Exploring STEM Pedagogy in the Mathematics Classroom: a

Exploring STEM Pedagogy in the Mathematics Classroom:a Tool-Based Experiment Lesson on Estimation

Allen Leung1

Received: 30 August 2017 /Accepted: 9 August 2018 /Published online: 24 August 2018# Ministry of Science and Technology, Taiwan 2018

AbstractHow to pedagogically integrate the four STEM (science, technology, engineering, andmathematics) disciplines is a main discussion issue in STEM education. Inquiry-basedlearning, mathematical modeling, and tool-based pedagogy could go hand in hand toprovide a possible integrated STEM pedagogical framework for mathematics andscience. This discussion paper explores such a boundary pedagogy via the proposalof an inquiry-based modeling pedagogical cycle. The cycle is used as a lens to analyzea sequence of tool-based mathematics school lessons conducted as a science experi-ment on the topic of estimation. The lesson design is seen through the proposedpedagogical cycles and selected students’ work is presented to illustrate students’different problem-solving tracks in a minimalist instruction environment. Pedagogicalfeatures that characterize a STEM boundary pedagogical approach are identified. Thepaper concludes with a discussion on some core values of STEM education.

Keywords Boundaryobject . Inquiry-basedlearning .Mathematicalmodeling .Pedagogy.

STEM education

Introduction

A major discussion in STEM Education is how to pedagogically integrate the fourSTEM disciplines: science, technology, engineering, and mathematics (see forexample, English, 2016, 2017). There is no common agreement on a model ofintegration since how these four STEM disciplines interact in a pedagogical situation

International Journal of Science and Mathematics Education (2019) 17:1339–1358https://doi.org/10.1007/s10763-018-9924-9

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10763-018-9924-9) contains supplementary material, which is available to authorized users.

* Allen [email protected]

1 Department of Education Studies, Hong Kong Baptist University, Hong Kong SAR, China

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depends on the levels of integration chosen; for example, disciplinary, multidisciplin-ary, interdisciplinary, and transdisciplinary (English, 2016). However, an importantperspective in STEM education suggests that teachers need to Bintegrate the correlatedSTEM disciplines in ways without losing the disciplines’ unique characteristics, depth,and rigor^ (Corlu, Capraro & Capraro, 2014). Some STEM models give mathematicsand science central roles (see for example, Corlu et al., 2014; Kertil & Gurel, 2016)while others put engineering as the major component of STEM. Engineering, in thecontext of secondary and primary schools, is interpreted as acts that includeresearching, designing, and producing (see for example, English & Mousoulides,2011; Hudson, English & Dawes, 2014). Capraro and Slough (2013) proposed thatsuch an engineering-focused integration could be achieved through interdisciplinaryproject-based learning where students engage in attacking ill-defined tasks via partic-ipating in interactive collaboration that involves hands-on activities to design andproduce artifacts (Han, Capraro & Capraro, 2015). A common technology-orientedSTEM education trend is to focus on robotic, coding, and computer programming. Aninteresting recent example for this approach is the ScratchMaths project (http://www.scratchmaths.org/) which sets out to explore how the SCRATCH environment can buildup mathematical knowledge and reasoning for primary school children in the UK(Benton, Hoyles, Kalas & Noss, 2016). Bryan, Moore, Johnson, and Roehrig (2015)proposed three forms of STEM integration: (i) content integration, (ii) integration ofsupporting content, and (iii) context integration. Content integration means to designlearning activity or unit that aims to teach concepts from all STEM disciplines.Integration of supporting content is about addressing an area (e.g. science) in supportof the learning objectives of the main content (e.g. mathematics). Context integration isabout putting the content of one STEM discipline in the center and teaching thatcontent by selecting relevant contexts from the other STEM disciplines. For example,teacher designs and implements mathematics lesson within the science, engineering, ortechnology contexts.

English (2016) remarked that BI have argued for a greater focus on STEM integra-tion, with a more balanced focus on each of the disciplines. Specifically, I havemaintained that mathematics and engineering are underrepresented in studies claimingto address STEM education^ (ibid., p. 6). Along this vein, research has shown that inthe mathematics classroom, context integration is a beneficial approach to practice(mathematical/engineering) modeling (see for example, English & Mousoulides, 2011;Hamilton, Lesh, Lester & Brilleslyper, 2008; Roehrig, Moore, Wang & Park, 2012),and modeling could serve as a pedagogical tool to forefront mathematics and engineer-ing in STEM education. An overarching pedagogical approach for STEM educationcould be a modified Scientific Inquiry-based Learning where students engage in aninvestigation and exploration process to postulate hypotheses or construct solutions forreal-life problems. Sanders (2009) developed a Bpurposeful design and inquiry^ forSTEM Education which includes Bproblem-based learning that purposefully situatesscientific inquiry and the application of mathematics in the context of technologicaldesigning/problem solving^ (p. 21). In another vein, using tools and artifacts inmathematics pedagogy has been researched (see for example, Leung & Bolite-Frant,2015). A tool-based pedagogy may bridge the classroom practices for the STEMdisciplines since all the four STEM disciplines could employ tools in shaping contentideas and concepts.

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http://www.scratchmaths.org

http://www.scratchmaths.org

Mathematical modeling, inquiry-based learning, and tool-based pedagogy could gohand in hand to provide a possible STEM pedagogy framework for mathematics andscience. The purpose of this discussion paper is to explore such pedagogy with respectto a sequence of tool-based mathematics school lessons conducted as a scienceexperiment on the mathematics topic of estimation. This is a discussion paper basedon classroom empirical evidences to initiate a research direction in STEM educationlooking for an overarching pedagogical approach that could bridge the pedagogicalpractices of the STEM disciplines. In the next section, theoretical backgrounds ofinquiry-based learning, modeling, and tool-based pedagogy are presented ending witha proposal of boundary pedagogy for STEM. An inquiry-based modeling cycle ispresented. It is followed by a description of an experiment mathematics lesson onestimation carried out in a science laboratory in a Hong Kong secondary school Form 1(Grade 7) class. The structure and design of the lesson are analyzed using the inquiry-based modeling cycle. A selection of students’ work for the experiment mathematicslesson is presented and discussed to illustrate students’ problem-solving strategies andmathematical and scientific thinking. Pedagogical features are then identified from theexperiment mathematics experiment lesson that may characterize a STEM boundarypedagogical approach. The paper concludes with a short discussion on the idea ofboundary pedagogy in STEM education.

Theoretical Background

An approach to integrate different domains of pedagogical practices is to discerncommonalities and differences among the domains. Commonalities serve to bridgeand establish communication; differences serve to maintain the integrity of the indi-vidual domain and provide multiple perspectives. In searching for a STEM boundarypedagogy, I start with considering the domains inquiry-based learning in scientificinquiry, mathematics modeling, and tool-based pedagogy. These three pedagogicaldomains share at least a common epistemological theme: making systematic inquiryin a problem situation. They may differ in the inquiry processes, but the differencesmay lie mainly on different representations of reality: physical reality, abstract reality,or tool-dependent reality. Furthermore, each of the three pedagogical approaches has itsown domain of methodological practice but they share a common major feature: theyare all about creating and implementing inquiry problem-solving processes. A multi-perspective on understanding the phenomenon under study would certainly enrich thesystematic enquiry process and the depth of knowledge acquisition.

Inquiry-Based Learning

In the Report on the Status of Science, Technology, Engineering, and MathematicsEducation in the United States (2006) by Pennsylvania State College of Education,under the section BDevelop New Instruction Methods,^ it was stated that BUsinginquiry requires the students to develop their own ideas and really think at a differentlevel than the way that they’re thinking in a traditional classroom^ (ibid., p. 8). Inquiry-based learning has always been the main constructivist pedagogy in science education.It is a problem-based teaching and learning approach. Research has confirmed that

Exploring STEM Pedagogy in the Mathematics Classroom: a Tool-Based... 1341

Bsustained use of an effective, research-based instructional model can help studentslearn fundamental concepts in science and other domains^ (Bybee et al., 2006, p. 1).The commonly used inquiry-based learning model is the BSCS 5E Instructional Modelwhich consists of five pedagogical phases: engagement, exploration, explanation,elaboration, and evaluation.

The European Fibonacci Project (2010–2013) aimed at a large dissemination ofinquiry-based science education and inquiry-based mathematics education (IBME)throughout the European Union. In the project, the construct of inquiry-based mathe-matics education was expounded and implemented.

Mathematical inquiry presents evident similarities with scientific inquiry … ..This inquiry process is led by, or leads to, hypothetical answers – often calledconjectures – that are subject to validation … This is rarely a linear process.(Artigue & Baptist, 2012, p. 4)

IBME’s domain of inquiry extends from natural phenomena and human artifacts (thesources of scientific inquiry-based learning) to mathematical objects in Bthe terrain formathematics experimentation^ (ibid, p. 5) under a diversity of contexts (e.g. mathe-matics embedded in digital technology environment).

Modeling

Mathematical modeling is usually about using mathematics to descriptively interpret,explain, and understand phenomena outside the mathematical world. Its process ofmaking systematic inquiry in a problem situation is to translate meanings between theextra-mathematical domain and the mathematical domain. Typically, mathematicalmodeling is represented by similar versions of the modeling cycle which consists ofthe key sub-processes: preparing the extra-mathematical domain, mathematizing theidealized situation and questions, dealing with the mathematical situation, de-mathematizing the mathematical outcomes, and validating the model (Niss, 2015).Extensive research has been done studying the pedagogical dimensions of modelingprocesses (see for example, Blum & Niss, 1991; Stillman, Brown & Galbraith, 2010).There is also a prescriptive purpose of modeling using mathematics Bto specify ordesign objects or structures that are meant to inhabit some extra-mathematical domainwhilst possessing (if possible) certain required or desired properties^ (ibid., p. 69). Thistype of prescriptive mathematically modeling (ibid.) follows the modeling cycle butdoes not have a direct validation process, rather it has a process of critique on the use ofthe model. Niss (2015) discussed examples of this type of modeling (e.g. body massindex, a paper format) and proposed Type I and Type II prescriptive modeling tasks.Type I focuses on invention of notions to describe or to measure something in the extra-mathematical domain (e.g. how to measure obesity) while Type II focuses on themathematical treatment of a prepared situation (e.g. how to design a paper format).Didactical, Type II appears to be more accessible for students in the mathematicsclassroom as students use mathematical knowledge to design and to construct(mathematical) objects.

Lesh and Doerr (2003) introduced the Models and Modeling Perspective(MMP) about teaching, learning, and problem-solving in mathematics and

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science education. This perspective is founded on problem-solving activities thatare referred to as model-eliciting activities which prompt students to produceBshareable, manipulatable, modifiable, reusable conceptual tools for constructing,describing, explaining, manipulating, predicting^ (ibid., p. 3). Students developmathematical ideas while working on meaningful, real-life situated problemssuch that the modeling process becomes the product of the problem-solvingactivity. Knowledge as interpretation, making connections, and stances/rolesadopted by the problem-solver during the problem-solving process are keyknowledge dimensions in MMP that drive the design of model-elicit activities(Brady, Lesh, & Sevis, 2015). In this connection, learning progression in MMPcould take on a topographic approach or an evolutionary approach. A topograph-ical view of learning metaphorically views learning as different routes andpassages through a disciplinary domain during a modeling task, students’ explo-ration processes form different learning trajectories when simultaneously puttogether, generating a knowledge map representing a collective multi-perspective modeling. On the other hand, an evolutionary approach Bhighlightsthe importance of the diversity of ideas within problem-solving groups … andaccumulation of changes in ideas over iterative cycles^ (ibid., p. 62). Therefore,MMP pedagogy embraces a dynamic and non-linear development of knowledgeacquisition where students co-construct problem solutions. The dynamism restson self- and co-evaluation of students (group or individual) Bmodel drafts,^resulting on selection of most suitable solutions to the posted problem. MMPwill play an important role in the subsequent discussion.

Tool-Based Pedagogy

Designing suitable pedagogical tasks is a key to implement inquiry-based learningand the use of tools is a major epistemic component in carrying out the systematicinquiry-based process. Instruments and tools are inseparable from scientific inqui-ry, even to the extent that sometimes the observation and even interpretation ofreality is tool-dependent (e.g. the quantum world). Thus, tools play an importantrole in knowledge acquisition. Recent research on inquiry-based experimentalapproach for mathematics pedagogy forefronts the use of tool, in particular digitaltechnology, as an epistemic medium (see for example, Leung & Baccaglini-Frank,2017). A tool-based mathematics task is one that and interactive tool-basedenvironment is activated where teacher, students, and resources mutually enricheach other in producing mathematical experiences. Leung and Bolite-Frant (2015)discussed in-depth tool-based mathematics task design under the following con-siderations: (i) use strategic feedback from a tool-based environment to createlearning opportunities for student, (ii) design activities to mediate between thephenomena created by a tool and the intended mathematical concept to be learned,(iii) make use of the affordances and constraints of a tool to design learningopportunities, (iv) switch between different mathematical representations or tools.These tool-based task design features could structure the pedagogical design of theinquiry-based learning process focusing on mediation between concrete empiricalexperience and conceptual understanding, a major boundary crossing in mathe-matics knowledge acquisition (Leung & Baccaglini-Frank, 2017).


Boundary Pedagogy

Inquiry-based learning and mathematical modeling are pedagogies in the science andmathematics knowledge domains respectively. On the boundary between these twodomains lie compatible epistemological approaches focusing on systematic and logicalinquiry activities. Hence, it may be possible to Btranslate or transfer^ pedagogiesbetween science and mathematics via a common boundary object. Such a boundaryobject, with respect to the definition coined by Star and Griesemer (1989), should havea plasticity to adapt to the individual domains and be robust enough to maintain thecoherence of a common identity (in this case, an epistemic process); furthermore, it hasa structure that is flexible when the two domains interact and is rigid when one domainis dominant. Juardak (2016) proposed different boundary objects for boundary crossingbetween problem-solving in school and problem-solving in the real world in whichmodeling, narrative discourse, and STEM mathematical practices are among them. Inthis connection, I search for a boundary pedagogy that can cross the boundaries of theSTEM disciplines and serves as a boundary object to connect the classroom practices ofthe disciplines. Since STEM lessons most often situate in tool-rich pedagogical envi-ronment for problem-solving, a possible integrative approach would be to develop aninquiry modeling tool-based STEM pedagogy.

For this paper, I explore a hybrid (boundary) pedagogy that connects the STEMdisciplines mathematics and science. In this connection, Carreira and Baioa (2018)carried out an experimental work on a mathematics modeling task setting where hands-on experimental approach was integrated under a STEM context. I concoct a tentativeinquiry-based modeling pedagogical cycle to bridge between scientific inquiry andmathematical modeling (Fig. 1). This concoction is inspired by the modeling cyclepresented in Artigue and Baptist’s (2012) discussion on inquiry-based mathematicseducation. Specifically, I add in the BSCS 5E Instructional Model in IBME. Thepedagogical cycle combines the essential elements of inquiry-based learning (theshaded parts in Fig. 1) and mathematical modeling (boxes with solid frame). It startswith an authentic problem and goes through a fine-grained problem-solving process.The boxes with dashed frame represent intermediate or transitory activities. Tool usage

Fig. 1 An inquiry-based modeling pedagogical cycle in which elements of mathematical modeling (boxeswith solid frame) and inquiry-based learning (shaded) are integrated

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permeates the cycle whenever it is pedagogically conducive and significant. This is apedagogy that intends to cross between scientific investigation and mathematicmodeling.

This proposed pedagogical cycle is a hybrid boundary object attempting to mergeinquiry-based learning and mathematical modeling with an intentional ambiguitybetween them which allows the opening of discourse space and hence translationbetween the two pedagogies. It aims to facilitate the designing of integrated STEM-based experiences (English, 2017). With respect to mathematics and engineering,English proposed STEM-based modeling is Ba cyclic, generative learning activity …where learning of content and/or processes is elicited by the student, rather thanprovided^ (ibid., p. 15). How the inquiry-based mathematical modeling cycle, withrespect to mathematics and science, is in-line with STEM-based modeling via using thecycle to measure by matching the pedagogy flow of a STEM lesson? In the followingsections, this question will be explored under an authentic secondary school STEMlesson sequence.

A Tool-Based Experiment Mathematics Lesson on Estimation

In this section, a lower secondary (Grade 7) mathematics lesson sequence conducted ina science laboratory is presented. The background of the lesson is the research projectAn Explorative Study on Designing Tool-based Task in the Teaching of School Math-ematics conducted by the author. The project’s main aim is to investigate how schoolmathematics teachers design teaching and learning tasks integrating different toolenvironments in the teaching of school mathematics, hence to explore possible tool-based pedagogical models in the mathematics classroom.

A group of 15 mathematics school teachers and mathematics education researchersformed a community of practice to discuss, implement, and evaluate the designs ofdifferent tool-based mathematics lessons regularly. Research data are collected in theform of classroom observation field notes, teacher and student clinical interviews, videorecord of all lessons, student-produced materials, and audio record of the community ofpractice meetings. At the beginning of the project, one of the teacher-researchers in thegroup shared a mathematics lesson based on scientific experimentation conducted inher school. After discussion and deliberation, it was decided that a re-run of this lessonwould be considered as a research lesson for the project. The inquiry-based mathemat-ical modeling cycle schematics presented in Fig. 1 did not exist at the time when thisresearch lesson was conducted; hence, neither the researchers nor the teachers had theFig. 1 integrative schematics in their minds during the design and the implementationof the research lesson. However, the teachers in that school were familiar with theBSCS 5E Inquiry-based Learning Instructional model and Pólya’s four-step problem-solving strategy (Pólya, 1945).

Description of the Lesson Sequence

This mathematics lesson was a sequence of five lessons for Hong Kong Secondary One(Grade 7) classroom. It was taught by four teachers in their own classes. Each class has20 to 24 students, and the students were divided into 5 to 6 groups with 4 to 5 students


in each group. Hence, a total of 20 lessons were recorded on video. The objective of thelessons was asking students to estimate how many marbles there are in a large conicalflask and judge the reasonableness of the estimation results (Fig. 2a).

The lesson sequence consists of five lessons and they are carried out in theclassroom and in the school’s science laboratory equipped with standard scientificmeasuring instruments (see examples in Fig. 2b). Students were asked to make theestimation three times progressively, each with more tool-based support, resulting insupposedly more accurate estimation than the previous one. In the last lessons, studentspresented their exploration using an iPad and the teacher discussed the errors instudents’ estimation strategies. The lesson sequence, conducted in consecutive schooldays, was a new and ambitious mathematics teaching project initiated by the schoolmathematics team to explore STEM education.

Teachers designed the teaching tasks based on Inquiry-based learning and Pólya’sfour-step problem-solving strategy: understand the problem, devise a plan, carry out theplan, and look back (Pólya, 1945). Each student group is given a booklet containing asequence of structured tool-based tasks where students record the whole experimentprocess. The following are brief descriptions of the consecutive lessons intended by theteachers.

Lesson 1: Introduce the estimation activity in the classroom Students observeand guess a range for the number of marbles. They are not allowed to touch theflask. By guessing an answer, students are motivated to understand the problemand suggest necessary mathematical knowledge needed to solve the problem.Students are asked to design a plan to solve the problem. They are encouragedto take photos using an iPad, count the marbles that are visible on the surface of theflask, search for useable formulas in the Internet, and try to apply them. At the endof the lesson, students are asked to suggest first estimated values for the number ofmarbles.Lesson 2: Go to the science laboratory to perform the first estimation exper-iment Scientific measuring instruments are provided for students to performestimation experiments (see Fig. 2b). Small flasks containing marbles are givento the student groups to simulate the real situation. Students discuss how to use thegiven instruments and to design tool-based methods to do the estimation. Thenthey carry out the designed plan and suggest second estimated values for thenumber of marbles. Students are asked to evaluate their tool-based methods afterclass and bring in their own tools and materials if necessary to perform a modifiedestimation experiment in the next class.

Fig. 2 a Estimate how many marbles there are in a large conical flask b Standard scientific measuringequipment used by the student in the science laboratory

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Lesson 3: Go to the science laboratory to perform the second estimationexperiment This is a continuation of Lesson 2. Students modify their previousmethod (they can bring in their own tools or materials) and perform the estimationactivities to arrive at third estimated values (if any).Lessons 4 and Lesson 5: Student groups look back at their solutions usingthe presentation software Keynote (before class) on their iPads to explainhow to get their estimated values. Each group goes around to other groupsto present to others and to solicit comments. Teacher observes and dis-cusses with students their solution models and possible errors in theestimation processes.

Analysis of the Lesson Design

The teachers designed the lessons based on the BSCS 5E Inquiry-based LearningInstructional model (engagement, exploration, explanation, elaboration, and eval-uation) and Pólya’s four-step problem-solving strategy as an attempt to integratethe scientific method into mathematics instruction. The following is my ownanalysis on how the lesson design can be seen through the lens of the inquiry-based mathematical modeling cycle depicted in Fig. 1 (the italics below refer tothe elements in the cycle), and specifically, on seeing the lesson sequence as acyclic, generative learning activity where learning is elicited by the student, ratherthan provided.

Lesson 1: The marble estimation problem is motivated by a teacher who saw asimilar type of display in a shopping mall asking the customers to guess howmany objects was there in certain container. The marble problem is posedwith limited information, requires students ask many questions, demandscommunication, utilizes estimation, and emphasizes process rather than Bthe^answer. In this way, the marble estimation problem can be regarded as a kindof model-eliciting activity where the pedagogical topographic or evolutionaryapproach could be realized (Brady et al., 2015, pp. 61–62). Students aredivided into groups and teacher monitors the group discussion progress.Teacher provides objective guidance and critical comment to students, thusgenerates a Domain of Inquire via teacher-students discourse. Students engagein the discourse that involves understanding the problem, discuss possiblemathematical ideas or techniques needed to explore the problem (an initialmathematization), search (using available technology) for information ormethods, and apply Apps to solve the problem. At the end of the lesson,students suggest initial models and give a first attempt to explain how to solvethe problem (making initial conjectures).Lesson 2: The class is conducted in a tool-rich (concrete and virtual) class-room (the science laboratory) where students can perform scientific experi-ments to solve an estimation (mathematical) problem. The discussion inLesson 1 continues and is Bre-cycled^ to a different level of inquiry-based,modeling activity with tool-based support. Teacher provides objective


guidance and critical comment to students generating tool-based teacher-students discourse. Students discuss how to use the given tools to inventand design mathematical models for a tool-based mathematical solution to theproblem (tool-based mathematical analysis). Students solve the problem usingtheir new models and compare/contrast the solutions of the first and thesecond models (model results, explore, explain, elaborate, and evaluate).Lesson 3: Students’ estimation experiences are deepened in this lesson as morefocus is on elaboration and validation of students’ tool-based models. Morediscussion is on interpretation and mathematical analysis and on how to presentclear explanations.Lesson 4 and Lesson 5: This is an important conclusive session of the whole lessonsequence where the teacher and student themselves play critical guidance roles tohelp students to elaborate, evaluate, and validate their solution models. Here,mathematical discussion should be at the center of the classroom discourse.

The lesson sequence is about solving an estimation problem. Students conductBscientific^ experiments to solve the problem using different types of tools(concrete and virtual). Students are free to do whatever they like and work ingroups to do collaborative investigations to produce negotiated solutions andpresentations. Teacher guides and gives critical comments to students but nevertells students how to do the experiment as there is no fixed ways of doing it.Students need to design their own strategies. Knowledge needed to investigate theproblems may lie beyond students’ prior knowledge and students are encouragedto surf the Internet to find what are suitable for them to use. Students areencouraged to make hypotheses and to build mathematical models to solve theproblem. Furthermore, student groups compete to arrive at the best solutionmethod; that is, the best way to guess the number of marbles. Teachers do notexpect students to get the correct answer and different student groups are expectedto produce different solution strategies resulting in different answers. Thesecharacteristics capture the pedagogical spirit of an inquiry-based modelingapproach.

Selected Student Work from the Implemented Lesson

The intended lesson design resonances a minimalist instruction motif within aself-determined learning environment where concepts and procedures are mainlyconstructed by students themselves using (prior) known knowledge. Teachers playa minimal role, as an inquisitor or an information provider in this case, in directingstudents’ knowledge construction process (Carroll, 1998; Haapasalo, 2007). Con-sequently, the implemented lesson sequence may diverge from the intended lessonsequence. Because the intended lesson design, though seems to be rigidly struc-tured, allows flexible student self-directed learning elements, the implementedlessons still capture the flow of the inquiry-based modeling cycle.

To illustrate the kind of investigation, experimentation, and reasoning that studentswent through, three representative student groups’ worksheets selected from twoclasses, taught by two teachers (A and B) will be presented. They are experienced

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mathematics teachers, one of them who is also a science teacher. The purpose of thisillustration is not to generalize but to show evidences on how under a minimalistinstruction environment seen through the inquiry-based modeling cycle, studentsconstruct diverse problem-solving paths using different strategies and tools. Theselected student work was illustrative in these pedagogical aspects.

After Lesson 1 when students finished proposing their designed plan, teacherscollected the group worksheets and made inquiry comments on them. Theworksheets were given back to the groups when students began their investigationin the science laboratory in Lesson 2. Observation notes and the actual content ofthese worksheets are presented below to highlight the inquiry modeling charac-teristics of students’ work.

Group Case 1 (Teacher A)

Figure 3 Part 1. In Step 1, students decided to find out the volume of 1 marble usingwater displacement method. They used 4 marbles in their plan and the teachercommented why. In Step 2, students showed their idea to find the volume of 1 marbleby using the calculation: 1 marble = volume of 4 marbles ÷ 4. Students were given theinformation (by teacher A) that the large flask containing the marbles had volume1000 ml. Therefore, in Step 3, students proposed that the calculation (1000 ml ÷ thevolume of 1 marble) should give a good guess. The teacher made a query on whetherall marbles have the same volume.

Figure 3 Part 2. Students followed the Steps outlined in Part 1 but instead of using4 marbles, 28 marbles were used. This may indicate that when students actuallycarried out the experiment, the water displacement for 4 marbles was too little fora reasonably accurate measurement, the larger the water displacement, the better.Also, 28 marbles fitted fully into the small flask. It is interesting to notice that tofind the number of marbles in the large flask, students did not use (1000 ml ÷ thevolume of 1 marble) as proposed in Part 1. Instead, they reasoned that since thesmall flask can only contain 28 marbles and the water volume displaced by the 28marbles in 70 ml; therefore, the large flask contained B700 ml marbles^. Theexpression B700 ml marbles^ is interesting even though ambiguous. It could havebeen resulted from the measuring tools that were used in the experiment whichfacilitated the calculation for the final guess. Apparently, 700 ml came from70 ml × 10, which may have been deduced from 1000 ml (volume of the largeflask) = 10 ml (volume of the small flask) × 10, a kind of proportional reasoning.

Figure 3 Part 3. The estimation of 280 marbles was 9 marbles more than the actualnumber 271 which is quite a good estimate. Students evaluated the estimationprocedure and came up with two possible factors for error in measurement. Theyrealized that there is air gap between the marbles, but they did not explain why theair gap accounted for more marbles being estimated. Another factor was the actualdisplaced water might be less than the expected displaced water, probably due towater was spilled or attached to the surface of the measuring tools. Students thensuggested a few real-life situations that they could apply what they learnt in thisestimation experiment.


Group Case 2 (Teacher A)

Figure 4 Part 1. In Step 1, students proposed to use a small flask (100 ml) as a scale-downmodel for the large flask and suggested that the volume of the 100-ml flask should equal tothe volume of themarbles that could be filled-in.Water was used as themean tomeasure thevolume of the 100-ml flask. It seems that the meaning of the 100-ml unit was not clear tostudents, but it is used as a key idea in the investigation just like Group Case 1. The teacherwas not sure what was the purpose of this and queried about it. In Step 2, students tried tomeasure the volume of 28marbles using water displacement method and hence deduced thevolume of 1 marble by division. The teacher was not sure how Step 2 is related to Step 1. InStep 3, students wrote down a procedure that seemed unclear and ambiguous: (a) Use thevolume of the remaining water of 100 ml flask and times 10; (b) 1000 ml flask isapproximately equal to 10 times of 100 ml flask. Therefore, we will know the total volumeof the 1000 ml flask; (c) The volume of the 1000 ml water is approximately equal to thevolume of all marbles in 1000ml flask; (d) The volume of all marbles in 1000ml flask ÷ Thevolume of 1marble equal to howmanymarbles are there in 1000-ml flask. LikeGroupCase1, proportional reasoning is at play here.

Figure 4 Part 2. Students did not carry out the plan in Part 1! Instead a different tool, anelectronic balance, was used. Students found it easier to use weight rather than volume toconceptualize the calculation. Only three weight measurements (the 1000-ml flask full of

Part 1

Part 2 Part 3

Fig. 3 Group Case 1: Part 1, design a plan; Part 2, carry out the plan; Part 3, evaluate the estimation process

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marbles, an empty 1000-ml flask, and a singlemarble)were needed to perform the estimationas presented in Step 1 and Step 2, which was interestingly stated as ≈ 283. Step 3 was arepetition of Step 1 and Step 2 answering a request in the worksheet for description andcalculationof theresult.This isanexampleofanabruptchangeofstrategyin the inquiry-basedmodelingprocess.Thismayduetothefact that theteacherprovidedanabundanceof resourcesfor students enabling students tomake ad hoc alternative decision on the choice of tool.

Figure 4 Part 3. As in Group Case 1, students realized there is air gap which wouldaffect the accuracy of the estimation and suggested measuring how many marbles canbe put into the air gap. But no method was suggested on how to do this.

Group Case 3 (Teacher B)

Figure 5. In Step 1, students weighted (using an electronic balance) a single marble, anempty small flask, and the big flask containing the unknown number of marbles. Teacher Bdid not allow students to have access to an empty big flask and did not suggest waterdisplacement tools for students to use. In Step 2, students planned to find out the weight ofthe empty big flask in an interesting way. They considered a Bcross section^ as a small flaskbeing cut vertically in half which has half of the weight of the small flask, and assumed thatthe large flask can be externally Bcovered^ by these cross sections. A calculation method

Part 1 Part 2

Part 3

Fig. 4 Group Case 2: Part 1, design a plan; Part 2, carry out the plan; Part 3, evaluate the estimation process


was then proposed: (small flaskweight)/2 × (total no. of cross section covering the big flask).In Step 3, this calculation is considered as the weight of the empty big flask using 15 crosssections as the covering, and by this theweight of the total number ofmarbles in the big flaskis deduced.Dividing this by theweight of a singlemarble computed in Step 1, the number ofmarbles in the big flask can be obtained. This is an ingeniousmathematical idea even thoughit has a lot of fault in this situation. Covering is a topological technique that is often used inadvanced mathematics. In contrast to Teacher A, Teacher B provided students with lessresource, consequently forcing students to be creative.

Figure 6. Students carried out exactly what they planned.

Figure 7. The estimate value 220 was quite far away from the actual value. Studentsreasoned that the (wall) thickness of the small flask and big flask may be different and

Fig. 5 Group Case 3: design a plan

Fig. 6 Group Case 3: carry out the plan

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this could cause error in estimation using their covering method (which they called thebenchmark method). They also reckoned the air gap problem and proposed a differentBtessellation^ for the covering to reduce the air gap. Tessellation is a mathematical idea.

Pedagogical Features of Boundary Pedagogy

Near the beginning of the paper, I asked the question how the inquiry-based mathematicalmodeling cycle with respect to mathematics and science is in-line with STEM-basedmodeling? The tool-based experiment mathematics lesson on estimation discussed aboveshows that Bcyclic, generative learning activity^ designedby the teachers stimulated Blearningof content and/or processes is elicited by the students.^ The student work presented abovecaptures the kinds of tool-based experimentation and mathematical reasoning of the imple-mented lessons. Major mathematical ideas used were proportional reasoning and tessellation.Mixture of scientific inquiry method and modeling was evident though not sophisticated.Interpret from this plus teacher post-lesson interviews and classroomobservation, pedagogicalfeatures of how students elicited learning of content, and process are suggested. These featuresare not generalizations, rather they are aspects inspired by the lesson that could contribute tothe development of STEM boundary pedagogy, the main purpose of this paper.

Contingency

The teachers indicated as they walked around observing and talking to the students duringthe lessons, they had to spontaneously respond to students’ tool-based strategies whichwere changing and developing, for example in Group Case 2, the intended plan and theimplemented plan were different. Hence, contingency is a dominant pedagogical mode indeveloping teacher-students discourse. Teacher needs to adjust teaching approach, evenchange the intended lesson plan, tomake an instant decision to facilitate students’ evolvinginquiry modeling process. However, in doing so, the teachers indicated that their expla-nation and student discussion were enhanced and deepened based on the developingteacher-student discourse. Furthermore, students’ tool usages are developing in diversepaths contingent to the teacher’s reaction and the availability of different resources.

Fig. 7 Group Case 3: evaluate the estimation process


Uncertainty and Ambiguity

Uncertainty and ambiguity should be embraced rather than avoided. In the inquirymodeling process, students’ uncertainty about how to proceed becomes a driving forcefor them to seek different possible (mathematical) strategies. Uncertainty is an essence ofrealistic problems. It motivates a need to make assumptions which is critical in the inquirymodeling process. Furthermore, uncertainty usually leads to ambiguity which sometimeinspires new concepts. During the inquiry process, students may create ambiguousexpressions to relate different ideas which might have been treated as unrelated before,for example, the expression Bvolume of all marbles^ was used repeatedly in the GroupCases. Ambiguous expressions create significant inquiry modeling discourse to develop aBsolution space^ for the problem (a metaphorical dimension where possible solutionsreside and relate to each other). The loosely used ideas of Bcovering^ and Btessellation^suggested in Group Case 3, even though not a viable strategy, illustrated how beautifulmathematical ideas can be inspired under uncertainty and ambiguity. Spatial tessellation isan advanced mathematical concept that is way beyond the students’ and the teachers’knowledge, but under this problem-solving environment, students were able to suggestsomething very mathematical, even though it was rough and imprecise.

Making Mistake

Mistake and failure are characteristic marks of the inquiry modeling process. Mistakecan be thought of as coarse but good (mathematical) model that needs further refine-ment. Learning from mistake and failure is a natural knowledge growing path in anykind of explorative activity. Error estimation is about how to Bmeasure mistake^ in alogical way. Students in Group Case 3 made a fantastic mistake.

Tool-Based Reasoning Discourse

Students think, reason, and talk while using tools during the inquiry modeling process andthus develop tool-based discourse. Whether such discourse restricts or expands students’mathematical thinking depends on how teacher enters the discourse and negotiatesmathematical meanings with the students. A tool may lead students to discover somethingthat may be different fromwhat is expected to learn, giving rise to unexpected conflict anduncertainty. Unexpected reality generated by a tool could stand against conventionaltextbook knowledge. Sometimes, the group who calculated a good estimation did notconduct a logical experiment, but the group who carried out the experiment logically andcreatively did not get a good answer. The teachers were surprised to hear from studentsthat it is necessary to do error reduction after students have had good experience in usingthe measuring tools. Error reduction is not part of the mathematics curriculum.

Tool-Based Task Design

It was observed and discussed by the teacher on whether tools should be given tostudents before or after they think about how to solve the problem, and how and whattools should be introduced to the students. This task design decision affects the kinds ofsolution students produce and the tool-based reasoning developed. Teacher A and

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Teacher B made different decisions on what tools were given to the students andresulted in different student mathematical models. It is interesting to note that the morerestricted the tool resource, the more likely students would resolve to mathematicalreasoning rather than relying solely on tool usage techniques.

Applicability and Extension

Question was raised by teachers on whether tool-based strategies/solutions are appli-cable or extendable in other daily life situations. This is a question about what havestudents actually learnt in the problem-solving process. Teacher A suggested a follow-up lesson. It is a reverse lesson in which students now are given a fixed number ofmarbles and the task is to design and construct, using 3D printing, an optimized box forthe marbles. Students will have to design and draw a blueprint for the box, learn thesoftware for 3D printing, and 3D print the box. This will be a mathematics, engineer-ing, and technology STEM lesson. The lesson was actually carried out later.

Refinement and Modification

Students’ strategies and plans were changing during the inquiry modeling process dueto the availability of resources. Sometime students changed strategy in the middle ofcarry out a plan without completing the original plan, for example, Group Case 2.Continuous, even abrupt, refinement and modification are common learning behaviorin the inquiry modeling process.

The above features serve as starting criteria to conceptualize the formation of STEMboundary pedagogy which act as a common discourse to bridge the pedagogicalpractices in the STEM disciplines.

Conclusions

Dillion (2008) developed generic ideas of pedagogy of connection and boundarycrossings between disciplines which take account of interventions, the use of tools,and the notion of changes in learning behavior. Furthermore, Akkerman and Bakker(2011), based on an understanding of boundaries and boundary crossing as dialogicalphenomena, expounded four potential learning mechanisms that can take place atboundaries: identification, coordination, reflection, and transformation. These genericconcepts on boundary crossing could be further explored in the context of STEMeducation. In this paper, I explore and search for boundary pedagogy that acts ascommunicator between the epistemological and pedagogical approaches in the math-ematics and science classrooms. An inquiry modeling cycle is proposed as an initiatelayer on which to build a boundary object connecting the two knowledge acquisitiondomains. The inquiry modeling cycle serves to design an intended mathematics-sciencelesson sequence. Nevertheless, the implementation and enactment of the lesson de-pends a lot on students’ Bhabit of mind^ in learning which may not be exactly in stepswith the intended inquiry modeling cycle, especially if students are given freedom to dowhatever they want in a problem-solving context. Students’ modeling trajectories maydiffer from teacher’s intention especially when refinement, recycling of ideas and


modification are involved during students’ exploration. This is particularly relevant in atool-rich pedagogical environment. Tool usage schemes can be personal (or collectivelynegotiated) and how these schemes facilitate knowledge acquisition may be contingentto students’ experiences. Nevertheless, the inquiry modeling cycle can serve as apedagogical frame on which students build their own situated mathematical knowledgewith teacher guiding them to Bre-invent^ mathematical knowledge.

Certain pedagogical features are identified based on a mathematics lesson that wasconducted as a scientific experiment. These features may inform us on how toconceptualize the structure of boundary pedagogy. They are mainly about teachingand learning in a fluid and flexible mode (minimal instruction), allowing students (andteacher) to try different things, to be uncertain, to make mistake, and to questionthemselves. These are characteristics of the state of mind of someone who is at theboundary between two domains. Boundary pedagogy should include these features.

The spirit of STEM education is problem-solving, a process that involves identifyingproblems and constraints, research, design, create ideas, experiment, explore, analyzeideas, build, express, test, and refine. These are common for the four STEM disciplines’pedagogical approaches: mathematical modeling, inquiry-based learning, computation-al algorithmic thinking, and engineering design. The proposal to develop boundarypedagogy crossing the four disciplines by capturing the disciplines common Bhabit ofmind^ opens a new direction to interpret what STEM education is. Exposure to risk andfailure in the classroom is essential for mathematics, science, engineering, and tech-nology education as it reflects real-world STEM career practices. Some core attitudinalvalues for STEM Education (or STEM Literacy) are about searching for uncertainty,recognizing ambiguity, and learning from failure. A paradigm shift is needed to makeroom and space for failure, to create a culture of allowing error, and letting students tocelebrate making fantastic mistakes.

Funding Information This research was supported by a grant from the General Research Fund (Projectnumber GRF12404614), University Grant Committee, Hong Kong SAR and in collaboration with themathematics teachers of True Light Middle School of Hong Kong

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