gorgorió 2015). method) should not be limited to just those students who will pursue careers in STEM fields, but rather to all students. 4.4 Final remarks and future research Acknowledgements Open access funding provided by Linköping University. The authors thank Helen M. Doerr for her comments and suggestions on an earlier draft of this work. Lluís Albarracín is a Since FPs originated in the context of physics to coordinate Serra Húnter Fellow in UAB and supported by the projects EDU2017- theoretical knowledge with experimental work, and transi- 82427-R (Ministerio de Economía, Industria y Competitividad, Spain), tioned into physics classrooms as a didactic tool, their usages 2017 SGR 497 (AGAUR, Generalitat de Catalunya) and José Castillejo program (CAS17/00289 reference). have expanded to other disciplines. However, much of the research on FPs is discipline specific and unconnected, Open Access This article is distributed under the terms of the Crea- with little cross-discipline referencing. In this review, we tive Commons Attribution 4.0 International License (http://creat​iveco​ have connected isolated results on FPs, which, when taken mmons​.org/licen​ses/by/4.0/), which permits unrestricted use, distribu- together, suggest a promising potential for integrating learn- tion, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the ing in the STEM disciplines with the development of estima- Creative Commons license, and indicate if changes were made. tion skills and modeling as ways of making complex phe- nomena and concepts accessible to students. However, more systematic research on the teaching and learning, using FPs that are cross-disciplinary in nature, is needed. Given the References results of the review, sequences of FPs might be a promising Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi line of inquiry. problems involving large numbers. 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