Vol.:(0123456789)1 3

ZDM Mathematics Education (2017) 49:785–798 DOI 10.1007/s11858-017-0862-8

ORIGINAL ARTICLE

Learning extrema problems using a non-differential approach in a digital dynamic environment: the case of high-track yet low-achievers

Assaf Dvir1  · Michal Tabach1 

Accepted: 12 April 2017 / Published online: 30 May 2017 © FIZ Karlsruhe 2017

Keywords Knowledge construction · Dynamic e-resource · Non-differential math · Creative reasoning · Abstraction thinking

1 Introduction

The field of mathematics education has identified sev-eral topics in the K-12 curriculum that pose serious chal-lenges to students (e.g., Yerushalmy 2009). One such topic in high school calculus is the solution of extremum prob-lems (Metaxas 2007; Kouropatov and Dreyfus 2014). High schools commonly adopt the classic differential approach to minima and maxima geometric problems. Although cal-culus serves as a systematic and powerful technique, this rigorous instrument might also hinder students’ ability to understand the behavior and constraints of the objec-tive function. This hurdle might be easier for students if they were to use a non-calculus of variations approach in a dynamic geometry environment. The current research seeks to examine this hypothesis, specifically among lower achieving students. In other words, we examine whether we can harness a purposefully designed e-resource to help stu-dents better understand extrema problems. This goal could not be achieved by relying only on the standard curricu-lum. To assist learners in making meaning of the subject content, the teacher had to do what Adler (2000) refers to as re-conceptualizing current sources. In our case, this re-sourcing is accomplished by generating new e-resources using spreadsheets and a dynamic geometry environment.

The study design reflects our dual interest in student learning processes and student understanding. To explore the first point of interest, we adopted the abstraction in context (AiC) framework for tracing students’ knowledge construction process as they worked in pairs on a set of

Abstract High schools commonly use a differential approach to teach minima and maxima geometric prob-lems. Although calculus serves as a systematic and pow-erful technique, this rigorous instrument might hinder stu-dents’ ability to understand the behavior and constraints of the objective function. The proliferation of digital environ-ments allowed us to adopt a different approach involving geometry analysis combined with the use of the inequal-ity of arithmetic and geometric means. The advantages of this approach are enhanced when it is integrated with dynamic e-resources tailored by the instructor. The current study adopts the abstraction in context framework to trace students’ knowledge construction processes while solv-ing extremum problems in an e-resource GeoGebra-based environment using a non-differential approach. We closely monitored the learning of 5 pairs of high-track yet low achieving 17-year-old students for several lessons. We fur-ther assessed the students’ understanding at the end of the learning unit based on their explanations of extrema prob-lems. Our findings allowed us to pinpoint the contributions (and pitfalls) of the e-resources for student learning at the micro level. In addition, the students demonstrated the abil-ity to solve extrema problems and were able to explain their reasoning in ways that reflect the e-resources with which they worked.

* Assaf Dvir assafdvir@mail.tau.ac.il; assafdvir@netvision.net.il

Michal Tabach Tabachm@post.tau.ac.il

1 Tel Aviv University, Tel Aviv, Israel

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problems in a customized dynamic geometry environment. We decided to work with the lower third of an 11th grade class of high-track students. The research questions are as follows: (1) what affordances and constraints can be iden-tified in processes of abstracting mathematical knowledge within the designed environment? (2) Can we point to traces of the learning processes in the students’ reasoning while they solve extrema problems?

2 Background

2.1 Solution methods for extrema geometric problems

Extrema problems are a mandatory part of high school calculus at almost all levels of mathematics (e.g., Israel Ministry of Education, High School Math Curriculum for 5 units–topics, p. 11). Extrema problems that call for find-ing the optimum point/s of a function enable students to link calculus and real life problems. Finding the optimum is a generic problem structure in many real life domains, among them science, economics, and technology. Prior to the symbolic calculation stage, a relatively complex extrema problem requires in-depth problem analysis and understanding, selecting the suitable variables, translating constraints and variable relations into a set of equations and generating a target function. After symbolic representation of the problem, the next stage is fairly straightforward and involves applying a well-defined and well-known techni-cal procedure of finding the absolute values of a function in a limited domain interval. In high school mathematics studies, this stage is directed at one common and classi-cal procedure based on the Lagrange principle: finding the function derivative, identifying the critical points using the resolution of a derivative-based equation and evaluating the target function over a set of candidate points—the critical and interval endpoints. As stated in the Israeli mathematics curriculum, “The first and second derivatives are required for solving extrema problems in open and closed intervals” (Israel Ministry of Education, High School Mathematics Curriculum for 5 units–topics, p. 11).

This typical learning sequence disregards a set of alter-nate solution processes with the potential to enrich and expand understanding of the problem domain and the func-tion behavior. Alternative methods that use geometric, symbolic and graphic analysis can extend students’ under-standing. The pitfalls of the classic procedure for solving extrema problems and the implications for students’ knowl-edge when they later proceed to advanced mathematics studies are described by Birnbaum (1982):

Ask a group of sixth formers how to find the maxi-

mum value of (

n

r

)

, for fixed n, and realising that cal-

culus is inappropriate, they are likely to look at you blankly. Mine did, and I began to wonder why, when calculus fails, intelligent pupils do not know how to proceed. The problem, I think, is that prior to calculus algebraic approaches to finding extrema are rarely considered, and so, when calculus is later applied to find maxima and minima, it seems to the pupil as if these concepts are inextricably bound up with differ-entiation. It is almost as if the words maximum and minimum produce differentiation as a stimulus response! (p. 8).

Among the alternative methods that do not belong to the calculus of variations are geometric methods, a general-ized arithmetic mean and geometric mean inequality, graph theory methods and general inequalities methods (Cauchy, Jensen). In the following sections, we describe the first 2 methods in detail as used in the current study with 11th grade students.

Mathematical optimization problems require finding the minimum or maximum of something (e.g., the area of a rectangle, the length of a fence, the cost of a product) based on a given set of constraints. Traditionally, as taught in high school and later on in tertiary education, these types of problems are solved using calculus. Tikhomirov (1991, pp.  109–117) provides a detailed theoretical background of the Lagrange principle for finding extrema of functions with single or multiple variables.

Solving extrema problems without calculus-based meth-ods dates back to ancient Greece. The Greeks described a few classical problems that could be resolved using theo-rems from Euclidean geometry. One of the most famous is Dido’s problem. As described by Tikhomirov (1991), Prin-cess “Dido negotiated the sale of land with the local leader, Yarb. She asked for very little—as much as could be ‘encir-cled with a bull’s hide.’ Dido managed to persuade Yarb, and a deal was struck. Dido then cut a bull’s hide into nar-row strips, tied them together and enclosed a large tract of land. On this land she built a fortress and, near it, the city of Carthage. There she was fated to experience unrequited love and a martyr’s death” (p. 9). Many historians think this was the first extremal problem discussed in scientific litera-ture. The problem may be specified mathematically as fol-lows: “Among all planar shapes with the same perimeter, which one encloses the larger area?” This well-known clas-sical problem is called the isoperimetric problem. It can be extended to three dimensions, and the circle and the sphere are the solutions for two and three dimensions respectively.

Even elementary school math includes optimum prob-lems in geometry, although they are not termed as such nor are they studied by means of rigorous techniques. A famous simple problem is to find the rectangle with the maximal area among all rectangles with a given perimeter.

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In elementary school, students are expected to use empiri-cal analysis to find the solution—a square. In high school, the problem is posed again as one of the first extrema prob-lems. It can be solved using pure geometrical means with-out the need for differential analysis.

The second non-differential method for solving extrema problems is based on the arithmetic-mean (AM) and geo-metric-mean (GM) Inequality. Let a and b be two nonnega-tive numbers. Their arithmetic mean and geometric mean are a+b

2and

√

ab, respectively. According to the AM-GM Inequality,

√

ab ≤a+b

2 for any two nonnegative numbers.

Equality is achieved when a = b. This implies (1) that the geometric mean does not exceed the arithmetic mean and (2) that the inequality is exact whenever a and b are differ-ent numbers and equality is actually attained if and only if both numbers are the same.

The general case of the AM-GM Inequality encom-passes the Harmonic Mean (HM) as well, as follows:

For any nonnegative numbers, x1, x2, ..., xn, the following inequality is always true:

The proof can be found in Niven (1981, pp. 20–23).Based on the above theorem, one can solve and under-

stand the solution of the extrema problem for the area of rectangles. The theorem explains how and why the square, a regular quadrilateral, becomes the optimal solution. The explanation is not just the solution of an equation. It is the result of the relations between the geometric mean and the arithmetic mean.

A few researchers suggested that using the AM inequal-ity for extremum problems might be even simpler than the classical methods. According to Klamkin (1992), “If one surveys the maximum and minimum problems in our cal-culus texts, one will find that a great many of them can be done much more efficiently just using the A.M.-G.M. ine-quality in two and three variables” (p. 113). Klamkin pro-vided a detailed description of three geometric problems of this type in the two-dimensional space that can even be extended to n-dimensional space using the same inequality methods. Additional geometric examples were provided by Freedman (2012).

2.1.1 Students difficulties with calculus

Calculus in general and extrema problems in particular pose serious challenges to high school students. Previous

min(x1, x2, ..., xn) ≤n

∑n

i=1

1

xi

≤n

�

�

�

�

n�

i=1

xi≤

∑n

i=1xi

n

≤

�

∑n

i=1xi2

n≤ max(x1, x2, ..., xn)

studies (Metaxas 2007; Kouropatov and Dreyfus 2014) describe the “cognitive load” associated with the number of elements that students must consider simultaneously during calculus studies. Swidan and Yerushalmy (2014) reported conceptual calculus challenges related to 11th grade stu-dents’ understandings of indefinite integrals. Another important factor is the symbolic algebra used in introduc-ing new notions, such as derivatives of various orders or integrals. Metaxas (2007) examined students’ understand-ing of integral concepts and commented that their “under-standing was mainly procedural and that they had difficulty handling the integral as an object” (p.  267). Thompson (1994) argued that “students’ difficulties with the Funda-mental Theorem of Calculus (FTC) stem from impover-ished concepts of rate of change and from poorly developed and poorly coordinated images of functional covariation and multiplicatively-constructed quantities” (p. 2). Thomas et al. (2009) provided additional evidence of students’ dif-ficulties and of technological solutions in constructing the anti-derivative graphs of a given static function graph.

The “cognitive load” and the derived challenges in stud-ying calculus constitute just one aspect that might call for using a different methodology. Another aspect is the desire to generate understanding that goes beyond the ability to arrive at the exact numerical results or the exact proof. Such understanding is assessed by the students’ ability to provide an in-depth explanation for their findings and not necessar-ily by solving equations per se. In Principles and Standards for School Mathematics (NCTM 2000), the chapter on prin-ciples begins as follows: “Students must learn mathemat-ics with understanding, actively building new knowledge from experience and prior knowledge” (p.  229). Conse-quently, an important challenge in mathematics education is to guide students to become skilled problem-solvers who clearly understand the underlying concepts.

Non-classical methods might assist in meeting this chal-lenge. Learning the concepts of function behavior, includ-ing the subset of finding extrema points, is an important part of high school mathematics curricula. We believe that in some cases, using alternate non-classical methods will enhance student understanding and result in a better con-ceptual grasp of the solutions and their properties.

2.2 Integrating e-resources into calculus study

The general interest and importance of technology and resources in mathematics education is reflected, for instance, by the evolution and progression of the CERME technology thematic working group (TWG). In 1999, the first CERME conference assigned a dedicated TWG for technology, while extending the TWG domain and con-tributions at subsequent conferences. Chapter  11 includes a survey of research focus and scope framed by various

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levels of relationships–tools, knowledge, learners, teachers and curriculum (Trgalova et al. forthcoming):

In the beginning, the focus of the research was on the effects of using technology on students’ learning and teachers’ practices. Now, as we know more about these effects, our attention has shifted to be concerned with researching how we can scale ‘successful’ inno-vations in mainstream education systems (p. 17).

Dynamic technological tools have played an impor-tant role within the wide spectrum of research integrating technology into mathematics. For the last several decades, many studies have investigated technology usage, integra-tion, and impact on students’ understanding of mathemati-cal concepts (e.g., Erbas and Yenmez 2011; Stahl et  al. 2011). The use of e-resources and digital curriculum mate-rials, specifically their visual property, enables “seeing the unseen” (Arcavi 2003, p.  216). In one sense, seeing the unseen in education refers to the ability to visualize the “abstract” world (ibid.) and helps transcend the limitations of the mind (Pea 1987).

The increased availability of e-resources has been cou-pled with growing research efforts to study the impact of such integration. A major aspect relevant to the current study is the use of dynamic geometry resources for student activities. According to NCTM, technology extends the range of opportunities and enriches the quality of mathe-matical learning by providing a platform for investigating mathematical objects from different perspectives (NCTM 2000, p. 25). Arcavi (2003) emphasized that.

Nowadays, the centrality of visualization in learn-ing and doing mathematics seems to become widely acknowledged. Visualization is no longer related to the illustrative purposes only, but is also being recog-nized as a key component of reasoning, problem solv-ing, and even proving (p. 235).

More specifically, research efforts focusing on the cal-culus domain, including function-graphs and derivatives, exemplify this trend. Swidan and Yerushalmy (2014) dis-cussed the practical implications of using dynamic artifacts in learning the notions of calculus graphically. They found that students used semiotic means to mediate connections between graphs and mathematical ideas, such as rate-of-change. Berry and Nyman (2003) demonstrated that the use of graphic technology combined with group discus-sion resulted in a move from an instrumental understanding of calculus as an algorithmic process toward a relational understanding of the graphical links between the derivative of a function and the function itself. The study participants were eight students enrolled in an intensive spring term course that used TI-83 Plus hand-held devices.

Berry and Nyman (2003) emphasized the link between technology and a potential remedy for the student dif-ficulties described above by implementing the “Rule of Three”—the ability to identify and represent the same mathematical object by numerical, graphical and algebraic representations in calculus education. Hughes-Hallett et al. (1994), wrote:

One of the guiding principles is the ‘Rule of Three’, which says that wherever possible topics should be taught graphically and numerically, as well as analyti-cally. The aim is to produce a course where the three points of view are balanced, and where students see a major idea from several angles (p. 121).

Nowadays, with the proliferation of dynamic geometry software (DGS) such as GeoGebra, Cabri and Geometer’s Sketchpad and the e-resources within each, implementa-tion of the Rule of Three becomes easier. DGS enables students to build mathematical objects and to investigate their properties and their dynamic connections quickly and easily. One of DGS’s major advantages is the ability to change objects continuously and to observe the effects of these changes on other representations while maintain-ing the geometrical properties of the object throughout the dynamic process. DGS has been in use for over 10 years and, as noted by Trgalova et al. (forthcoming), it provides “opportunities for new problem strategies; discovering geometrical theorems as invariants whilst varying objects (points, straight lines,…); supporting conjecturing; and providing or discovering proofs” (p. 3). For extrema prob-lems and analysis of a target function in a limited constraint environment, DGS provides an important advantage. The student can, for example, change a rectangle’s sides gradu-ally and observe the changing behavior of the area function while keeping the given periphery as a constant.

Research has revealed many cases and methods that sup-port students’ conceptual understanding by utilizing availa-ble DGS advantages. Granberg and Olsson (2015) discuss the use of tools like GeoGebra that combine geometric and algebraic representations. In their study, students in the upper grades of secondary school dealt with interpreting and analyzing linear functions graphs. The case included tasks that students were unlikely to solve using mere imita-tive reasoning1 and situations related to interpretation of linear functions. The results revealed that DGS had a sig-nificant impact on students’ understanding and ability to generate creative reasoning. Cullen et al. (2013) examined

1 Reasoning is seen as “a product that appears in the form of a sequence of reasoning that starts in a task and ends in an answer” (Lithner 2008, p.  256). Imitative reasoning is either memorized, where the student’s answer is based on recalling a complete similar answer, or else it uses algorithmic reasoning, where the answer is based on a pre-specified procedure.

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the extrema points of functions. They used GeoGebra to study the reasoning of preservice secondary school teach-ers in various cases of local extrema for high-degree poly-nomials. Cullen et  al. emphasized that “these types of authentic mathematical explorations are closely aligned to the work of mathematicians and a valuable component of our students’ educational experience.” (p. 68).

2.3 Abstraction in context and RBC model

The current research adopts the AiC theoretical framework for examining processes of constructing and consolidat-ing abstract mathematical knowledge (Hershkowitz et  al. 2001). Abstraction is defined as the process of vertically reorganizing previous mathematical constructs within mathematics and using mathematical means to interweave them into a single process of mathematical thinking that leads to a construct that is new to the learner. According to AiC, the genesis of abstraction passes through three stages: (1) the need for the new construct; (2) the construction of the new construct; and (3) the consolidation of the new construct. The process of knowledge construction (stage 2) can be analyzed by using the nested epistemic actions model–recognizing, building-with, construction (RBC) for studying AiC. The RBC model identifies three epistemic actions in the process of knowledge construction: (a) recog-nizing—the learner recognizes a known element relevant to solving the problem at hand; (b) building-with—an action by which the learner uses recognized elements to achieve a goal, such as solving a problem or justifying a statement; and (c) constructing—the main action in the knowledge construction process, in which the learner vertically reor-ganizes pieces of known constructs into a new construct (Schwarz et al. 2009).

Several studies have used the AiC framework. We describe those most relevant to our case. Kidron and Dreyfus (2010) analyzed the case of a solitary learner, an experienced mathematician. They used the AiC frame-work and the RBC model as a platform for analyzing the learner’s mathematical justification processes while inves-tigating bifurcation points in dynamic systems, using an e-resource for drawing bifurcation diagrams. A solitary learner presents a real challenge for analysis because indi-viduals rarely need to report on their learning. The RBC model enabled the researchers to identify that the learner’s constructing actions and justifications were being enriched. Kouropatov and Dreyfus (2014) also used AiC in a research study geared to examining the study processes of four pairs of high school students while learning integral concepts in calculus. The study environment was not computer based. The researchers showed that within a limited time most of the eight students succeeded in attaining most of the

intended new constructs and that many of their construc-tion processes became observable via the RBC model.

A major part of our data analysis was based on AiC using the RBC model. We chose this framework since our objective was to carry out an in-depth analysis of knowl-edge construction in a very specific context, the context of a digital environment using new e-resource materials. This framework makes it possible to inspect the evolution and progression of mathematical ideas based on combining the learner’s existing knowledge with empirical visual observa-tions that the digital environment provides.

2.4 Research questions

The present study was conducted in a digital resources environment explicitly designed for 11th grade students to learn minima and maxima geometric problems as part of the curricular requirements. The research questions are as follows:

1. What affordances and constraints can be identified in processes of abstracting mathematical knowledge within the designed environment?

2. Can we point at traces of the learning processes in stu-dents’ reasoning regarding extrema problems?

3 Methodology

3.1 Participants

Twenty-nine 11th grade high school students participated in the current study. All came from one organic mathemat-ics class comprising students with relatively high com-petence that were aiming to qualify for the highest high school mathematics level in Israel. The study took place at the beginning of the second semester of their second year, when according to the Israeli curriculum, students study the topic of calculus extrema problems. All students were familiar with the concepts of function analysis and differ-entiation. In addition, all of them had some experience with GeoGebra as part of various prior activities.

The students were divided into two groups based on their achievements in mathematics as reflected by their final grades at the end of the first semester. One group (Team A) consisted of 11 students ranked as the low achievers in class, i.e., the lower third. Team A learned a non-classical approach to solving extrema problems. The second group (Team B) consisted of the remaining 18 students. Team B learned the classical approach for solving extrema prob-lems according to the standard high school curriculum.

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3.2 Teacher-designed resources

Students’ difficulties, previously described, motivated the teacher, who is the first author, to seek a modified curricu-lum. As emphasized by Swidan and Yerushalmy (2014), “the role of the teacher is to foster the evolution of mean-ing through activity with the artifact and to mediate the personal meanings that emerge through the activity, so as to make them recognizable as mathematical meanings” (p. 530). The teacher’s goal was to design activities that use non-differential means to solve extrema problems and that have the potential to expand the students’ abilities to exam-ine and understand the process of solving extremum prob-lems. With this in mind, the teacher harnessed the available technological environment, augmenting the existing curric-ulum with specific, dedicated digital curriculum materials. The digital-based activities were designed to minimize the teacher’s intervention, and students were expected to cope independently with each task.

Most of the activities were based on geometric extrema problems taken from a standard calculus textbook. How-ever, the AM-GM inequality method is not part of the standard curriculum and consequently required designing a special activity. In order to adopt a more learner-centered pedagogy, the teacher needed to re-source and rebuild exist-ing teaching practices. The first activity (Activity 1), which was comprised of 6 tasks named 1a to 1f, enabled the stu-dents to investigate the behavior of the AM-GM inequal-ity both graphically and numerically and to formulate the connections between the means (see examples in Sect. 3.4). The next two activities (Activities 2–3, see examples in Sect.  3.4) converted standard geometric problems that can be solved non-differentially into the GeoGebra envi-ronment, creating a specific e-resource for each problem. These activities, comprised of five geometric tasks each (tasks 2a–3e), were accompanied by predesigned GeoGe-bra resources. Each resource presented the various param-eters of the constraints and the target values using graphi-cal and algebraic means and actually explicated the unseen. The three activities encompassed a combination of paper-and-pencil and digital curriculum resources for the student learning process.

3.3 Research description

Both teams studied the topic in 7 double lessons (90 min). Both began with two standard introductory lessons taught according to customary methods. During the next three consecutive lessons, each team received different instruc-tion. Team B held a whole-class discussion based on the classical approach to differential analysis of extrema prob-lems, with one exception—the addition of a theoretical presentation and limited practice of the AM-GM inequality.

Team A worked in pairs on a sequence of inquiry activities on purposefully designed spreadsheets and GeoGebra envi-ronments (described in the previous section). Students first considered the AM-GM inequality and then solved geomet-rical extrema tasks using a non-differential approach. The teacher acted as a silent observer who could answer techni-cal questions only.

The two teams came together for the last two lessons, which were dedicated to non-geometric extrema problems presented using standard whole-class pedagogy. At the end of the learning unit, all students took a standard paper-and pencil test designed as part of the high school testing program.

3.4 Research tools

As mentioned, Activities 1, 2 and 3 took place in a tech-nological environment. In the following paragraphs, we provide several examples of tasks that involve qualitative inquiry-questions as opposed to calculations or geometric proofs.

Activity 1 focused on the AM-GM inequality. The objective of this activity was to introduce the inequality in various forms, both numerical and geometric. The activ-ity began by defining the three means and then presenting various inquiry tasks for self-study and self-investigation of the relations between the means. In the first task, the stu-dents were asked to analyze the three means—arithmetic, geometric and harmonic—of two positive numbers and to propose conjectures of any sort regarding the relations between these means. The students were instructed to use Excel to examine various combinations of positive pairs of numbers using both numerical and graphical means. They wrote their answers on the activity sheet with no need to provide explanations or proofs.

The second task in Activity 1 focused on graphic repre-sentation of the AM and GM of two numbers (see Fig. 1). The students were asked to analyze the arithmetic and geo-metric means of two segments in a circle and provide expla-nations for the relations they had observed in the previous task by means of a predefined GeoGebra e-resource. The general GeoGebra environment, including the graphic and algebraic panes, is an explicit implementation of the Rule of Three mentioned above (Hughes-Hallett et al. 1994).

Activity 2 comprised five simple geometric extrema tasks. Tasks 1 and 2, for example, involved finding the rec-tangle with the maximal area among a group of rectangles with a constant perimeter and solving the question of mini-mal perimeter for constant area. Although these problems can be easily solved using calculus, the students were asked to find the optimum in the GeoGebra environment using a graphical representation of the target function and to try to explain their findings based on previous knowledge from

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Activity 1 regarding the AM-GM inequality. The other tasks dealt with various simple geometric scenarios of tri-angles and rectangles.

In Activity 3, the students performed more complicated tasks. A typical task is the Heron problem. In this case, the analytic solution is difficult and the students were directed to find a geometric-based solution, including an explana-tion that is not a direct algebraic result. Figure 2 shows a screenshot of the GeoGebra e-resource. In order to inves-tigate the problem, the student can move point C along the blue line and observe the sum of distances AC and AB. The pane on the right side displays a graph of the target function. Sliders a and b on the top left allow the student to change the problem constraints, e.g., the position of points A and B.

The second research tool was a standard test of the extremum problem. It consisted of four geometric questions and lasted 2  h. The questions spanned various geometric shapes and scenarios. For each question, students could use any method, whether calculus or non-calculus, to find an optimum. Each question ended with a request to provide a qualitative explanation for the result.

3.5 Data analysis

The study had two main data sources. The first source included the video-recorded observations of Team A’s work in the technological environment. We transcribed all student activities verbatim and collected the students’ writ-ten responses to the worksheet. The second data source included all the students’ written responses to the test.

Data analysis of the first data source consisted of two steps: (1) a priori analysis of knowledge elements expected to be constructed (based on AiC, as further elaborated below), and (2) analysis of the dyads’ transcript discussions and written documents.

3.5.1 Step 1: A priori analysis

During this step, we analyzed and mapped each activ-ity task into a series of knowledge elements (KEs). Each knowledge element included a basic piece of information or a conclusion the students were expected to construct. The KEs were expected to be built in a gradual step-by-step process. Table 1 provides an example of the first ten prede-fined KEs for Activity 1.

Fig. 1 Two segments controlled by sliders, combining a circle diam-eter

Fig. 2 GeoGebra e-resource for analyzing the Heron problem

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After listing all the expected KEs for each task in each activity, we built a map of expected KEs showing the con-nections between the KEs (Fig. 3).

3.5.2 Step 2: Transcript analysis based on RBC model

Two dyads from Team A were selected for in-depth anal-ysis of the knowledge construction process: the dyad that exhibited the best relative improvement in its final achieve-ment (D&R dyad) and the dyad with the lowest improve-ment (R&L dyad). The researcher analyzed the transcripts and documents of the students’ work in search of the KEs. A matrix generated for each activity illustrated the KE con-struction process compared to the expected process. To assure inter-rater reliability, 2 researchers independently analyzed and mapped the KEs for about 10% of the tran-script, with 85% agreement.

As for the second data source, each student response was categorized as correct, incorrect or partially correct. The response grade was a combination of two factors—mathematical correctness and qualitative explanation of the numerical results. Each student was asked to provide a written explanation and justification for the optimal solu-tion, and the teacher assessed the clarity, correctness and broadness of the explanation. The grade comprised two components: the mathematical correctness of the response (75%) and the qualitative explanation of the numerical results (25%), taking into account partial explanations as well.

4 Findings

The findings related to the first research question—affor-dances and constraints of the designed study environment on the learning processes—are presented in the first two

Sects.: (4.1) differences between expected knowledge ele-ments and actual knowledge construction process as a whole, and (4.2) actual knowledge elements whose con-struction is likely associated with the use of e-resources. Findings related to the second research question—traces of learning processes in students’ reasoning—are provided in Sect. 4.3.

4.1 Constructed knowledge elements–actual vs. planned

We analyzed the KE construction process for all dyads and activities. Figure  4 shows a matrix used to map the end results of the process. The rows denotes the six tasks in Activity 1. The columns indicate all the expected KEs for the activity. The shaded cells indicate whether the KE was expected and its content indicates whether the dyad actually constructed it. Showing all the tables is beyond the scope of this paper. The aggregation of the tables for all Team A dyads for the 3 activities indicates that the students

Table 1 A subset of predefined KEs for Activity 1

No. Knowledge element

E1.1 AM/GM/HM of two positive numbers is a positive numberE1.2 AM/HM of two negative numbers is a negative numberE1.3 GM of two negative numbers is a positive numberE1.4 GM of a pair of negative and positive numbers is undefinedE1.5 HM of two numbers of which at least one of them is zero is undefinedE1.6 All three means for a pair of positive numbers are not less than the minimal number in the pair and not more

than the maximal number in pairE1.7 AM/GM/HM of two equal positive numbers is equal to these numbersE1.8 A pair of two positive numbers and their AM generates an arithmetic sequence where AM is the second elementE1.9 A pair of two positive numbers and their GM generates a geometric sequence where GM is the second elementE1.10 When two given unequal positive numbers in two sequences, arithmetic and geometric, are expanded by adding

a middle number, the added number for the arithmetic sequence is greater than the one added for the geomet-ric sequence

Fig. 3 KE map example for Activity 1, Task 1

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managed to perform 80% of the planned tasks and con-structed 72% of the expected KEs.

In addition to identifying KE construction in the tran-scripts, we analyzed the students’ reasoning and actions that led them to construct a specific KE. We searched for student reasoning and explanations that were highly likely to be associated with information obtained from the use of the digital environment, as described in the next section.

4.2 Episodes for KE construction while working with the e-resources

As mentioned above, we conducted the in-depth analysis on two focal dyads. The findings below are common to both dyads, unless otherwise stated.

Activity 1 was devoted to investigating the AM-GM ine-quality using an Excel spreadsheet. In the first task, the stu-dents empirically identified the relations between AM, GM and HM after about 7 min. They generated the necessary formulas themselves, performed around a dozen iterations to check various combinations of numbers and came up with correct conjectures about the three means. Although the task requirement referred only to positive numbers (as the results were to be used for magnitudes of geometric segments), all Team A students used the available platform to check zero and negative numbers as well. We believe this is a direct affordance of working with the easy-to-use Excel instrument, which minimized excessive calculations in the students’ extended investigation.

None of the students expressed any concern about the fact that these conjectures were empirically based and required proofs. They appeared confident about the inequal-ities and had no problem moving on to the next tasks based solely on empirical results. We see this as a constraint of

the e-resource. The need to prove the results mathemati-cally came up later, in response to a leading question by the teacher. Next, we provide a few episodes to exemplify the students’ knowledge construction process while work-ing with the spreadsheet or the dynamic geometry resource.

D&R constructed the knowledge about the inequality in three major steps. First, they explored each mean separately and recognized that it falls between the minimum and the maximum of the given pair. Second, they explored the rela-tions between the means. D&R managed to construct the relations fully, while R&L, the second dyad, constructed them only partially. Third, both dyads continued their exploration beyond the activity requirements and decided to check the validity of the conjectures for all kind of inte-gers, not just positive. We attribute this voluntary expan-sion to the affordances of the working environment.

The following episode, Episode 1, is from the D&R dyad. It refers to their use of Excel to investigate the rela-tionships between the AM, GM and HM of two numbers.

125 R: Exactly, this one (AM) is always the largest.126 D: This one (GM) is in the middle, and this (HM) is

the smallest.127 R: And if we exchange the numbers, it (GM) always

remains in between.128 D: Right, it is always between the smallest and the

largest. HM remains the smallest of the three, no matter what we do with the numbers.

Episode 1: D&R exploring relations between HM, GM and AM.

D&R entered different combinations of number pairs into the appropriate cells and observed the Excel results for the behavior of the three means. They did not perform any algebraic calculations and reached their conclusion based on the Excel results. The instant Excel feedback accelerated

Fig. 4 Constructed KEs–actual vs. expected for specific dyad and activity. Shaded cells indicate that KEj was expected in activity task i. Indicates that KEj was constructed by the dyad in activity question i. Indicates that KEj was partially constructed by the dyad in activity question i

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the process and increased the number of iterations and sam-ples. Consequently, D&R constructed KEs E1.11, E1.12 and E1.13 that refer to GM always being the intermediate value between HM and AM, which are the lower and upper bounds, respectively.

The need to provide analytical proof for the conjecture was prompted by the second task of Activity 1, when the students began working with the GeoGebra e-resource. In this and the following task, the students were instructed to connect between the various means of two numbers and geometric objects in the GeoGebra environment. The instrument (see Fig. 1) depicted a general circle as a base for the inquiry. At this point, the students switched from the numerical and empirical working mode to the geometrical and analytical mode. This is probably an outcome of the DGS e-source affordances, which Excel lacks. It may also be related to students’ stipulation that proof belongs with geometry. In the geometric environment, they managed to construct the following KEs:

• The correspondence between the AM and GM of two numbers and the relevant segments in the circle (KEs E1.16–E1.19).

• The relations: equality, inequality and upper bound between AM and GM (KEs E1.20–E1.22).

• The move from empirical conjecture to analytical deduction.

Analysis of the transcripts based on the RBC model reveals the various epistemic actions of knowledge con-struction toward the above constructs. The findings of the detailed analysis are beyond the scope of this paper. Yet note that one of the students explicitly expressed the dif-ferent orientation modes, moving from empirical to ana-lytical analysis: “Yes, we managed to verify and prove the relations, it is a proof!” This statement illustrates the

ability to interpret and integrate visual GeoGebra con-structs into a complete proof, based on dynamic observa-tions using the e-source. The student managed to formu-late a proof using the drawing terms that appeared in the GeoGebra graphics pane.

In Task 1e, the students failed to generate the link between the AM-GM inequality and the extrema prob-lem. There is no evidence of constructing any kind of connection. For all dyads, the two topics seemed to be totally unrelated. The task itself may not have highlighted the connection or perhaps the mathematical concept was too abstract at this stage.

Activity 2 involved investigating simple geometric extrema problems using the GeoGebra resource (see Fig.  5). The resource enabled the students to set a con-stant value for the rectangle’s perimeter and then to change the size of one edge using the slider tool. This changed the shape of the rectangle while its perimeter remained constant, allowing the students to observe the value of the resulting area. The right side of the graph depicts the value of the area as a function of edge sizes.

As the question about the rectangle with the max area was central to understanding the link between the geo-metric extrema problem and the AM-GM inequality, we next show Episode 2 from the second dyad, R&L.

127 R: Look, I can see that. When the edges get close to each other, I mean when their lengths tend to be more and more similar, I mean, when the difference between them gets smaller, like in this case, I have the feeling that the area is increasing.

(Changes the rectangle edges using GeoGebra sliders).128 L: Let’s see, it looks like this is true.129 R: Look, I change the size again from 3 to 4, and

the rectangle shape becomes more rounded; I mean, it is not a circle, it is not round, how can I say, it is “fatter.”

Fig. 5 GeoGebra resource for analysis of rectangle with max area

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130 L: So, if we can cause the rectangle to be fatter and fatter, we can gain more area.

131 R: Yes, I think that it absorbs, it accumulates more area when it is more rounded and does not have a cigar shape.

132 L: So, when you reduce one edge, you lose area, but you compensate with the other edge. And when the edges become more and more similar, the gain is larger than the loss.

133 R: Perfect.Episode 2: Dyad R&L working with GeoGebra resource

on max area rectangle problem.Our major findings in Activity 2—using the GeoGebra

resource for specific simple geometric extrema problems—indicate that by using visual dynamic representation and based on the environment affordances, the students realized that the absolute difference between the rectangle’s edges is the major factor affecting the changes in the rectangle’s area. We noticed a gradual knowledge construction process of understanding that a rectangle with a constant perimeter surrounds a larger area when its edges become more and more similar. The knowledge construction occurred while the visual shapes were shown dynamically on the screen. Additionally, another aspect of GeoGebra affordances, the fact that the digital resource provided a graph representing the value of the area target function, made the analysis vis-ible and hence easier. The students exploited the dynamic visual link between changes in the free parameters and the resulting change in the objective function. The graph pro-vided the ability to combine multiple observations into one coherent picture.

4.3 Students’ explanations and justifications in the test responses

The following analysis focused on the students’ reason-ing about their numerical optimum solutions for each test question. Students from both teams provided explana-tions and justifications for their answers, both for correct and for incorrect answers. Table 2 shows the percentage

of students who were able to provide a full or almost full and correct explanation of the solution result, per test question and by team. Team A students gave explanations for 100% of the cases. Not all of these explanations were necessarily valid or complete, as indicated in Table  2 below. Team B students gave explanations for 74% of the cases. Moreover, Team A’s explanations were more detailed and reasoned than those of Team B. This find-ing is evident in the average length of the explanations: six lines for Team A compared to three to four lines for Team B.

Table  2 shows that relatively more Team A students managed to provide qualitative answers for the first three questions. One exception is question four, which only one student from Team B explained. The fact that the Team A students were the low achievers in this class did not affect their relative performance compared with the Team B students.

Next we provide four examples of explanations. The first two examples are explanations of question 2 by Team A and Team B students. The third and fourth exam-ples are explanations of question 3 by Team A and Team B students.

According to the first example in Table 3, the student was able to explain the optimum of the area target func-tion using pure geometric considerations without mak-ing any mention of the function graph, its derivative or local extrema points. The student made use of a rela-tively rare theorem regarding the multiplication equality of two internal circle segments that intersect each other. We assumed that this might result from the context of the geometric environment he used.

The second and fourth examples in Table  3 are from Team B students and offer only vague, unspecified expla-nations. Neither of the students had a clear idea that could be formalized into a detailed explanation, or perhaps they just vaguely understood the potential connection between the geometric target function and the AM-GM inequality and took a shot in the dark. The explanations are obvi-ously not sufficiently detailed and coherent.

The answer of the Team A student in the third row of the table is an example of a partially correct answer. The student did not rely on calculus, and he provided geome-try-based reasoning. Nevertheless, the explanation based on the graphical drawing was not sufficiently accurate due to a few unnecessary assumptions.

Finally, we quantitatively analyzed changes in the stu-dents’ relative rankings based on end-of-term pre-inter-vention and post-intervention test grades. In Team A, 6 out of 11 students (66%) improved their class ranking. In Team B, 7 out of 18 students (35%) improved their ranking.

Table 2 Percentage of qualitative explanations for optimal test answers

Team A (11 students) Team B (18 students) Team % Question

78 60 166 30 278 65 3– 5 4

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5 Summary and discussion

The current study sought to explore an alternative method of teaching a learning unit on minima and maxima prob-lems in an inquiry e-resources environment. In addition to the most popular method of setting target functions in a single variable and using the derivatives, there is room for geometric methods, AM-GM inequality-based methods and others. We explored whether these non-dif-ferential methods combined with purposefully designed e-resources can influence students’ understanding. The research setting consisted of two parallel groups of 11th grade students from the same high-track class. One group (Team B), comprised of the upper two-thirds of students, studied the entire learning unit about extrema problems using the classic calculus-based method. The other group (Team A), comprised of the lower third of students in the class, devoted three out of seven lessons to studying the geometric subset of extrema problems in a computer-ized environment with purposefully designed GeoGebra e-resources and Excel, using non-calculus inquiry meth-ods. At the end of the learning unit, we evaluated the stu-dents’ knowledge using a written test.

Our research questions were as follows:

1. What affordances and constraints can be identified in the processes of abstracting mathematical knowledge within the designed environment?

2. Can we point to traces of the learning processes in stu-dents’ reasoning regarding extrema problems?

The knowledge construction processes of the Team A students were analyzed using the RBC model (Schwarz et  al. 2009). The analysis and findings presented above allowed us to propose an answer to the first research question. During the first activity, working both with Excel and with GeoGebra e-resources, all 5 dyads man-aged to construct about 80% of the expected KEs regard-ing the relations between AM, GM and HM, which is the basis for posing the AM-GM inequality theorem. Although Team A students did not fully formalize the theorem, they empirically discovered and formulated it based on dozens of trial-and-error experiments in the Excel environment. Excel affordances, the ease-of-use of the spreadsheet and its ability to generate and repeat the built-in formulas for the various means facilitated fast construction of ideas and knowledge about the order relation of the three means and the conditions for equal-ity and inequality relations. It seems that the ease-of-use enabled the students to extend their experimentations and realizations beyond the scope of the specified tasks and to initiate further explorations. Yet, we observed a major constraint in the process of developing the mathematical ideas. The ability to scan a large sample of numbers inter-fered with the need for providing mathematical proofs for the conjectures.

Table 3 Examples of student reasoning regarding optimum points for extrema problems

Question Student reasoning

Question 2: A triangle is bounded in a circle of a given radius, with one edge on the diameter. Find the perpendicular edges for the max area triangle

Since one edge is the diameter, the triangle is a right-angled triangle. The circle diameter is its hypotenuse. Each height from the vertex is actually half of a circle segment. The segment is split into two equal parts by the diameter. The triangle area is half of the product of the diameter and the height. Since the diameter is fixed, we can move the free vertex. The longest height is generated when the height is a radius. Consequently, the optimal result is for a right-angled isosceles triangle

[Student #3, Team A]The optimal point is achieved for two reasons. First, it is the local maxima of the area functions

according to the first and second derivative. Second, the area will be maximal when the two perpendiculars are equal. In this case, the area is half of the GM squared. Now, since the GM is bounded by the AM, the maximum is achieved when both are equal

[Student #14, Team B]Question 3: Given a rectangle with area of 80

square centimeters and edge length of 10 cm. Draw two right-angled isosceles triangles on opposite vertexes of the rectangle and create an internal parallelogram. What are the triangle edges for a parallelogram with maximal area?

The area of the rectangle is fixed (=80 cm2). So the maximal area of the parallelogram is obtained when the areas of the four triangles at the edges are minimal (complementary rule). The sum of the triangle areas is

x2 + (10 − x)(8 − x) = 80 − 18x − 2x2

The maximal value is obtained when x is 4.5. (Here the studen provided an explanation based on a drawing, using graphical reasoning to demonstrate the maximal x that can still generate a bounded parallelogram/)

[Student #7, Team A]The largest height of a parallelogram is achieved when the parallelogram turns into a rectangle;

this is the optimum based on the AM-GM inequality[Student #17, Team B]

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In the next task, the students moved from empirical con-jectures to formal proofs using the GeoGebra e-resource. The dynamic geometric environment enabled the students to visually identify a geometric representation for two of the three means, AM and GM. The RBC-based findings suggest that in a visual and dynamic environment a student can discover and prove properties that otherwise would be very difficult if not impossible to assume. This conclusion is consistent with other research studies (Erbas and Yenmez 2011; Granberg and Olsson 2015). In addition, the students constantly used Excel for testing, referencing and visualiz-ing for negotiating shared knowledge.

In the second activity, the Team A students did not man-age to find the connection between the AM-GM inequality and the extrema problem. While there may be various rea-sons for this, we propose that such a connection is too com-plicated and too abstract for 11th grade students. This could be the subject of future research entailing a longer, more extensive and more gradual e-resourced-based inquiry unit.

After constructing the infrastructure for solving extrema problems with the AM-GM inequality, the Team A stu-dents gained some experience with pure geometric analy-sis of extrema problems in the designated GeoGebra-based e-resources. Activities 2 and 3 involved geometric extrema problems of gradually increasing complexity. The chal-lenge for the students was to find the optimum and to try to explain or even prove why that optimum is attained at cer-tain values. The inquiry was based on the students inspect-ing the related geometric objects in the e-resource environ-ment, playing with the constraints using GeoGebra sliders and viewing the graph of the objective functions in a paral-lel pane.

Our research findings indicate that students’ construc-tion processes in this e-resource environment resulted in their ability to identify the optimum, raise various hypoth-eses and in most of the tasks prove/explain the identified optimum using AM-GM considerations. The students did not reach the level of providing a full rigorous proof, but they were able to provide arguments that can be character-ized as visual descriptive statements. As researchers, we were able to trace those descriptive utterances back to the relevant e-resources in major parts of Team A’s discus-sions. We believe that this finding is evidently a result of working in a visual, dynamic e-resource geometry environ-ment. Additionally, the various episodes demonstrate that in most cases students’ arguments were constructed as a result of their teamwork within the environment.

The students’ argumentation was mainly a process of building collaborative conjectures. This result supports other studies and observations regarding the strength of a computerized environment for cultivating small team dis-cussions. When comparing the five dyads in Team A, we observed considerable similarities between the students’

construction processes (especially with the first simpler tasks in Activity 1 and 2). The dyads moved along simi-lar paths of knowledge element construction (e.g., starting with AM unequal positive numbers, continuing to equal samples, adding negative numbers, etc.) This similarity might be afforded by the design of the activities and the relatively closed GeoGebra e-resource.

Our second research question referred to traces of the learning processes in students’ reasoning regarding the test problems. As seen in Tables 2 and 3, the students from Team A provided more specific, explicit and detailed justi-fications for the extrema optimum solutions. This was not observed in the written explanations of students from Team B, even though the Team A students comprised the lower third of this class. Other studies (Berry and Nyman 2003) exhibited similar differences between students who used computer-based scaffoldings and those who did not, though this conclusion is not necessarily general. Yerushalmy (2006) examined the impact of intervention using a graphic software environment for less successful students. Rela-tively slow seventh-grade students studied algebra prob-lems with functional contexts while accessing graphic soft-ware tools when useful. The result was a delay in moving towards the use of symbolic formalization, changes in solu-tion patterns and emphasis on search solutions of a numeric and graphic nature. There was no impact on improved success.

In our study, we selected a very specific and relatively complicated learning unit that lent itself to visual and geometric representations. The results of our study may suggest that a non-calculus learning approach should be considered as a valid and useful alternative for teaching extrema problems in an e-resource environment and may lead to relational understanding (Skemp 1987). On a more general level, we speculate that learning topics that are problematic as established by research in mathematics edu-cation may be precisely the topics in the curriculum where the advent of e-resources will make the difference between relational and instrumental learning, making this a fertile avenue for further study.

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