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Connecting Algorithmics to Mathematics Learning: A Design Study of the Intermediate Value Theorem and the Bisection Algorithm

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Title: Connecting Algorithmics to Mathematics Learning: A Design Study of the Intermediate Value Theorem and the Bisection Algorithm
Language: English
Authors: Lagrange, Jean-Baptiste (ORCID 0000-0003-1789-877X), Laval, Dominique
Source: Educational Studies in Mathematics. Feb 2023 112(2):225-245.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 21
Publication Date: 2023
Document Type: Journal Articles
Reports - Research
Education Level: Secondary Education
Descriptors: Algorithms, Mathematics Education, Mathematical Concepts, Design, Secondary School Students, Foreign Countries, Mathematics Skills, Programming
Geographic Terms: France
DOI: 10.1007/s10649-022-10192-y
ISSN: 0013-1954
1573-0816
Abstract: Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16-19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1364086
Database: ERIC
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  Value: <anid>AN0161580831;esm01feb.23;2023Feb01.02:23;v2.2.500</anid> <title id="AN0161580831-1">Connecting algorithmics to mathematics learning: a design study of the intermediate value theorem and the bisection algorithm </title> <p>Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16–19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions.</p> <p>Keywords: Algorithmic and mathematical working spaces; Mathematics-computer science connections; Bisection algorithm; Intermediate value theorem; Programming tasks; Algorithmic thinking; Adidacticity</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0161580831-2">Introduction</hd> <p>Worldwide, many school systems are introducing computational thinking (CT) courses that include programming, in some cases as part of the mathematics curriculum.[<reflink idref="bib1" id="ref1">1</reflink>] This reflects the dual nature of computer science: while it has developed as an autonomous field, with significant implications for science and society, it also has deep connections with mathematics. This introduction analyzes how, over the years, educators have considered the connections between the two domains, identifies the specific issues that motivate our study, and then introduces our concrete focus.</p> <p>Papert ([<reflink idref="bib23" id="ref2">23</reflink>]) noted that programming can provide a meaningful introduction to other sciences, especially mathematics. He introduced the concept of CT to refer to how learners think. There is, however, no consensus on the meaning of CT. For example, for Wing ([<reflink idref="bib28" id="ref3">28</reflink>]), it consists of cognitive abilities that are specific to computer science and Fraillon et al. ([<reflink idref="bib9" id="ref4">9</reflink>]) define it with respect to a range of skills that are only partially related to programming. There is also no consensus regarding what students learn through programming. In the 1980s, against the background of CT, educators put great emphasis on programming, but authors such as Tetenbaum and Mulkeen ([<reflink idref="bib27" id="ref5">27</reflink>]) and Bower ([<reflink idref="bib2" id="ref6">2</reflink>]) found little evidence that students' cognitive abilities were, in fact, improved. Crahay ([<reflink idref="bib5" id="ref7">5</reflink>]) attributed this result to the naïve constructivism that is often prevalent in classroom use of computers, as teachers confuse Papert's instructionless learning with non-intervention.</p> <p>In the 1990s, authors such as Ruthven ([<reflink idref="bib24" id="ref8">24</reflink>]) considered that the emphasis should shift to the use of software packages and mathematical applications, rather than programming. However, in the 2000s, things changed again, and some mathematics curricula started to reintroduce programming. In France, Kahane ([<reflink idref="bib11" id="ref9">11</reflink>]) noted, in a government report, that the use of software as a tool did not help to understand the underlying computational principles. He recommended introducing some computer science into mathematics teaching, through programming activities.</p> <p>The above observations raise three issues, namely: (<reflink idref="bib1" id="ref10">1</reflink>) the conceptual foundations of programming; (<reflink idref="bib2" id="ref11">2</reflink>) the conditions needed for students to develop cognitive abilities through programming; and (<reflink idref="bib3" id="ref12">3</reflink>) the value of programming for learning mathematics. Since there is no consensus on the concept of CT, and to clarify the conceptual foundations of programming (<reflink idref="bib1" id="ref13">1</reflink>), we refer to the notion of algorithmic thinking (AT) defined by Knuth ([<reflink idref="bib13" id="ref14">13</reflink>]). The latter author associates AT with computer science, and identifies its contribution to reasoning in various mathematical fields, using a method that helps him to distinguish algorithmic thinking from mathematical thinking. Two types of thinking (...) separate mathematicians from computer scientists. In the first place, there is almost no notion of 'complexity' or economy of operation in (mathematics). The other missing concept (...) is the dynamic notion of the state of a process (...) Changing states of affairs (...) seem to be intimately related to algorithms and algorithmic thinking. (p. 181)</p> <p>Building upon AT, we see programming as the production of a text or a diagram that defines a systematic procedure, in other words, as a way to organize changes in the state of a system for automatic execution (Lagrange & Rogalski, [<reflink idref="bib16" id="ref15">16</reflink>]). The text or diagram can be seen as a program or as an algorithm, depending on the point of view. The system may be a computer, a robot controlled by a formal language, a virtual device such as a Turing machine, or a human being who follows operational rules. With this understanding, we consider algorithmics (seen as designing and thinking about algorithms) as the conceptual foundation of programming, and we develop our theoretical approach in the next section. We also address the conditions needed for students to develop cognitive abilities through programming (issue 2) from a theoretical point of view in the next section.</p> <p>Regarding the value of programming for learning mathematics (issue 3), we adopt a pragmatic stance. In France, where algorithmics has been formally reintroduced into the mathematics curriculum, Couderette ([<reflink idref="bib4" id="ref16">4</reflink>]) found that little progress has been made in real-life teaching. This suggests that algorithmics in mathematics education has not been conceptualized in a way that addresses the constraints and opportunities of everyday classrooms. The content to be taught is an important starting point. Nowadays, curriculum developers challenge the ability of content to explain a world in which digital applications are ubiquitous. As a piece of content, <emph>real analysis</emph> is culturally important, and undeniably necessary for applications, but also the links to algorithmics renew its relevance for education.</p> <p>The focus of our study is the intermediate value theorem (IVT)[<reflink idref="bib2" id="ref17">2</reflink>] for two reasons: (<reflink idref="bib1" id="ref18">1</reflink>) it has potentially deep connections to the bisection algorithm, and (<reflink idref="bib2" id="ref19">2</reflink>) the French upper secondary school curriculum includes preparation for real analysis in Grades 10 and 11, and specifically for the IVT in Grade 12. Algorithmics is prescribed not as stand-alone content, but as an activity to be integrated into other content (MEN, [<reflink idref="bib18" id="ref20">18</reflink>]).[<reflink idref="bib3" id="ref21">3</reflink>] This situation offers an opportunity to not only carry out an epistemological analysis from both algorithmic and mathematical points of view, but also to develop tasks that take into account potential connections between the two domains, and to evaluate the relevance of these tasks in everyday French upper secondary school teaching. In the next section, we provide a theoretical framework for our epistemological analysis, and the design and evaluation of tasks.</p> <hd id="AN0161580831-3">Theoretical framework</hd> <p>As indicated above, this section addresses the conceptual foundations of programming (issue 1), and the conditions needed for students to develop cognitive abilities through programming (issue 2).</p> <hd id="AN0161580831-4">Mathematical working spaces and algorithmic working spaces</hd> <p>Our approach to algorithmics and programming in mathematics education builds upon the theory of mathematical working spaces. According to Kuzniak et al. ([<reflink idref="bib15" id="ref22">15</reflink>]), a mathematical working space (MWS) is an abstract space that is organized to support mathematical work in an educational setting. The theory of MWS contributes to the understanding of teaching/learning situations as a function of multiple factors, including the domain (geometry, algebra, etc.), and the level of the student (basic, advanced, etc.). It has been adopted in many studies, some of which take into account the articulation of several working spaces (WSs). For example, Minh and Lagrange ([<reflink idref="bib20" id="ref23">20</reflink>]) studied the articulation between WSs related to three domains: geometry, measures, and mathematical functions. In the present study, we follow the latter line of research, and consider the articulation of two WSs: an algorithmic working space (AWS), which supports algorithmic thinking, and a mathematical working space (MWS), which supports mathematical thinking.[<reflink idref="bib4" id="ref24">4</reflink>] Both AWS and MWS should be considered with respect to how they appear and evolve in a given situation. It is, nevertheless, possible to contrast them at a general level. To do so, we use the structure of WSs in three dimensions:</p> <p></p> <ulist> <item> Semiotic: the use of symbols, graphics, and concrete objects understood as signs.</item> <p></p> <item> Instrumental: constructing and experimenting with models representing the relations between objects in a given configuration (geometric figures, graphs, programs, etc.). Artefacts are used.</item> <p></p> <item> Discursive: justification and proof using a theoretical frame of reference.</item> </ulist> <p>In an AWS, <emph>the semiotic dimension</emph> is characterized by notation that is similar to mathematical symbol systems, although the similarities are often misleading. For example, the concept of a variable exists, but differs from the mathematical notion. A particular difficulty concerns the assignment[<reflink idref="bib5" id="ref25">5</reflink>] statement, by which a value is given to a variable that replaces a previous value. Assignment is specific to AT, since it is linked to the idea of state change. In this context, Samurçay ([<reflink idref="bib25" id="ref26">25</reflink>]) found that these differences between computer and mathematical variables are confusing for beginners.</p> <p> <emph>The instrumental dimension</emph> in a MWS is different to that in an AWS. For example, geometry uses traditional or digital instruments to construct figures, while activities focused on functions involve creating graphs to be able to experiment with properties. In contrast, the instrumental dimension of an AWS entails the execution of an algorithm, either on a device or mentally. Execution does not imply passively running a program, instead, it is a process of construction, where implementations are run with selected data, in order to carry out an empirical exploration.</p> <p>Turning to <emph>the discursive dimension</emph>, reasoning and demonstrating are important in every domain of mathematics. In algorithmics, these discursive practices address the following specific questions: (<reflink idref="bib1" id="ref27">1</reflink>) Termination: does the algorithm terminate in a finite number of steps? (<reflink idref="bib2" id="ref28">2</reflink>) Effectiveness: in the case of termination, does the output actually correspond to the goal? And (<reflink idref="bib3" id="ref29">3</reflink>) Complexity: how many steps are necessary for termination, depending on the input? Computer scientists such as Dijkstra ([<reflink idref="bib6" id="ref30">6</reflink>]) propose methods to address these questions.</p> <p>In order to understand work in specific domains from an educational point of view, the theory of MWS distinguishes the following three levels:</p> <p></p> <ulist> <item> A <emph>reference</emph> WS is meant for people who are familiar with the domain.</item> <p></p> <item> A <emph>suitable</emph> WS helps to manage the work for beginners.</item> <p></p> <item> A <emph>personal</emph> WS is specific to individuals.</item> </ulist> <p>Menares Espinoza and Vivier ([<reflink idref="bib19" id="ref31">19</reflink>]) argue that beginners tackle a new domain by drawing upon their previous knowledge. Therefore, they cannot start at the reference level. Instead, learners must be set tasks that help them to develop a personal WS. A <emph>suitable</emph>[<reflink idref="bib6" id="ref32">6</reflink>] WS is a key element in the design and analysis of these tasks, and it helps to evaluate the appropriateness of the tasks that students are asked to undertake. Hence, this paper focuses on the determination of suitable AWS and MWS in the context of the IVT, and the bisection algorithm.</p> <hd id="AN0161580831-5">Iteration and adidacticity</hd> <p>We draw upon a French study (Nguyen and Bessot, [<reflink idref="bib21" id="ref33">21</reflink>]) of a situation in which 10th grade students were learning iteration.[<reflink idref="bib7" id="ref34">7</reflink>] The situation is also of interest with regard to the conditions needed for students to develop cognitive abilities through programming (issue 2, see above) because, in order to move away from naïve constructivism, the authors were inspired by the theory of didactical situations (Brousseau, [<reflink idref="bib3" id="ref35">3</reflink>]). Brousseau developed the notion of <emph>adidacticity</emph> to characterize students' actions when interacting with a <emph>milieu</emph> independently of the teacher's intentions regarding knowledge construction<emph>.</emph> It is important to note that adidacticity should not be understood as non-intervention, because the adidactical interaction is managed by the teacher, based on the knowledge that he/she intends to teach, and unfolds in a broader didactic situation that includes <emph>institutionalization:</emph> throughout the situation, students develop knowledge that remains closely associated with the adidactical interaction; in the institutionalization process, elements of knowledge are identified and associated with the body of knowledge that constitutes the learning objective of the situation.</p> <p>Table 1 Algorithms</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Guess my number: (simulation of player A)</p></th><th align="left"><p>Binary search: search for a <italic>hidden</italic> integer within interval</p><p>[0: 100] (simulation of player B)</p></th><th align="left"><p>Bisection algorithm: Computation of intervals [<italic>u;v</italic>] with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>v</mi><mo>-</mo><mi>u</mi><mo>≤</mo><mi>e</mi><mspace width="0.277778em" /><mspace width="0.277778em" /><mspace width="0.277778em" /><mspace width="0.277778em" /><mspace width="0.277778em" /><mspace width="0.277778em" /><mspace width="0.277778em" /><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>·</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq3.gif" /></p></th></tr></thead><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>h</mi><mi>i</mi><mi>d</mi><mi>d</mi><mi>e</mi><mi>n</mi><mo stretchy="false">←</mo><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>o</mi><mi>m</mi><mfenced close=")" open="("><mrow><mn>1</mn><mo>,</mo><mn>100</mn></mrow></mfenced></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq4.gif" /></p><p>enter <italic>guess</italic></p><p>while <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>g</mi><mi>u</mi><mi>e</mi><mi>s</mi><mi>s</mi><mo>≠</mo><mi>h</mi><mi>i</mi><mi>d</mi><mi>d</mi><mi>e</mi><mi>n</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq5.gif" />:</p><p> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>g</mi><mi>u</mi><mi>e</mi><mi>s</mi><mi>s</mi><mo><</mo><mi>h</mi><mi>i</mi><mi>d</mi><mi>d</mi><mi>e</mi><mi>n</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq6.gif" />:</p><p> display "too small"</p><p> else:</p><p> display "too big"</p><p> enter <italic>guess</italic></p><p>display "found"</p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>u</mi><mo stretchy="false">←</mo><mn>0</mn></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq7.gif" /></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>v</mi><mo stretchy="false">←</mo><mn>100</mn></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq8.gif" /></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>m</mi><mo stretchy="false">←</mo><mtext>int</mtext><mfenced close=")" open="("><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq9.gif" /></p><p>while <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>m</mi><mo>≠</mo><mi>h</mi><mi>i</mi><mi>d</mi><mi>d</mi><mi>e</mi><mi>n</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq10.gif" />:</p><p> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>m</mi><mo><</mo><mi>h</mi><mi>i</mi><mi>d</mi><mi>d</mi><mi>e</mi><mi>n</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq11.gif" />:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="17.0pt" /><mi>u</mi><mo stretchy="false">←</mo><mi>m</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq12.gif" /></p><p> else:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="17.0pt" /><mi>v</mi><mo stretchy="false">←</mo><mi>m</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq13.gif" /></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="8.5pt" /><mi>m</mi><mo stretchy="false">←</mo><mtext>int</mtext><mfenced close=")" open="("><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq14.gif" /></p><p>display <italic>m</italic></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>u</mi><mo stretchy="false">←</mo><mi>a</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq15.gif" /></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>v</mi><mo stretchy="false">←</mo><mi>b</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq16.gif" /></p><p>while <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>v</mi><mo>-</mo><mi>u</mi><mo>></mo><mi>e</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq17.gif" />:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="8.5pt" /><mi>m</mi><mo stretchy="false">←</mo><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq18.gif" /></p><p> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>f</mi><mfenced close=")" open="("><mi>u</mi></mfenced><mo>·</mo><mi>f</mi><mfenced close=")" open="("><mi>m</mi></mfenced><mo>></mo><mn>0</mn></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq19.gif" />:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="17.0pt" /><mi>u</mi><mo stretchy="false">←</mo><mi>m</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq20.gif" /></p><p> else:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mspace width="17.0pt" /><mi>v</mi><mo stretchy="false">←</mo><mi>m</mi></mrow></math><inline-graphic href="10649_2022_10192_Article_IEq21.gif" /></p><p>display [<italic>u;v</italic>]</p></td></tr></tbody></table> </ephtml> </p> <p>The algorithms presented here are illustrative, and do not use any specific programming language. Blocks are indented. The students used various programming environments, each with its own syntax, depending on the teacher's choice.</p> <p>The adidactical component in Nguyen and Bessot ([<reflink idref="bib21" id="ref36">21</reflink>]) consisted of three tasks that were inspired by an epistemological examination of the development of digital computers. Students were given a calculator with an erasable memory to store data, and the results of intermediate calculations. In the first task, they were asked to compute the values of simple, then more complex polynomial expressions and, finally, composite functions. In the second task, they were asked to tabulate similar functions, and then identify a block of instructions that could be repeated. The third task was to develop a structure for the block of instructions, along with a dedicated repetition instruction, and memory initialization.</p> <p>Nguyen and Bessot ([<reflink idref="bib21" id="ref37">21</reflink>]) found that the students associated the calculator's erasable memory with computer variables that were different to mathematical variables. Then, the notion of iteration emerged, along with the idea of iterative variables. Students had to think about mathematical objects, and structures such as the variation of a function, discretization, and composite functions. The situation proved to be effective in helping students understand basic programming concepts, and required them to develop their mathematical knowledge. It is therefore a source of inspiration for this study.</p> <hd id="AN0161580831-6">Research questions and method</hd> <p>Drawing on the above observations, our research focuses on the determination of suitable AWS and MWS, and the connections between them, based on the design, implementation, and evaluation of appropriate learning tasks.</p> <p>RQ1: What <emph>suitable</emph> AWS and MWS provide the conceptual foundations for tasks that engage secondary school students in adidactical work on the bisection algorithm and the IVT?</p> <p>RQ2: Based on these <emph>suitable</emph> AWS and MWS, what are the a posteriori characteristics of students' work? What does this work tell about the connections between algorithmics and mathematics learning?</p> <p>Addressing these two RQs should allow us to conceptualize the connections between algorithmics and everyday mathematics teaching/learning, which is the overarching objective of our research.</p> <p>The above RQs, and the issues outlined in the introduction lead us to select a design-based research (DBR) method. According to Godino et al. ([<reflink idref="bib10" id="ref38">10</reflink>]), the paradigmatic issues addressed by DBR include the identification of ways to improve learning in realistic contexts, and the validation of resources and emerging theories in the following three phases: (<reflink idref="bib1" id="ref39">1</reflink>) preliminary analysis and task design; (<reflink idref="bib2" id="ref40">2</reflink>) implementation; and (<reflink idref="bib3" id="ref41">3</reflink>) retrospective analysis. Godino et al. ([<reflink idref="bib10" id="ref42">10</reflink>]) include didactical engineering (DE) in DBR, stressing the centrality of epistemological questions. In DE, a study begins with an epistemological examination, and the task design is complemented by an a priori analysis of students' work. The a posteriori analysis examines the actual processes that emerge during implementation, which supplement the a priori analysis.</p> <p>Inspired by DE, we begin with an epistemological and curricular investigation to identify reference WSs. Then, we design a series of tasks for students, and determine a priori suitable WSs. The aim here is to answer RQ1. To address RQ2, we implement these tasks in classrooms, and analyze the work in the class. This analysis makes it possible to carry out an a posteriori examination of suitable WSs, whose method is detailed in Section 6.</p> <hd id="AN0161580831-7">From epistemology to reference WSs</hd> <p>We start by examining the relationship between the bisection algorithm and the IVT in mathematics. Then, we look at the role played by binary search in computer science. Finally, we study the IVT and the bisection algorithm both in the curriculum, and in current practice.[<reflink idref="bib8" id="ref43">8</reflink>]</p> <hd id="AN0161580831-8">Epistemological aspects of the IVT and the bisection algorithm</hd> <p>As footnote 2 states, we limit our study to the zeros of a function. This is consistent with how mathematicians have historically approached the IVT, namely by focusing on algorithms to approximate zeros of functions. Stevin (1585, quoted in Katz & Katz, [12]) considered the zeros of polynomial functions as valid numbers, since an algorithm is able to find as many digits as required in the decimal format.[<reflink idref="bib9" id="ref44">9</reflink>] Two centuries later, Cauchy introduced the IVT in his course at the <emph>Ecole Polytechnique</emph>, and provided a proof using the bisection algorithm (Barany, [<reflink idref="bib1" id="ref45">1</reflink>]).[<reflink idref="bib10" id="ref46">10</reflink>] This proof is based on two adjacent sequences[<reflink idref="bib11" id="ref47">11</reflink>] made up of successive values of the boundaries of intervals computed by bisection. It relies on the geometrical idea of continuity, rather than the application of theorems. Like Stevin and Cauchy, for many years, mathematicians took for granted the existence of a zero for a function that changes sign,[<reflink idref="bib12" id="ref48">12</reflink>] using the argument that an algorithm returns arbitrarily small intervals with the same property. The introduction of the concept of continuity was a response to challenges to the validity of this argument. An elementary course in real analysis should highlight the issue, and our aim is to design tasks that would illustrate this point.</p> <hd id="AN0161580831-9">Binary search in computer science</hd> <p>Binary search operates on a discrete, strictly ordered dataset. For instance, it can be used to search for a word in an alphabetically ordered list. Table 1 presents the algorithms used in the implementation. A comparison of the second and third columns of this table highlights that binary search and the bisection algorithm have the same structure: alternatives (<emph>if</emph> < <emph>condition1</emph> > : <emph>...else:...</emph>) are nested within an iteration (<emph>while</emph> < <emph>condition2</emph> > <emph>:...</emph>). The treatment of the two iterative variables <emph>u</emph> and <emph>v</emph> (footnote 7) is almost the same, with two differences. First, they do not operate on the same datatype: in binary search, the use of the <emph>int</emph> function ensures that all values are integers,[<reflink idref="bib13" id="ref49">13</reflink>] while the bisection algorithm returns non-integer numbers for <emph>e</emph> < 1. Second, the conditions are different (equality of two numbers in binary search, length of an interval below a threshold in bisection). We use this opportunity to introduce students to the bisection algorithm adidactically. Specifically, we propose a game in which we expect them to build a binary search, and then adapt it to the zeros of functions for a bisection algorithm.</p> <hd id="AN0161580831-10">The IVT and the bisection algorithm in the curriculum and practice</hd> <p>The French curriculum does not require a rigorous presentation of real numbers and continuous functions. However, adjacent sequences and the continuous mapping theorem (continuous functions preserve limits) are covered, which makes a proof of the IVT feasible. This proof is not compulsory because real analysis is oriented towards intuitive approaches and applications, rather than proof. While the bisection algorithm could be seen as a theoretical tool to address continuity, as in Stevin and Cauchy's work, it is limited in current practice to an application to continuous monotonic functions. Laval ([<reflink idref="bib17" id="ref50">17</reflink>], pp. 219–237) carried out a comprehensive study of the textbooks used in the French curriculum, and found that all include exercises like the following:Prove that the function defined by <emph>f(x)</emph> = <emph>x</emph><sups><emph>3</emph></sups>+<emph>x</emph><sups><emph>2</emph></sups>+<emph>x-2</emph> over the interval [0;2] has a unique zero. Write and execute an algorithm using bisection to obtain an interval of length smaller than 10<sups>−5</sups> that includes this zero.</p> <p>Textbooks usually include this exercise after introducing the algorithm using graphical examples of continuous monotonic functions. Typically, there is no discussion of the algorithm itself. This means that the bisection algorithm is <emph>presented</emph>, rather than <emph>constructed</emph>, and that connections between the underlying MWS and AWS are restricted to applications. With regard to the AWS, effectiveness (whether or not the interval contains one zero) is not discussed and, with regard to the MWS, students are unaware of preconditions (continuity, monotonicity). Binary search is not part of the mathematics curriculum although students are familiar with searching an ordered set. However, it is one thing to be able to carry out an informal search, and another thing to be able to program this search. According to our definition, the latter would involve producing a procedure that organizes changes in the state of a system for automatic execution. The task, namely designing a systematic procedure is, thus, useful preparation for the bisection algorithm.</p> <hd id="AN0161580831-11">Reference MWS and AWS</hd> <p>The study of epistemological aspects of the IVT and the bisection algorithm leads us to consider a reference MWS, where the algorithm can be used not only as a practical way to obtain a zero, but also as a theoretical tool to question the existence of a zero. A reference AWS should support the construction of a binary search algorithm, and its adaptation to the bisection algorithm. This AWS should include the use of appropriate programming structures and notation, as well as discussing its properties with respect to termination, effectiveness, and complexity. As evidenced in the preceding paragraph, these reference WSs break with current practice. However, they are consistent with other elements of the curriculum such as theorems in real analysis, and algorithmics as an activity to be integrated into other content. Thus, these reference WSs form the basis for designing tasks, and determining suitable WSs.</p> <hd id="AN0161580831-12">Tasks and suitable WSs</hd> <p>The overall task design is composed of three parts. The aim of the first part is to introduce students to the notion of binary search, and the bisection algorithm adidactically. It is based on a <emph>guess my number</emph> game: player A thinks of an integer in the range 0 to 100, and player B tries to guess what the integer is. Each time B proposes a number, A can answer either: "found" (then the game stops), "too big," or "too small." The second part questions the effectiveness of the bisection algorithm with regard to the properties of functions. Although, for a given <emph>e</emph>, the algorithm always terminates, returning an interval [<emph>u</emph>;<emph>v</emph>] with the property: <emph>v − u</emph> ≤ <emph>e</emph> and <emph>f</emph>(<emph>u</emph>)<sups>.</sups><emph>f</emph>(<emph>v</emph> ≤ 0), it is not necessarily effective. In other words, the interval may contain no zero, or not all of the zeros. The third part is focused on using the algorithm as a theoretical tool to find a proof in elementary real analysis. We outline the tasks and then determine a priori suitable WSs. As we will show, not all of the features of a WS are new with respect to the corresponding task. Some can be assumed to be standard at this grade, and others are adopted from a previous task in the series. When this is the case, we refer to components of a <emph>previous</emph> WS.</p> <hd id="AN0161580831-13">Part 1: from binary search to the bisection algorithm</hd> <p>In the first task, students are asked to program a simulation of player A (Table 1, column 1). In the second task, they have to do the same for player B (Table 1, column 2). In the third task, they are asked to adapt player B's algorithm to approximate the zero of a monotonic function.</p> <p>The simulation of player A is similar to binary search in structure, but no variables are involved. The simulation of player B is a binary search, and draws upon the notion of iterative variables to express the successive boundaries of the search interval. All values must be integers. The algorithm that is the focus of the third task is a bisection algorithm and, as noted above, the conditions are different from binary search.</p> <hd id="AN0161580831-14">Part 1: a priori suitable WSs</hd> <p>A suitable MWS is one where students can think mathematically about binary search and the bisection algorithm without the idea of automatic execution. This MWS includes calculating and expressing successive boundary values and conditions using a mathematical formalism (discursive and semiotic dimensions). A suitable AWS is one where students can think about developing a systematic procedure. Thus, the discursive dimension of the AWS involves iteration, and its semiotic dimension includes the use of iterative variables and structure markers.</p> <p>Iterative variables differ from the mathematical variables previously used by students (unknowns in equations, mute variables of functions, etc.) and are typical of an AWS, in the sense that a name is given to a data item whose value changes as the algorithm is executed. Moreover, this change is the result of a systematic calculation. In the semiotic dimension, this calculation uses the mathematical formalism. Part of this formalism is assumed to be new for students. For instance, the guess in the simulation of player B has to be an integer, and its expression uses a function (<emph>int</emph>), which is new, both with respect to the function and its notation.</p> <p>In the instrumental dimension, the AWS is where students can execute their algorithm by choosing sample data to test, or reflect on a procedure.</p> <hd id="AN0161580831-15">Part 2: effectiveness and properties of functions</hd> <p>In part 2, the task is to execute a bisection algorithm and answer the question "do the intervals computed by the algorithm contain the unique zero of the function?" for two hidden functions (Table 2). Students are also asked to discuss their answer after adding a loop to the program to graph the function. The first function is chosen in order to have two zeros, while the intervals computed by the algorithm contain only one. The second function is chosen in order to have no zero, and is not defined for a given value. The intervals returned by the algorithm contain this value.</p> <p>Graph</p> <p>As explained above, the challenge is twofold: (<reflink idref="bib1" id="ref51">1</reflink>) to break with the idea that termination means effectiveness for any algorithm, and (<reflink idref="bib2" id="ref52">2</reflink>) to draw attention to the preconditions for the existence of a zero.</p> <hd id="AN0161580831-16">Part 2: a priori suitable WSs</hd> <p>Beginning with the suitable AWS, in the discursive dimension students have to consider that termination and effectiveness are two different properties of algorithms. In the suitable MWS, they must consider that the existence and uniqueness of zeros depend on the function's properties. These new WSs build upon previous WSs for algorithms and functions at this grade. However, previous WSs do not have the reflective components needed for the new, suitable WSs because, as shown above, current practice only introduces continuous monotonic functions that change sign,[<reflink idref="bib14" id="ref53">14</reflink>] and therefore have one and only one zero; in which case, bisection is effective. We can thus expect that students will think that the intervals computed by the algorithm contain the unique zero of the function.</p> <p>We expect that the discursive dimension of the MWS, which students are familiar with, will enable them to consider the properties of a function, and that the semiotic dimension will help them to see a graph globally, depicting these properties. Thus, they should be able to reconsider the above answer after graphing, and relate this reconsideration to the function's properties. After completing the task set in part one, students should be able to follow the execution of an algorithm (the AWS's instrumental dimension). This ability could support their analytical examination of the function's properties, and help them to understand how these properties explain the algorithm's functioning. Thus, the execution of the algorithm is expected to have an effect on the discursive dimension of both WSs.</p> <hd id="AN0161580831-17">Part 3: a proof in elementary real analysis</hd> <p>Here, the task is to prove the existence of at least one zero for a continuous function that changes sign. This means proving an important IVT lemma.[<reflink idref="bib15" id="ref54">15</reflink>] As in Cauchy's work, the expected proof uses the sequences (<emph>u</emph><subs><emph>n</emph></subs>) and (<emph>v</emph><subs><emph>n</emph></subs>), which represent successive boundary values of the intervals computed by the bisection algorithm. A recurrence relation[<reflink idref="bib16" id="ref55">16</reflink>] can be deduced from the calculation of these values by the algorithm, and adjacent sequences can be proved by induction, which are then shown to converge towards a common limit <emph>c</emph>. The continuous mapping theorem states that <emph>f</emph>(<emph>u</emph><subs><emph>n</emph></subs>) and <emph>f</emph>(<emph>v</emph><subs><emph>n</emph></subs>) both converge towards <emph>f</emph>(<emph>c</emph>) as, for all <emph>n: f</emph>(<emph>u</emph><subs><emph>n</emph></subs>)<sups>.</sups><emph>f</emph>(<emph>v</emph><subs><emph>n</emph></subs>) ≤ <emph>0</emph>, passing to the limit <emph>f</emph>(<emph>c</emph>)<sups>2</sups> ≤ <emph>0</emph> and, since the square of a number cannot be strictly negative, <emph>f</emph>(<emph>c</emph>) = 0.</p> <hd id="AN0161580831-18">Part 3: a priori suitable WSs</hd> <p>Sequences and their properties are the components of the discursive dimension of the suitable MWS in part three. Adjacent sequences and the associated theorem (footnote 11) can be assumed to be available to students from their work in the previous MWS, where they were used to prove the convergence of given sequences. In the suitable MWS, the existence of a limit is not a goal in itself; here, a proof that the limit is a zero of the function is required. Adjacent sequences must be used more flexibly. In addition, the discursive dimension of the suitable AWS includes the bisection algorithm as a theoretical tool. This differs to part 2, where the focus was on the algorithm's properties.</p> <p>Thus, the discursive dimensions of a suitable MWS and AWS should be connected, and this connection is based on the values of iterative variables in the AWS, and adjacent sequences in the MWS. The semiotic dimensions of the MWS and the AWS should also be connected, to convert the iteration introduced in the bisection algorithm into a recurrence relation. The connection is already part of the previous MWS since this type of conversion is a standard task, but a relation involving two sequences is unusual. In the discursive dimension of the suitable MWS, completing the proof requires the use of theorems and proof techniques. Complementary to this, the instrumental dimension of the AWS allows students to visualize the properties of sequences by executing the algorithm for a function.</p> <p>In this section, suitable WSs are determined a priori as those that allow students to complete the tasks. Table 3 presents a summary. The next section evaluates these WSs, by analyzing students' work as they attempt the tasks.</p> <p>Table 3 Summary of the a priori analysis</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" colspan="2"><p>Dimensions</p></th><th align="left"><p>Suitable MWS</p></th><th align="left"><p>Suitable AWS</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p>Part 1</p></td><td align="left"><p>Discursive</p></td><td align="left"><p>Calculation of successive values</p></td><td align="left"><p>Iteration</p></td></tr><tr><td align="left"><p>Semiotic</p></td><td align="left"><p>(Integer) number "in the middle" of an interval</p><p>Functional notation</p></td><td align="left"><p>Markers of iteration and alternative</p><p>Iterative variables</p></td></tr><tr><td align="left"><p>Instrumental</p></td><td align="left" /><td align="left"><p>Execution</p></td></tr><tr><td align="left" rowspan="3"><p>Part 2</p></td><td align="left"><p>Discursive</p></td><td align="left"><p>Existence and unicity of zeros</p><p>Properties of functions</p></td><td align="left"><p>Distinction between termination and effectivity</p></td></tr><tr><td align="left"><p>Semiotic</p></td><td align="left"><p>Graph (global)</p></td><td align="left" /></tr><tr><td align="left"><p>Instrumental</p></td><td align="left"><p>Graph (analytic)</p></td><td align="left"><p>Execution</p></td></tr><tr><td align="left" rowspan="3"><p>Part 3</p></td><td align="left"><p>Discursive</p></td><td align="left"><p>Properties of sequences</p><p>Induction</p><p>Theorems and techniques for finding limits</p></td><td align="left"><p>Treatment inside iteration</p></td></tr><tr><td align="left"><p>Semiotic</p></td><td align="left"><p>Formalism of infinite sequences</p></td><td align="left"><p>Iterative variables</p></td></tr><tr><td align="left"><p>Instrumental</p></td><td align="left"><p>Properties of sequences</p></td><td align="left"><p>Execution on a given function</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0161580831-19">Implementation and a posteriori analyses</hd> <p>In France, at the time of the implementation, algorithmics was introduced in 10th grade. Functions were taught in 11th grade and the IVT in 12th grade. To be as realistic as possible, tasks related to parts 1, 2, and 3 were prepared respectively for 10th, 11th, and 12th grade students. In order to complete the study in one school year, we selected three classes (around 30 students each, with mixed abilities), one for each level (10th, 11th, and 12th grade).[<reflink idref="bib17" id="ref56">17</reflink>] The first part lasted 100 min, and the second and third parts lasted 45 min each. The class of 10th graders knew little about algorithmics, especially iteration. The class of 11th graders had been introduced to the bisection algorithm, but this was limited to current practice as exposed above. The class of 12th graders had been introduced to the IVT, as a way to prove the existence of zeros of given functions, but this did not include theoretical work.</p> <p>The teachers were volunteers responsible for the class, and were aware of the purposes of the research study. Accordingly to our choice of adidacticity, we asked them to present the tasks, have their students work in pairs, and possibly give some hints, without directing the class towards a solution. We expected that any hints would be particularly important in the first part, notably to introduce syntax, but only after students had identified the need for it. Teachers were also asked to encourage their students to start by writing down their solution on paper, before implementing and executing algorithms on a computer, but the class was allowed to alternate between paper and the computer (e.g., to check syntax). All sessions were audio-recorded and the students' written work was collected. A detailed report can be found in the second author's thesis (Laval, [<reflink idref="bib17" id="ref57">17</reflink>]).[<reflink idref="bib18" id="ref58">18</reflink>]</p> <p>Our aim is to answer RQ2 through an a posteriori interpretation of students' work. Based on the components of the suitable WSs that had been determined a priori, the a posteriori analysis focuses on how students adopt or mobilize these components. In this context, "adopt" a component means that the student's work is, at a given stage, consistent with what is expected from the a priori analysis. "Mobilize" a component means that the student had already adopted this component, or that it is part of a previous WS. In general, our analysis is based on the work carried out by the whole class. However, we also report on individual students, where these cases provide additional insight.</p> <hd id="AN0161580831-20">Part 1: the work done by the class (10th grade)</hd> <p>After being familiarized with the <emph>guess my number</emph> game (Section 5), students were asked to complete the following assignment:</p> <p></p> <ulist> <item> 1. Write a program simulating the actions of player A.</item> <p></p> <item> 2. Write a program simulating the actions of player B.</item> <p></p> <item> 3. Provide evidence of a zero for the function defined over [0;1] by <emph>f(x)</emph> = <emph>x</emph><sups><emph>3</emph></sups>+<emph>x</emph>+1 and adapt player B's program to calculate an interval of length smaller than 10<sups>−5</sups> containing this zero.</item> </ulist> <p>The students moved from naïve repetition to iteration in three steps. To simulate player A, they initially produced a program that contained a series of three or four alternatives followed by "..." or "etc.," before asking for the teacher's help. The teacher asked one of the students to play the game, and to record all of the actions, emphasizing their repetitive nature. At this point, the class switched to <emph>While</emph> or <emph>Repeat</emph> markers. To simulate player B, some began once again with a series of alternatives, but quickly switched to iteration.</p> <p>The class succeeded in creating a systematic procedure in the following way. First, they decided that player B had to guess a number that was in the previous interval in order to decrease the number of iterations, but were initially unable to come up with a proposal that could be implemented in a program. Then, after much reflection, the groups interacted with the teacher. The following excerpt illustrates a group discussion:<bold>S1</bold>: We shouldn't pick a number at random, we should pick the border between the two sub-intervals.<bold>Teacher</bold>: What do you mean by the border?<bold>S1</bold>: (after hesitation): the middle of the interval<bold>S2</bold>: Sir, does the software have a 'Middle' function?<bold>Teacher</bold>: Do you think you need a 'Middle' function to calculate the middle of an interval?<bold>S3</bold>: We don't need it, because ultimately the middle of the interval is the mean of its limits.</p> <p>When S1 says, "pick a number at random," he meant "any number." By discarding this possibility, and instead choosing the mean, the group demonstrated their awareness of the need for a systematic choice.</p> <p>Following a discussion involving the whole class, the students started to use iterative variables: <bold>S1</bold>: For each sub-step in the process, there are three values to consider: the lower bound, the upper bound and the center.<bold>Teacher</bold>: How would you name these values?<bold>S1</bold>: VI1 is the value of the lower bound, VS1 is the value of the upper bound and VC1 is the central value at the start of the procedure. Then they become VI2, VS2 and VC2 after the first proposition, then VI3, VS3 and VC3 after the third proposition, and so on until we get to the secret number.<bold>Teacher</bold>: In computer science, a variable does not necessarily function as it does in mathematics. In fact, it can change its value along the execution, but it always keeps the same name (...)<bold>S2</bold>: I think we should keep the same names for the three variables VI, VS and VC because, as the teacher said, variables are understood in the sense of storage.</p> <p>This discussion about different naming systems for values in mathematics and computer science allowed the class to make the transition from a series of variables to a single iterative variable for each data, the two bounds and the center.</p> <p>In the player B simulation, the need for values to always be integers throughout the iteration was tackled differently among the groups. Some implicitly assumed that the formula (<emph>a</emph> + <emph>b</emph>)/2 returns an integer whatever the values of <emph>a</emph> and <emph>b</emph>. Others were aware that this formula could return a non-integer, and one student proposed rounding the number "like in physics". Some students ignored the problem, and produced a program that did not terminate for some hidden integer values (footnote 13). When implemented on the computer, execution stopped after the maximum number of iterations allowed by the language. These students then had to investigate successive values of the interval to fix the problem.</p> <p>The functional notation was difficult for the class to grasp. After the class understood the need to round the average of two values in the player B simulation, the teacher mentioned the existence of the <emph>int</emph> function, and indicated its syntax, which the class did not immediately understand. This issue arose again in the third task. The programming environment allowed the direct implementation of the condition <emph>f(u)</emph><sups>.</sups><emph>f(m)</emph> > <emph>0</emph>, following a declaration similar to <emph>function f(x): return x</emph><sups><emph>3</emph></sups> + <emph>x</emph> + 1. This declaration makes <emph>f</emph> a user-defined function, and allows the computer to calculate values of <emph>f</emph> at any point during program execution, replacing the parameter <emph>x</emph> with the data found in brackets. Some students did not understand this, and persisted in writing s<emph>tring f(x)</emph> = <emph>x</emph><sups><emph>3</emph></sups> + <emph>x</emph> + 1, a declaration that creates a string of characters and not a function. This resulted in a syntax error that they were not able to correct.</p> <hd id="AN0161580831-21">A posteriori analysis of suitable WSs in part 1</hd> <p>Naïve repetition, a component of the MWS's discursive dimension, led the class to develop a semiotic expression ("..." or "etc.") that is unfeasible in an algorithm. In the simulation of player A, the teacher's intervention highlighted the discursive dimension of repetition, and the class adopted this component of the AWS, together with the associated semiotics. Then, the group was able to mobilize these components in the following tasks. Interplay between discursive and semiotic dimensions of the AWS and the MWS also unfolded as students progressively adopted iterative variables, once their teacher had emphasized the differences between mathematics and computer science.</p> <p>An interplay between the discursive dimensions of the two WSs occurred as the class was asked to express player B's guess in the form of a calculation (MWS), that supported a systematic procedure (AWS). Their choice of a value within the preceding interval was related to the complexity of the algorithm and, mathematically, rounding the mean to an integer was appropriate. In the semiotic dimension, students were required to change how they used the mathematical formalism, particularly regarding functional notation. While all adopted the notation <emph>int(...)</emph> in the binary search, some could not adopt the syntax of a user-defined function in the bisection algorithm. Understanding functional notation is an objective in 10th grade and, although not all of the students achieved it, we could interpret their efforts as the adoption of components of the MWS's semiotic dimension to comply with the AWS's constraints. Termination was a challenge, particularly as a failure to round the mean could create an infinite loop. The problem was addressed empirically: students adopted execution, a feature of the AWS instrumental dimension to reflect on, and correct their program.</p> <hd id="AN0161580831-22">Part 2: the work done by the class (11th grade)</hd> <p>Both tasks that make up this part consisted of executing the bisection algorithm on a hidden function, i.e., a function called by the program using an identifier <emph>f</emph>, but whose formula was not visible. In both tasks, the question to be answered was, "does the interval computed by the program contain the unique zero of the function?" The assignment was to provide an initial answer after executing the program, then to program the graph of the function and discuss the initial answer. In the first task, the hidden formula was <emph>f(x)</emph> = <emph>x</emph><sups><emph>3</emph></sups><emph>-3 x</emph><sups><emph>2</emph></sups> + <emph>2</emph> and the initial interval was [0;10]. In the second task, the hidden formula was <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mtext>x</mtext></mrow><mn>2</mn></msup><mtext>-2</mtext></mrow></mfrac></mrow></math> </ephtml> and the initial interval was [1;2] (Table 2).</p> <p>In all cases, the students' first answer was <emph>yes</emph> for both functions, but their reconsideration of this answer, after they had programmed the graph, differed. For the first function, two zeros were visible on the graph, while the intervals computed by the algorithm contained only one; in this case, the students concluded that the algorithm was wrong. Some stated that the program should return two intervals, one for each zero. In the case of the second function, which was not defined for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow></math> </ephtml> and had no zero, it was harder for the class to reconsider their answer because the graph built by the program did not accurately represent the function close to the <emph>x</emph>-axis. Instead, the program drew a line with a sharp positive slope crossing the <emph>x</emph>-axis. The students suspected a mistake, and looked at the values of the function computed by the algorithm at the boundaries <emph>u</emph> and <emph>v</emph> of the intervals. One student noted, for instance: "The values are smaller and smaller for <emph>f</emph>(<emph>u</emph>), and larger and larger for <emph>f</emph>(<emph>v</emph>), so the intervals obtained by the algorithm cannot contain a zero of <emph>f</emph>. Maybe it is a prohibited value. The graph might have a hole there." Finally, the class concluded that there was no zero. For instance, one student stated: "The program gave intervals around the prohibited value because <emph>f</emph> changes sign before and after this value."</p> <hd id="AN0161580831-23">A posteriori analysis of suitable WSs in part 2</hd> <p>The wrong answers given by the class confirms that the a priori AWS was not spontaneously adopted and, as also expected, they mobilized the semiotic dimension of a previous MWS to recognize a graph that contradicted this answer. For the first function, their conclusion that the algorithm was wrong is not relevant, since it actually computes intervals around a zero. Students were surprised to see that this zero was not unique, and implicitly assumed that when bisection computes intervals around a zero, this zero is unique. This is further evidence of their difficulty in adopting the suitable AWS. At this stage, they did not carry out the analytical examination of the algorithm's functioning that our a priori analysis predicted would happen. In contrast, the fact that the representation of the second function was less conclusive encouraged the class to adopt the analytical examination, a component of the instrumental dimension of the suitable AWS. They were then able to recognize that an arbitrary small interval that contains a value does not mean that the value is a zero, rather that it is a consequence of the sign change. This understanding clearly resulted in the effect on the discursive dimension of both WSs that we expected. Vocabulary such as "prohibited value," which is standard in mathematics teaching in France, along with the recognition of the function's properties close to this value, indicates that components of the previous MWS were mobilized.</p> <hd id="AN0161580831-24">Part 3: the work done by the class (12th grade)</hd> <p>In this last part, the assignment was as follows: Prove the existence of one or more zeros for a continuous function that changes sign[<reflink idref="bib19" id="ref59">19</reflink>] using adjacent sequences and the bisection method.</p> <p>Here, we analyze how students recognized that bisection and adjacent sequences could be used in a proof. The class was initially surprised to see the introduction of adjacent sequences and the bisection method in the assignment. First, they wondered if the IVT could be applied to sequences. Then, they asked their teacher for permission to print a bisection program from a previous session, and began to execute it manually for a particular function, looking for a clue. During this process, they started to think of successive boundary values <emph>u</emph> and <emph>v</emph> as two sequences, (<emph>u</emph><subs><emph>n</emph></subs>) and (<emph>v</emph><subs><emph>n</emph></subs>). To observe the behavior of these sequences, one student proposed running the program with an additional line that would display successive values of variables. After the class adopted this proposition, they were able to make the connection with the adjacent sequences theorem (footnote 11). They then inferred the preconditions of this theorem from observation of the particular function without a formal proof. Trying to go beyond observation, some students analyzed the algorithm in order to write a recurrent definition of the sequences, but they could not achieve a proof by induction.</p> <p>The following typical answer illustrates how the class completed the assignment.The sequences (<emph>u</emph><subs><emph>n</emph></subs>) and (<emph>v</emph><subs><emph>n</emph></subs>) are adjacent because (<emph>u</emph><subs><emph>n</emph></subs>) is increasing, (<emph>v</emph><subs><emph>n</emph></subs>) is decreasing and (<emph>u</emph><subs><emph>n</emph></subs>−v<subs><emph>n</emph></subs>) becomes closer to zero when <emph>n</emph> becomes bigger and bigger. Then these two sequences converge towards a common limit <emph>c</emph>. Because <emph>f</emph> is continuous, <emph>f</emph>(<emph>u</emph><subs><emph>n</emph></subs>) and <emph>f(v</emph><subs><emph>n</emph></subs>) converge towards <emph>f</emph>(<emph>c</emph>), which is zero. The theorem is proved with the computer for a particular function.</p> <p>This example contains a hybrid collection of facts (properties of sequences, the value of <emph>f</emph>(<emph>c</emph>)) observed during the execution of the algorithm, and deductions (convergence of the sequences). The last sentence indicates that the class was unsure of the status of their proof.</p> <hd id="AN0161580831-25">A posteriori analysis of suitable WSs in part 3</hd> <p>Students were initially surprised that the assignment mentioned bisection as a way to develop a proof of the IVT. This indicates that, although it had been the object of a discussion in part 2 that mobilized the discursive dimensions of both WSs, they did not spontaneously adopt the algorithm as a theoretical tool. First, they considered the algorithm in the AWS's discursive dimension by consulting a printout. However, they quickly switched to the instrumental dimension, mobilizing the same observation of values as they did in part 2. This had an effect on the MWS's discursive dimension, as they were able to recognize adjacent sequences, and deduce a common limit. Some students mobilized the semiotic dimensions of both WSs in connection, to convert from iteration to a recurrent definition of sequences.</p> <p>Returning to the MWS's discursive dimension, although the class did not prove the preconditions for adjacent sequences, they were able to mobilize the continuous mapping theorem to prove the convergence of the sequence of values of the function. We see here a connection with the AWS's discursive dimension since, after part 2, the group became aware that a sign change is not a sufficient condition for effectiveness, and therefore that continuity is an important precondition. Mobilizing the AWS's instrumental dimension to visualize values of <emph>f</emph>(<emph>u</emph><subs><emph>n</emph></subs>) and <emph>f(v</emph><subs><emph>n</emph></subs><emph>)</emph>, they concluded that the common limit of the sequences was zero, but overlooked a formal proof.</p> <p>Table 4 presents a summary of the a posteriori analysis.</p> <p>Table 4 Summary of the a posteriori analysis</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Dimensions</p></th><th align="left" colspan="3"><p>The work in the class</p></th></tr><tr><th align="left" /><th align="left"><p>Part 1</p></th><th align="left"><p>Part 2</p></th><th align="left"><p>Part 3</p></th></tr></thead><tbody><tr><td align="left"><p>Discursive</p></td><td align="left"><p>Discard naive repetition (AWS)</p><p>Distinguish between math and computer variables (AWS–MWS)</p></td><td align="left"><p>Question both effectiveness and properties of functions (AWS–MWS)</p></td><td align="left"><p>Link adjacent sequences and bisection</p><p>(AWS–MWS)</p><p>Use theorems (MWS)</p></td></tr><tr><td align="left"><p>Semiotic</p></td><td align="left"><p>Use iteration markers (AWS)</p><p>Understand functional notation (AWS–MWS)</p></td><td align="left"><p>Assess globally the functions' properties on graphs (MWS)</p></td><td align="left"><p>Convert between math and computer formalism (AWS–MWS)</p></td></tr><tr><td align="left"><p>Instrumental</p></td><td align="left"><p>Use feedback of execution to question termination in relation with numbers (AWS–MWS)</p></td><td align="left"><p>Assess analytically the functions' properties through execution (AWS–MWS)</p></td><td align="left"><p>Visualize properties of functions by execution (AWS–MWS)</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0161580831-26">Discussion and conclusion</hd> <p>RQ1 was: What suitable AWS and MWS provide the conceptual foundations for tasks that engage secondary school students in adidactical work on the bisection algorithm and the IVT?</p> <p>To answer this question, we started by the determination of reference WSs. This enabled us to design a series of tasks that break with current practice, but are feasible in the French curriculum. Then, suitable MWS and AWS were determined for the tasks in the series. The suitable MWS included algebraic and functional notation, properties of functions associated with the existence and uniqueness of zeros, the use of graphs at both global and analytic levels, the properties and formalism of sequences, and theorems and techniques for finding limits. The suitable AWS included an understanding of iteration, the use of specific markers and iterative variables, key properties of algorithms (complexity, termination, and effectiveness), and the use of feedback. These two suitable WSs were organized as a function of the three dimensions (discursive, semiotic, and instrumental) and established connections between algorithmics and mathematics (Table 3). The fact that they allowed in-depth, a priori analyses suggests that they provide the conceptual foundations mentioned in RQ1.</p> <p>More generally, we argue that suitable AWS and MWS determined by our method are able to effectively capture the essence of mathematical and algorithmic thinking, with respect to a given content.</p> <p>RQ2 was: Based on these suitable AWS and MWS, what are the a posteriori characteristics of students' work? What does this work tell about the connections between algorithmics and mathematics learning?</p> <p>We answered this question by investigating how students adopted and mobilized the components making up the suitable WSs (Table 4). Our findings demonstrate that suitable WSs were not spontaneously adopted. We observed that 10th graders needed time to discard naïve repetition, and to develop a systematic way to change the value of variables. The class of 11th graders initially seemed to think that the bisection algorithm should always return the unique zero of the function, regardless of its properties. The class of 12th graders struggled to grasp how an algorithm can be used in a proof. Our tasks gradually introduced students to the components of the suitable WSs.</p> <p>Table 4 outlines the distribution of the components as a function of the semiotic, instrumental, and discursive dimensions. Interplays can be seen between the AWS and the MWS in each dimension. For example, we noted that 10th graders eventually understood the distinction between mathematical and algorithmic variables, and were able to recognize that although the two semiotic systems share similar signs, the meaning is different. They were also challenged by the mathematical formalism, particularly functional notation, because of the need to express calculations and conditions in a form that could be written as an algorithm. In the 11th grade, the class related an algorithm that terminated with a wrong answer to the properties of the function.</p> <p>While building their proof, 12th graders had to make the connection between mathematical sequences and iteration, in both discursive and semiotic dimensions. The interplay between the instrumental dimensions of the AWS and the MWS consisted of the link that the students made between a mathematical idea (for instance, a "prohibited value" for a function), and what they saw during the execution of their program (increasing absolute values at the boundaries of the intervals around the missing value). We found that this interplay had a subsequent effect on the discursive dimension, as the visualization of the algorithm gave students new insight into the mathematical idea. Then, the a posteriori analysis of the students' work in the light of the two suitable WSs provides evidence of connections between mathematical and algorithmic thinking as they operate in students' work, and of potential new opportunities in curricula that combine computer science and mathematics.</p> <p>With respect to previous research, the outcomes of the first part of our study are consistent with results obtained by Nguyen and Bessot ([<reflink idref="bib21" id="ref60">21</reflink>]). In this earlier study, as in our work, students were asked to solve tasks that required the implementation of an iterative structure, and the use of an appropriate functional formalism. In Nguyen and Bessot's study, the iterative structure was a loop with a fixed number of repetitions: <emph>repeat</emph> < <emph>n</emph> > , < <emph>block</emph> >. In our study, the iterative structure was more general, since continuation depends on a condition that is independent of the number of iterations. Moreover, the second and third parts of our work go beyond Nguyen and Bessot's study, as we directly target mathematical content. In addition, as used in our study, the suitable WSs provide a comprehensive overview of the connections between mathematics and algorithmics, taking advantage of the three dimensions of WSs.</p> <p>More recently, Noss et al. ([<reflink idref="bib22" id="ref61">22</reflink>]) reported a study that aimed to connect programming and mathematics, but at different grades, and using a different methodology. The authors discuss the ScratchMaths project, which involved 3000 students aged 9–10. Although the project was found to have a positive and significant impact, measured by a computational thinking (CT) test, statutory tests found no impact on standard mathematics attainment. However, as we noted in our introduction, there is currently no consensus regarding what CT is, precisely. In practice, an in-depth analysis of the CT test shows that many items relate to what we consider to be the instrumental and discursive dimensions of the AWS, while some items relate to the MWS, especially those that address spatial and logical reasoning. These observations suggest that, from a WS perspective, an analysis of student performance that is based on the CT test used in Noss et al. ([<reflink idref="bib22" id="ref62">22</reflink>]) could confirm the contribution of programming to mathematics learning, although no impact may be detected by statutory tests.</p> <p>Our study has two limitations. First, our analyses highlight that, in an adidactical situation, students adopted components of the two WSs, but neglect the question of institutionalization. In theory, institutionalization should have made students aware of the new elements of knowledge associated with these WS components (e.g., the properties of algorithms, such as termination or effectiveness). In the context of our theoretical framework, both institutionalization and adidacticity are important in order to move away from the naïve constructivism discussed in our introduction. Further work should focus on institutionalization. This would include proposals for learning objectives involving both mathematics and algorithms, and empirical study of the process by which these objectives are pursued, in conjunction with the adidactic situation.</p> <p>Second, our study avoids the question of students' cognitive development when adopting or mobilizing the components of WSs. Inspired by Kuzniak and Nechache ([<reflink idref="bib14" id="ref63">14</reflink>]) in geometry, further work should identify individual trajectories, and investigate students' personal AWS and MWS with respect to different profiles of mathematical and algorithmic thinking. Much remains to be explored in this domain, with implications for the development both of theory, and of appropriate tasks and learning objectives, in the present context of curricula that bring together mathematics and computer science.</p> <hd id="AN0161580831-27">Acknowledgements</hd> <p>The authors thank the anonymous referees and the editor who provided useful and detailed comments on earlier versions of the manuscript.</p> <hd id="AN0161580831-28">Data availability</hd> <p>Data that support the findings of this study are available in Laval ([<reflink idref="bib17" id="ref64">17</reflink>]) on the repository https://tel.archives-ouvertes.fr/ with the identifier tel-01943971.</p> <hd id="AN0161580831-29">Declarations</hd> <p></p> <hd id="AN0161580831-30">Conflict of interest</hd> <p>The authors declare no competing interests.</p> <hd id="AN0161580831-31">Publisher's note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0161580831-32"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Barany MJ. 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Other data are reported in Stephens ([26]).</bibtext> </blist> <blist> <bibtext> The IVT guarantees, for a continuous function <emph>f</emph> defined on an interval [<emph>a;b</emph>] and for every <emph>m</emph> in the interval [<emph>f(a); f(b)</emph>], the existence of one or more values <emph>c</emph> in [<emph>a;b</emph>] such that <emph>f(c)</emph> = <emph>m</emph>. In this paper, we consider a lemma that guarantees, for a function <emph>f</emph> defined on an interval [<emph>a;b</emph>] the existence of one or more zeros under the following sufficient conditions: <emph>f</emph> is continuous and <emph>f(a)</emph><sups>.</sups><emph>f(b)</emph> ≤ 0. A corollary states that the zero is unique under the supplementary sufficient condition that the function is strictly monotonic. The general IVT can be deduced from the lemma, using a simple algebraic manipulation which is not considered here, due to our focus on real analysis.</bibtext> </blist> <blist> <bibtext> The French curriculum includes algorithm design as one of four overarching skills, but does not indicate any particular learning goals. It reflects the idea that teachers should avoid teaching content that is too different to ordinary mathematics, and the fact that specifying learning goals would not be easy given the lack of tradition and experience in this domain.</bibtext> </blist> <blist> <bibtext> One could argue that algorithmics in mathematics education involves a single WS, with features that are associated with either algorithmic or mathematical thinking. This position would ignore the fundamental coherence of each WS. The metaphor of a professional with two specialties illustrates our point: A French elementary teacher teaches both mathematics and French in the same classroom; situations such as problem solving involve objectives in both domains, and the teacher has to act in two WSs (mathematics and French) that cannot be merged, but have to be coordinated.</bibtext> </blist> <blist> <bibtext> Many notation systems use the <emph>equal to</emph> sign. In this case, for example, <emph>x</emph> = <emph>x</emph> + 1 is a valid instruction for incrementing a variable. In this paper, we use an arrow rather than the <emph>equal to</emph> sign: <emph>x</emph> ← <emph>x</emph> + 1.</bibtext> </blist> <blist> <bibtext> Throughout this paper, "suitable" refers to a WS level as understood in MWS theory, and not in the general sense of  "appropriate.".</bibtext> </blist> <blist> <bibtext> See Table 1 for examples of iteration and iterative variables. An iteration starts with a specific marker (<emph>while</emph> in the examples shown), followed by a block of instructions. An iterative variable is a variable whose value changes with each iteration. In the bisection algorithm, <emph>u</emph> and <emph>v</emph> are iterative variables, in contrast to <emph>a</emph> and <emph>b</emph> (constants), and <emph>m</emph> (a local variable). Unlike bisection, iteration in Nguyen and Bessot ([21]) is a loop with a fixed number of repetitions: <emph>repeat</emph> < <emph>n</emph> > <emph>,</emph> < <emph>block</emph> > .</bibtext> </blist> <blist> <bibtext> In principle, identifying reference WSs would begin by reviewing previous research. However, we found no study that directly investigated the IVT and the bisection algorithm. Douady ([7]) implicitly considers the use of the IVT by students aged 8–10 in the generation of decimal numbers to approach a solution. However, while the class is able to figure out new numbers by bisection, there is no systematization. The goal is to encourage them to construct decimal numbers, rather than to raise, and investigate questions relative to the IVT. Other studies on the IVT concern undergraduate students in relation to the completeness of the set of real numbers, and do not include the bisection algorithm. Furthermore, we were unable to identify any studies on binary search, as research on computer science education is in its infancy.</bibtext> </blist> <blist> <bibtext> Unlike the bisection algorithm, Stevin's algorithm divides each interval into ten steps, adding a new decimal place at each step.</bibtext> </blist> <blist> <bibtext> Barany ([1]) refers to "a method of approximating roots." We do not discuss methods and algorithms here, as it is clear to us that the method was systematic, and can therefore be qualified as an algorithm.</bibtext> </blist> <blist> <bibtext> Two sequences of real numbers (<emph>u</emph><subs><emph>n</emph></subs>) and (<emph>v</emph><subs><emph>n</emph></subs>) are adjacent when (<emph>u</emph><subs><emph>n</emph></subs>) is increasing, (<emph>v</emph><subs><emph>n</emph></subs>) is decreasing, and the sequence (<emph>u</emph><subs><emph>n</emph></subs><emph>—v</emph><subs><emph>n</emph></subs>) converges towards zero. The theorem states that two adjacent sequences converge towards a common limit. It is equivalent to the idea of the completeness of the set of real numbers, and is a trivial corollary of the nested closed intervals theorem.</bibtext> </blist> <blist> <bibtext> A function changes sign when it is negative at one boundary of the interval of definition, and positive at the other: <emph>f</emph> defined on [<emph>a;b</emph>] and <emph>f(a)</emph><sups>.</sups><emph>f(b)</emph> ≤ 0.</bibtext> </blist> <blist> <bibtext> Without this constraint the algorithm may not terminate. For instance, searching for a hidden integer inside the interval [0;100], at the <emph>n</emph><sups>th</sups> iteration <emph>u</emph> = 100<emph>p</emph>/2<sups><emph>n</emph></sups><emph>; v</emph> = 100<emph>(p</emph> + 1<emph>)</emph>/2<sups><emph>n</emph></sups>,<emph>p</emph> is an integer. This can be proved by induction, and it follows that only 25, 50, and 75 can be found by a bisection algorithm that would overlook this constraint.</bibtext> </blist> <blist> <bibtext> See the definition in footnote 12.</bibtext> </blist> <blist> <bibtext> Footnote 2 explains why this lemma is considered, and not the full IVT.</bibtext> </blist> <blist> <bibtext> The relationship can be expressed in compact form, using an alternative statement inside a recurrence between ordered pairs, as follows: (<emph>u</emph><subs><emph>n</emph>+1,</subs><emph>v</emph><subs><emph>n</emph>+1</subs>) = if <emph>f(u</emph><subs><emph>n</emph></subs><emph>f(v</emph><subs><emph>n</emph></subs><emph>)</emph> > 0: (<emph>m</emph>, <emph>v</emph><subs><emph>n</emph></subs>) else (<emph>u</emph><subs><emph>n,</emph></subs><emph> m</emph>), <emph>m</emph> being the mean of (<emph>u</emph><subs><emph>n</emph>,</subs><emph>v</emph><subs><emph>n</emph></subs>).</bibtext> </blist> <blist> <bibtext> This means that our results for the three parts refer to different classes. The 11<sups>th</sups> graders had to be prepared for part 2 by completing part 1. Similarly, 12<sups>th</sups> graders completed parts 1 and 2 in preparation for part 3 (Laval, [17]). We do not report here on this preparation. In the context of our framework, we assume that after this preparation, 11<sups>th</sups> graders had adopted the components of suitable WSs covered in part 1, and that 12<sups>th</sups> graders had adopted the components of suitable WSs covered in part 2 (Table 3).</bibtext> </blist> <blist> <bibtext> In particular, see pp. 278–345 for part 1, pp. 377–416 for part 2, and pp. 450–464 for part 3. The assignments and the transcripts of classroom discussion are translated from the thesis.</bibtext> </blist> <blist> <bibtext> Being aware of the equivalence, the students identified the IVT and this lemma.</bibtext> </blist> </ref> <aug> <p>By Jean-Baptiste Lagrange and Dominique Laval</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib23" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib28" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib27" firstref="ref5"></nolink> <nolink nlid="nl4" bibid="bib24" firstref="ref8"></nolink> <nolink nlid="nl5" bibid="bib11" firstref="ref9"></nolink> <nolink nlid="nl6" bibid="bib13" firstref="ref14"></nolink> <nolink nlid="nl7" bibid="bib16" firstref="ref15"></nolink> <nolink nlid="nl8" bibid="bib18" firstref="ref20"></nolink> <nolink nlid="nl9" bibid="bib15" firstref="ref22"></nolink> <nolink nlid="nl10" bibid="bib20" firstref="ref23"></nolink> <nolink nlid="nl11" bibid="bib25" firstref="ref26"></nolink> <nolink nlid="nl12" bibid="bib19" firstref="ref31"></nolink> <nolink nlid="nl13" bibid="bib21" firstref="ref33"></nolink> <nolink nlid="nl14" bibid="bib10" firstref="ref38"></nolink> <nolink nlid="nl15" bibid="bib12" firstref="ref48"></nolink> <nolink nlid="nl16" bibid="bib17" firstref="ref50"></nolink> <nolink nlid="nl17" bibid="bib14" firstref="ref53"></nolink> <nolink nlid="nl18" bibid="bib22" firstref="ref61"></nolink>
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  Data: Connecting Algorithmics to Mathematics Learning: A Design Study of the Intermediate Value Theorem and the Bisection Algorithm
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  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Lagrange%2C+Jean-Baptiste%22">Lagrange, Jean-Baptiste</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-1789-877X">0000-0003-1789-877X</externalLink>)<br /><searchLink fieldCode="AR" term="%22Laval%2C+Dominique%22">Laval, Dominique</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Educational+Studies+in+Mathematics%22"><i>Educational Studies in Mathematics</i></searchLink>. Feb 2023 112(2):225-245.
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 21
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2023
– Name: TypeDocument
  Label: Document Type
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  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Design%22">Design</searchLink><br /><searchLink fieldCode="DE" term="%22Secondary+School+Students%22">Secondary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Programming%22">Programming</searchLink>
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  Label: Geographic Terms
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  Data: <searchLink fieldCode="DE" term="%22France%22">France</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1007/s10649-022-10192-y
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  Data: 0013-1954<br />1573-0816
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  Label: Abstract
  Group: Ab
  Data: Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16-19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions.
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  Data: 2023
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  Data: EJ1364086
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        Value: 10.1007/s10649-022-10192-y
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      – Text: English
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      Pagination:
        PageCount: 21
        StartPage: 225
    Subjects:
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematical Concepts
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      – SubjectFull: Design
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      – SubjectFull: Secondary School Students
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      – SubjectFull: Foreign Countries
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      – SubjectFull: France
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      – TitleFull: Connecting Algorithmics to Mathematics Learning: A Design Study of the Intermediate Value Theorem and the Bisection Algorithm
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