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Meaning Generation in the Interplay between Problem Solving and Posing in a Constructionist Environment

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Title: Meaning Generation in the Interplay between Problem Solving and Posing in a Constructionist Environment
Language: English
Authors: Ioannis Papadopoulos (ORCID 0000-0001-6548-1499)
Source: Computers in the Schools. 2025 42(3):233-256.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 24
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Grade 10
High Schools
Secondary Education
Descriptors: Problem Solving, Concept Formation, Comprehension, Grade 10, High School Students, Constructivism (Learning), Mathematics Education, Mathematics Skills, Mathematics Achievement, Computer Software, Troubleshooting
DOI: 10.1080/07380569.2024.2412300
ISSN: 0738-0569
1528-7033
Abstract: In this paper the potentiality of meaning generation in a constructionist environment through an interplay between problem solving and problem posing is examined. The meaning-making process is examined by following how a Grade-10 student is engaged in an iterative cycle of reformulating and solving the initial problem within a Logo-based microworld. The findings show that the contribution of this iterative cycle is in favor of the generation of mathematical meaning. The shift from reformulating to solving and vice versa supported the student to initially use his mathematical knowledge for achieving certain goals, then to discriminate and use parts of his Logo program through generalizing and debugging efforts to achieve new purposes.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1489575
Database: ERIC
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  Value: <anid>AN0187842866;cit01jul.25;2025Sep11.06:15;v2.2.500</anid> <title id="AN0187842866-1">Meaning Generation in the Interplay Between Problem Solving and Posing in a Constructionist Environment </title> <p>In this paper the potentiality of meaning generation in a constructionist environment through an interplay between problem solving and problem posing is examined. The meaning-making process is examined by following how a Grade-10 student is engaged in an iterative cycle of reformulating and solving the initial problem within a Logo-based microworld. The findings show that the contribution of this iterative cycle is in favor of the generation of mathematical meaning. The shift from reformulating to solving and vice versa supported the student to initially use his mathematical knowledge for achieving certain goals, then to discriminate and use parts of his Logo program through generalizing and debugging efforts to achieve new purposes.</p> <p>Keywords: Problem solving; problem posing; constructionism; half-baked microworlds; meaning-making</p> <hd id="AN0187842866-2">Introduction</hd> <p>According to Lotman ([<reflink idref="bib29" id="ref1">29</reflink>]), the role of a problem's text is twofold: "to convey meanings adequately, and to generate new meanings" (p. 34). Abramovich and Cho ([<reflink idref="bib3" id="ref2">3</reflink>]) refer to these roles as the univocal and dialogic function of the text. In the latter case, it seems that the boundaries of the problem are flexible which are easily crossed making thus the given problem a "thinking device or a generator of new meaning that can animate problem posing followed by problem-solving activity" (p. 316). In general, there is a close connection between mathematical knowledge and the interplay between problem solving and problem posing. This connections stems from the fact that most of the mathematicians work mainly by formulating their own questions and problems and then try to solve them rather than by spending time to solve problems posed by others (Borwein, Liljedahl, and Zhai [<reflink idref="bib5" id="ref3">5</reflink>]). And it is in this interplay that flexible knowledge occurs, and meaning is generated (Silver [<reflink idref="bib41" id="ref4">41</reflink>]). Liljedahl and Cai ([<reflink idref="bib28" id="ref5">28</reflink>]) in their survey paper emphasize the significant contribution of technology in problem solving and mention how the use of digital environments has an impact in the way we solve problems (see also the work of Kaufmann and Stenseth ([<reflink idref="bib18" id="ref6">18</reflink>]) who investigate the role of programming in students' mathematical problem-solving). However, for the problem-posing part they still invite researchers to examine how technology can be used in problem-posing activities. There are studies in the traditional environment of paper-and-pencil that investigate the role of problem solving and problem posing as a meaning-making process. The research shows that problem solving provides a basis for meaning making (Wheatley [<reflink idref="bib48" id="ref7">48</reflink>]) for both students (Koichu [<reflink idref="bib21" id="ref8">21</reflink>]) and teachers (Sevinc et al. [<reflink idref="bib39" id="ref9">39</reflink>]). Similar claims for problem posing can be found in the work of Lavy and Bershadsky ([<reflink idref="bib27" id="ref10">27</reflink>]), Cifarelli and Cai ([<reflink idref="bib9" id="ref11">9</reflink>]), or the work of Gade and Blomqvist ([<reflink idref="bib13" id="ref12">13</reflink>]). But, in the context of technology, the number of such studies is very limited and most of them rely mainly on the use of Digital Geometry Software, as it is in the work of Santos-Trigo, Moreno-Armella, and Camacho-Machín ([<reflink idref="bib36" id="ref13">36</reflink>]) or the use of spreadsheets (Abramovich and Cho [<reflink idref="bib3" id="ref14">3</reflink>], [<reflink idref="bib4" id="ref15">4</reflink>]).</p> <p>The focus in this paper is on examining the interplay between solving and posing in a constructionist environment. In the context of Constructionism, Hoyles and Noss ([<reflink idref="bib17" id="ref16">17</reflink>]) examine students' mathematical meaning-making processes while the students are working with expressive digital media in constructionism environments, such as Logo-based microworlds. Very often, these microworlds involve purposefully placed bugs in an artifact, they are called 'half-baked' and the students are engaged in the role of de-buggers (Kynigos [<reflink idref="bib22" id="ref17">22</reflink>], [<reflink idref="bib24" id="ref18">24</reflink>]). Students, in their efforts to solve the problem discover the bug. So, to overcome the difficulty they reformulate the problem according to how they think the correct solution might be achieved. Then, they attack to the new version of the problem. Very often, this first reformulation does not guarantee the general solution of the problem. Silver et al. ([<reflink idref="bib42" id="ref19">42</reflink>]) talk about the considerable time experts spend engaging in problem formulation and reformulation since "it has become increasingly common to view problem solving as a process involving establishing a series of successively more refined problem representations" (p. 294). Therefore, the students must be engaged in a series of iterative cycle of review—reformulate—solve. The act of reformulation of the problem assimilates to problem posing (Silver [<reflink idref="bib40" id="ref20">40</reflink>]). So, in this paper we accept that this action of finding the bug and reformulating the initial problem is considered an act of problem posing. Our aim is to highlight the use of half-baked microworlds as an environment triggering an interplay between problem solving and problem posing (mainly through the application of the what-if-not technique) thus facilitating students' meaning-making.</p> <hd id="AN0187842866-3">Theoretical background</hd> <p>The theoretical framework of this study is based mainly on the notion of meaning-making during the interplay between problem solving and problem posing when half-baked microworlds in the context of constructionism are used. Hoyles ([<reflink idref="bib16" id="ref21">16</reflink>]) claims that the use of digital tools opens windows on to students' meaning. Technology powerfully supports construction of knowledge by the students through a variety of opportunities to select relevant information, manipulate different (interconnected) representations of knowledge, and conduct systematic experimentation (Davis [<reflink idref="bib11" id="ref22">11</reflink>]). As Noss and Hoyles ([<reflink idref="bib31" id="ref23">31</reflink>]) claim, meaning-making in mathematics is supported when diverse, interconnected affordances are integrated in a learning environment. van Oers ([<reflink idref="bib47" id="ref24">47</reflink>]) defines personal meaning-making as the process of attaching personal value to actions and goals in the mathematical activity. Alternatively, Goldberg ([<reflink idref="bib14" id="ref25">14</reflink>]) defines it as the ability to read, process, and solve mathematical situations, thus constructing personal meaning of mathematical experiences. The issue of generating mathematical meaning in a rich digital context is multidimensional. In Drier's ([<reflink idref="bib12" id="ref26">12</reflink>]) exact words: "[...] the tools available in a computer environment, meaningful instructional and playful activities, students' understanding, and their social and computer interactions all operate interactively as potential meaning-making agents for students' construction of concepts" (p. 691). Lantz-Andersson, Linderoth, and Säljö ([<reflink idref="bib26" id="ref27">26</reflink>]) working with upper secondary school students found that the digital environment co-determines the meaning-making process that students engage in while solving problems by introducing new dimensions for the students to consider. Khan and Mason ([<reflink idref="bib19" id="ref28">19</reflink>]) in their work on proof discuss how digital technology provides exciting opportunities for generating meaning and constructing an understanding through acquired knowledge. The key affordances in a digital environment that support the generation and expression of mathematical meaning are mathematical figures and multiple representations (Carreira et al. [<reflink idref="bib8" id="ref29">8</reflink>]). The former is now an instance of a class of objects rather than an object. The latter resolves the confusion between representation of a concept and the concept itself. The students are able now to co-vary more than one linked representation and this supports the above-mentioned resolution (Kynigos [<reflink idref="bib23" id="ref30">23</reflink>]).</p> <p>In this paper we investigate how students' meaning-making is evolving while they are engaged in a constructionist activity. Constructionism is a pedagogical movement stemming from the work of Papert and Harel ([<reflink idref="bib34" id="ref31">34</reflink>]). The key idea of perceiving digital technology as expressive media used by students to generate mathematical meaning goes back to Papert ([<reflink idref="bib33" id="ref32">33</reflink>]) and the work of Abelson and DiSessa ([<reflink idref="bib1" id="ref33">1</reflink>]) who mainly refer to Logo-based environments. Indeed, in such environments the students are constructing, tinkering with, and remixing graphical models having thus the opportunity not only to express but also to generate mathematical meaning (Kynigos [<reflink idref="bib23" id="ref34">23</reflink>]). So, in this sense constructionism can be seen as a special case of constructivism. In constructivism learning is perceived as the generation of meaning from students while they try to bring cohesion to the way they see the world, whereas constructionism concerns a situation where the learners make or tinker with an object or entity. This was considered by Papert as one of the ways for the learners to manifest and made public their thinking. Healy and Kynigos ([<reflink idref="bib15" id="ref35">15</reflink>]) present the option of microworlds as computational environments embedding a coherent set of scientific concepts and relations that are designed in purpose to engage students in mathematical activities rich in generation of meaning (Sarama and Clements [<reflink idref="bib37" id="ref36">37</reflink>]). The students' mathematical ideas are progressively shaped as the students interact with tools aiming to construct digital artifacts within a microworld. Kynigos ([<reflink idref="bib22" id="ref37">22</reflink>]) uses the term half-baked for describing a microworld explicitly designed to ensure as the main activity of the students the change of this microworld. In Kynigos's ([<reflink idref="bib22" id="ref38">22</reflink>]) words:</p> <p>Half-baked microworlds are pieces of software explicitly designed so that their users would want to build on them, change them or de-compose parts of them in order to construct an artifact for themselves or one designed for instrumentation by others. They are meant to operate as starting points, as idea generators and as resources for building or decomposing pieces of software. This constructionist activity is seen as part of inquiry and argumentation leading to the generation of mathematical and scientific meaning (pp. 336–337).</p> <p>This allows students to keep having the feeling of ownership which comes from the way a half-baked artifact is changed and the ways the new artifact is being put to use. Given that this process of 'reformulating' a buggy, fallible artifact is approached in a broader sense as an activity of problem posing, the issue of ownership is significant. As Kilpatrick ([<reflink idref="bib20" id="ref39">20</reflink>]) claims, in the context of problem posing the re-formulation of an existing problem is very often connected to the sense of ownership of the new problem.</p> <p>As problem posing is approached in this study in the sense of reformulation of an existing problem, the main approach to do this is to apply the "what-if-not" technique of Brown and Walter ([<reflink idref="bib6" id="ref40">6</reflink>]). In this approach, the main attributes of the given problem are identified, and then the solver starts negating them, asking what would happen if these elements were different. Then, an alternative to the negated attribute can be made. This alternative, however, creates a new problem situation. Of course, when solvers use the "what-if-not" technique, they can change more than one attribute to get a new situation. Considering this in the constructionist context, the starting point is the half-baked artifact (given problem for solution). One of its attributes is bugged. Then the negation takes the form "what if that attribute was not like this?" (problem posing). The alternative answer signals the re-writing of the programming code after making the necessary corrections (which is solving the new problem). We will show that sometimes this process does not end after the re-writing of the code. According to the complexity of the problem, there can be an iterative cycle of solving, reviewing, and revising steps caused by a buggy artifact. And it seems that in this interplay between problem solving and problem posing (in the context of constructionism as explained above) that meaning making occurs (Papadopoulos et al. [<reflink idref="bib32" id="ref41">32</reflink>]).</p> <p>So, in this context, our research question becomes: In what ways the interplay between problem solving and problem posing in a constructionist environment including half-baked microworlds, facilitates students' meaning making?</p> <hd id="AN0187842866-4">Material and methods</hd> <p>This is a case study following one Grade-10 student (16 year old) while reading an e-book unit. The student's performance in Mathematics was above average in his classroom according to his mathematics teacher. According to the curriculum of Informatics, in his last year he had been taught basic programming skills (the suggested duration was 14 teaching hours) using Logo-like environments (Scratch, Snap!, K-turtle). The study took place in parallel to the normal teaching at the beginning of the school year and it was not integrated within the content of his Mathematics or Informatics lessons.</p> <p>The e-book is based on the storyline of Don Quixote and includes diverse widgets (hyperlinks, videos, but mostly instances or activities from a range of educational digital tools) between the lines of the narrative. The digital medium used in this study to design such a half-baked microworld is a well-established freely available online programming tool, called MaLT2. MaLT2 is a web-based Turtle Geometry environment which affords the design through Logo programming along with dynamic manipulation of 3D geometrical objects using sliders[<reflink idref="bib1" id="ref42">1</reflink>]. The specific e-book is inspired from the relevant book of Miguel de Cervantes and refers to the tale of Don Quixote and his adventures. The part presented here includes the following: Don Quixote is attacking to 30–40 windmills he mistakenly considers giant enemies. But, after being close to them he realizes that they are damaged windmills and he wants to repair one of them. Half-baked Logo codes in MaLT2 represent the windmills' fans and wings in various geometrical figures and Don Quixote has to modify the codes so as to repair and reconstruct the fans and the wings. There are several instances calling for repair concerning both the shape of the wings (that might be an isosceles triangle, an equilateral triangle or a parallelogram) and the angle between the wings of the fan.</p> <p>In the first case (1<sups>st</sups> task) Don Quixote has initially to close an open broken line, so as to get the wing in the shape of an isosceles triangle by modifying a given half-baked Logo code (Figure 1). The aim is to construct a complete wing. There are three variables concerning the length of the sides of the triangle (variables 'a' for the two equal sides of the triangle and 'b' for its third side), and the size of the turn (variable 'c'). After correcting the wing, the next step is to construct the whole fan. A new procedure is given including now an additional variable which is the number of repetitions (variable 'n') (Figure 1). Experimentation with the sliders that represent the above mentioned four variables, in conjunction with the bird's trajectories on the canvas, leads to the modification of the code, using the properties of right triangles and/or trigonometric functions, through a continuous process of making and checking conjectures.</p> <p>PHOTO (COLOR): Figure 1. Broken line instead of isosceles triangle.</p> <p>In the second case (2<sups>nd</sups> task) Don Quixote is challenged to repair a windmill's fan, consisted of k equilateral triangles (Figure 2) that have not been joined in a proper way. In the given Logo-code the critical variables are the 'k' variable for the total number of triangles (wings) in the fan and the 'b' variable for the angle between two consecutive wings. The first procedure (i.e., 'wing') uses the variable 'a' to construct an equilateral triangle with side length 'a'. Then, the next procedure (e.g., 'fan') aims to construct the whole fan by using the 'wing' as sub-procedure. There is a bug in the code making thus the code an open-ended problem with a variety of solutions. The challenge for the solvers is that they have to examine carefully both the number of repetitions (variable 'k') and the size of turn (variable 'b') in order to avoid overlapping between the wings (for example, a feasible solution would be to consider 'b' as 360 ÷ k).</p> <p>PHOTO (COLOR): Figure 2. Fan with equilateral triangles.</p> <p>In the same spirit, the next case invites Don Quixote to modify the 'wing' procedure to get a parallelogram and then to discover or recall angular and side properties of parallelograms and use them to allow all the variables functioning correctly. Moreover, it is necessary to use repeatedly the sub-procedure 'wing' to get a complete fan (Figure 3) by correcting the bug in the 'fan' procedure. There are four variables in the 'wing' procedure concerning the length of the sides of the parallelogram and the right turn of the bird in order to create the whole shape. In the 'fan' procedure a fifth variable has been added aiming at the number of the 'wing' repetitions.</p> <p>PHOTO (COLOR): Figure 3. Fan with parallelogramms.</p> <p>The student had as much time as he wanted to "read" this part of the book. It took him about 70 min working on these tasks. The student was asked to vocalize his thoughts while performing the tasks. At the same time a screen capturing software was used to record, the cursor movements, menu selections, pop-up windows, typing and everything else seen on the screen, in a video format. There was audio-recording for the student's aloud thinking which later was transcribed for the purpose of this paper. The student's notes in paper-and-pencil, the transcribed protocols, and the videos with the student's actions on the screen became our data. All of them were examined and analyzed in the context of qualitative content analysis (Mayring [<reflink idref="bib30" id="ref43">30</reflink>]), first in terms of an a priori coding (Stemler [<reflink idref="bib43" id="ref44">43</reflink>]) by identifying the successive transitions from problem solving to constructionist activities (posing) and vice versa, and then in terms of emergent coding by recording instances of meaning-making in mathematics. For the first, the protocol was parsed in episodes based on the transition from problem solving (writing the required ode) to problem posing (using the what-if questions) and vice versa. For example, the student constructed an equilateral triangle (problem solving) and then he posed a what-if question that negated one of the elements of his solution, i.e., the fact that all the three sides were equal. This is an instance of a problem-posing activity that leads to a new problem and signals the transition from one episode to the next. For the second, in the spirit of content analysis, the effort was to identify instances that indicate occurrence of meaning making. For example, the student's realization that when constructing triangular wings, the size of the angle opposite to the base of the triangle determines the number of the wings in the fan and that there is an inverse proportion between them, is an instance of mathematical meaning making. Finally, it is confirmed that the data used in this study was collected after the corresponding Research Ethics Committee approved the project. We obtained consent from the participating student.</p> <hd id="AN0187842866-5">Results</hd> <p>The student spent some time to read the page of the e-book unit and examine the Logo-code. He realized that 'a', and 'b' represent side lengths for the isosceles triangle while 'c' is related with turn and angles. He also recognized 'wing' as a sub-procedure that contributes to the construction of the fan. His comment was:</p> <p>Y.1.7 If it was me, I would design it differently. May I?</p> <p>And immediately erased the content of the 'wing' procedure. His aim was to start in a simpler way loosening the conditions, in the spirit of the Polya's ([<reflink idref="bib35" id="ref45">35</reflink>]) advice to examine and solve an analogous simpler problem. He decided to construct an equilateral triangle, reducing thus the number of variables from three to one since all the sides have equal length and the size of its angles is by definition 60°. His whole effort was based on the definition of the equilateral triangle (equal sides and angles).</p> <p>His thought was that according to the definition all the lines of the code (Figure 4, left) but the last would give an equilateral triangle and then the last one would turn the bird 30 degrees to the left in order to construct the next wing of the fan.</p> <p>PHOTO (COLOR): Figure 4. Effort to construct an equilateral triangle.</p> <p></p> <ulist> <item> Y.1.10 I will ask to always move forward 'a' and turn right 60 degrees to get an equilateral triangle.</item> <p></p> <item> Y.1.14 At the end I will ask to turn left 30 degrees and.... Lets' see.</item> <p></p> <item> He run the code, and his reaction was:</item> <p></p> <item> Y.1.15 I didn't get what expected.</item> <p></p> <item> Y.1.16 Oh! This (<emph>the right turn</emph>) should be 120 degrees instead of 60.</item> <p></p> <item> Y.1.17 But I keep the left turn (Figure 4, right).</item> </ulist> <p>He is not so sure however about the size of the left turn. He used 90 and 100 degrees but as he said:</p> <p></p> <ulist> <item> Y.1.23 The aim is for the bird to be in the proper position to repeat the next wing. But what is its proper position?</item> </ulist> <p>At this point he realized that the decision to work with equilateral triangles determines to a great extent the number of the wings in the fan. The first of a series of "what-if" questions occurred.</p> <p></p> <ulist> <item> Y.1.25 <emph>What if</emph> it was an isosceles instead of equilateral triangle?</item> <p></p> <item> Y.1.28 This would mean that I should change every time the size of the angle opposite to its base and therefore I could manage to have more wings in the fan.</item> <p></p> <item> Y.1.29 The smaller the angle the more the number of the wings.</item> <p></p> <item> Y.1.30 Which means that this angle must be a variable.</item> <p></p> <item> Y.1.32 There must be a relationship between the number of the wings and this angle.</item> <p></p> <item> Y.1.35 I will use one variable for the angle of the triangle and another variable for the left turn at the end.</item> </ulist> <p>The new problem has been formed and in order to solve it the student used finally 4 variables: 'a' for the two equal sides of the triangle, 'b' for its base, 'n' for the size of the angle opposite to its base, and 'c' for the left turn. He missed, however, that in an isosceles triangle the size of the angle opposite to its base determines the size of the other two angles. He found it difficult to write the correct code, especially for determining the size of the angle opposite to the triangle's base as well as the base length (Figure 5).</p> <p>PHOTO (COLOR): Figure 5. Working with 4 variables.</p> <p>It was now time for the next what-if.</p> <p></p> <ulist> <item> Y.1.49. <emph>What if</emph> the isosceles triangle is not the proper shape for the wing? <emph>What if</emph> it was a trapezium?</item> </ulist> <p>To deal with the new problem he turned initially to the paper-and-pencil environment to draw before writing the code.</p> <p></p> <ulist> <item> Y.1.51. I will first make the drawing on a piece of paper.</item> </ulist> <p>He decided to work with a right trapezium as the most suitable option for his fan (Figure 6). The work on paper-and-pencil was considered necessary because he had first to clarify in his mind the aimed shape and then to identify possible relationships between its side lengths. He considered the distance between the two bases being 'c'. Then he chose the bases of the trapezium to be 'b' and 'b + c'. The length of the big base was a deliberate choice. A more general choice for this length in the form of 'b + d' ('d' an arbitrary length) calls for an extra variable that will increase the complexity of the required calculations. So, he chose the difference in the length between the two bases to be equal with the distance between them (e.g., 'c') in order to simplify the code. Therefore, the four sides of the trapezium are 'b', 'c', 'b + c', and 'a'. At this point he did not notice the necessity to identify the relationship between 'a' and 'c' and wrote a procedure with these three variables 'a', 'b', 'c' to construct the right trapezium (Figure 7, left). This resulted in an 'open' trapezium. His code had a bug that should be fixed. In the MaLT2 environment, with the 'variation tool' you may provoke dynamic constant change to the shapes created through the use of a parametric command. By clicking on the shape, the sliders become activated, and the user can drag the cursor over the variation slider changing thus the shape dynamically. This could help the student to fix the bug in his code (Figure 7, right). By dragging the cursor over the slider 'c' the length of the 'c' side can be reduced accordingly so as to reach the edge of the side 'a' thus closing the trapezium. Unfortunately, the student was not aware of this affordance of the environment, and he had somehow to overcome alternatively the obstacle of the open trapezium.</p> <p>PHOTO (COLOR): Figure 6. Paper-and-pencil presentation.</p> <p>PHOTO (COLOR): Figure 7. Open trapezium and the use of sliders.</p> <p></p> <ulist> <item> Y.1.66 There is a mistake here...</item> <p></p> <item> Y.1.67<emph> What if</emph> I try to eliminate one more variable?</item> <p></p> <item> Y.1.69 There will be a relationship between 'a' and 'c'</item> </ulist> <p>Applying the Pythagorean theorem to the right triangle in Figure 6, the length of the 'c' is calculated as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mrow><msup><mi>c</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>⇔</mo><mi>c</mi><mo>=</mo><mrow><mfrac><mrow><mi>a</mi><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></math> </ephtml> .</p> <p></p> <ulist> <item> Y.1.74 Now I don't need the 'c' variable.</item> </ulist> <p>So, he was now able to solve correctly the new problem of constructing the right trapezium (Figure 8)</p> <p>PHOTO (COLOR): Figure 8. Closing the open trapezium.</p> <p>His next step then was to deal with the bigger issue, the construction of the fan.</p> <p></p> <ulist> <item> Y1.77 The number of wings is important.</item> <p></p> <item> Y1.79 I rather must use a 'repeat' command.</item> <p></p> <item> Y1.80 Let's say I want 3 wings.</item> <p></p> <item> Y1.81 This angle is 45° (<emph>He refers to the angle with the bird on its vertex, see Figure 8</emph>)</item> <p></p> <item> Y1.83 This means 135° in total (45°+45°+45°, for each one of the three wings).</item> <p></p> <item> Y1.84 This leaves 360–135 = 225 degrees.</item> <p></p> <item> Y1.88 Therefore, if 'n' is the number of the wings the left turn should be 225/:n</item> </ulist> <p>He missed the fact that this turn is valid only in case <emph>n</emph> = 3 and therefore this is a wrong generalization. He applied his idea in a Logo-code for <emph>n</emph> = 4 (Figure 9, left and middle).</p> <p>PHOTO (COLOR): Figure 9. Attempt to construct the fan for n = 4 (middle) and n = 5 (right).</p> <p>His reaction while looking at the result on the screen was that 'something went wrong'.</p> <p></p> <ulist> <item> Y1.91 The mistake has something to do with angles.</item> <p></p> <item> Y1.92 But it is not the angle of the quadrilateral. It is rather the left turn.</item> </ulist> <p>So, he started using his finger to follow on the screen the path of the bird while executing mentally one-by-one the commands of the code. He stopped at the left turn. He was not sure about the denominator in the fraction 225/:n.</p> <p></p> <ulist> <item> Y1.95 What if n = 5?</item> <p></p> <item> Y1.100 No! It is wrong (<emph>Figure 9, right</emph>)</item> </ulist> <p>According to this code only two visible wings can be constructed. Every new wing coincides with one of the initial two.</p> <p>He then began to control the reasonableness of his arguments which is a significant problem-solving action.</p> <p></p> <ulist> <item> Y1.103 I started by the fact that the angle with the bird on its vertex is 45 degrees. And then I took as given that the sum will be always 135<sups>°,</sups> but this is not the case. The number of wings will not always be 3. I suppose if I apply the code for n = 3 it will work.</item> <p></p> <item> Y1.104 So... It is not correct to subtract 360–135. The correct is (360–45*:n)/:n</item> </ulist> <p>So, he made the necessary modification to his code and run it for <emph>n</emph> = 5 (Figure 10, left and middle).</p> <p>PHOTO (COLOR): Figure 10. Modified code for n = 4 (middle) and n = 5 (right).</p> <p>He still was not satisfied, and he chose to go back to the starting point for <emph>n</emph> = 3 (Figure 10, right) which was also problematic.</p> <p>He followed again step-by-step the execution of each command on the screen, and he realized that the wrong shape of the fan was because of two elements: (i) the left turn and (ii) the orientation of the bird after completing a wing.</p> <p>The solution would be to force the bird to turn left 180 degrees after executing the last fd command (from A to B, Figure 10, right)). Therefore, the bird will be looking at the opposite direction, that is from B to A, and only then the student should be able to determine a right turn to signal the position of the next wing.</p> <p></p> <ulist> <item> Y1.113 When completing the last forward move it looks at this direction (<emph>he shows on the screen the direction of the last movement, from A to B</emph>).</item> <p></p> <item> Y1.115 At the end, what would be the proper direction for the bird to be heading for?</item> <p></p> <item> Y1.116 I want it looking toward that vertex (<emph>He shows the direction from B to A</emph>).</item> <p></p> <item> Y1.118 Therefore, I will ask the bird to turn left 180°</item> </ulist> <p>Then he had to decide the size of the right turn of (360–45*:n)/:n to determine the position of the new wing.</p> <p>The modified code worked fine for <emph>n</emph> = 3 and <emph>n</emph> = 6 (see Figure 11)</p> <p>PHOTO (COLOR): Figure 11. Running the correct code for n = 3 (middle) and n = 6 (right).</p> <p>However, the situation became more complex when he decided to use other values for 'n', such as <emph>n</emph> = 10, <emph>n</emph> = 13, or <emph>n</emph> = 24 aiming to reach the general solution (Figure 12, left, middle and right respectively).</p> <p>PHOTO (COLOR): Figure 12. The case of fan for n = 10 (left), n = 13 (middle), n-24 (right).</p> <p></p> <ulist> <item> Y1.134 <emph>What if</emph> I decide to use more wings such as 10, or 13 or 24?</item> </ulist> <p>This experimentation with gradually increased values for 'n' made him realize that there must be a certain range of values for 'n' to avoid overlapping between the wings to get a well-functioning fan.</p> <p>He was sure that 3 should be the lower limit for 'n'. To find the maximum number of wings he used the acute angle of the trapezium that was 45°. This number is factor of 360 and thus the value of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>÷</mo><mn>45</mn><mo>=</mo><mn>8</mn></math> </ephtml> , should not be an option.</p> <p></p> <ulist> <item> Y1.140 Since</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>÷</mo><mn>45</mn><mo>=</mo><mn>8</mn></math> </ephtml> , this number cannot be assigned to 'n' because in this case there will be not an empty space between successive wings which is the necessity for a fan to function properly.</p> <p></p> <ulist> <item> Y1.141 I think that the best option for a functional fan is to choose values for n in the range [<reflink idref="bib3" id="ref46">3</reflink>, 7].</item> </ulist> <p>He run again his code to check all the cases from <emph>n</emph> = 3 to <emph>n</emph> = 7. Accidentally he realized the existence of sliders and he spent some time to understand their functionality. This solved his problem how to put forward this certain range of values. The sliders allow you to choose values for the "From" and "To" fields. These numbers correspond to the limits for the specific variable. So, by putting the numbers 3 to the left ("From") and 7 to the right ("To") he determined the possible number of wings. Moreover, he changed the "Step" field to '1′ to specify that the variation tool can take values which differ by one between the limits he defined. So, by dragging the slider, all the possible fans with 3, 4, 5, 6 and 7 wings can be constructed (Figure 13).</p> <p>PHOTO (COLOR): Figure 13. Adjusting the n-variable limits.</p> <p>This allowed him to check the accuracy of his general solution and signaled the accomplishment of the task.</p> <hd id="AN0187842866-6">Discussion</hd> <p>From the very first moment (Y.1.7) the student made the decision to take an autonomous path in his exploration. He was not limited to what the narrative of the e-book unit suggested and worked in a way that reminds the suggestion of Polya ([<reflink idref="bib35" id="ref47">35</reflink>]): Could you think of a simpler analogous problem? Therefore, the student by loosening the conditions exhibited a sense of ownership of the problem (Kilpatrick [<reflink idref="bib20" id="ref48">20</reflink>]) that guided him during his effort to arrive to a solution. Interestingly his choices resemble to a great extend to the content of the tasks in the e-book unit's pages.</p> <p>In this effort the student was navigating (but not in a linear way) through an interplay between reformulating (debugging) and solving that seem to be dynamically related.</p> <p>Initially the student used parts of the code (Y.1.14) without awareness of the expected outcome. The left turn in his code (Figure 4) plays significant role but he was not able to determine the proper size of the turn. So, he used the affordance provided by the MaLT2 environment to run the code several times with different values for the left turn (<reflink idref="bib30" id="ref49">30</reflink>, 90, 100 degrees). At this point, despite that this behavior maintains a close relationship between the shape of the fan and the symbolic code, it does not guarantee an awareness of the exact role of the left turn to the construction of the fan.</p> <p>His first decision to start with an equilateral triangle resulted in a simple problem loosening the initial condition, e.g., two out of three sides of the triangle must be of equal length (isosceles triangle). Therefore, all the three sides should have the same length. He tried to solve this problem by writing (not successfully) the relative code.</p> <p>His first 'what-if' question (Y.1.25) negates the attribute of the shape. The alternative option resulted in a modified version of his last decision. He turned again to the case of isosceles triangle. This decision helped him to make explicit in his mind the different parts of the code structure (Y.1.28–29). He realized that the size of the angle opposite to the base of the triangle determines the number of the wings in the fan and that there is an inverse proportion between them: The less the size of this angle the more the number of the wings. This explicit understanding of the covariance between these two entities made him to think that the size of the angle must be a variable (Y.1.30). This awareness of the variability of numbers as inputs to commands and procedures can be considered an instance of perceiving variable as a general number.</p> <p>His next 'what-if' question negates the option of the shape being a triangle. What if the shape of the wing is not a triangle? The alternative led him to another constructionist activity based on his choice to use a right trapezium (Y.1.49) as the proper shape of the wing. This was a demanding decision, and the student felt the need to work in parallel in both environments, paper-and-pencil and MaLT2 (Y.1.51). This decision is indicative of an attempt to try things out in different contexts synthesizing thus his Logo-code experience with his 'other' mathematical work. The result was two different representations of the same mathematical situation (geometrical one in paper-and-pencil and its equivalent representation in the form of a Logo-code in the MaLT2 environment) which is considered an instance of conceptual understanding (Stohlmann et al. [<reflink idref="bib44" id="ref50">44</reflink>]). However, the final result on the screen was an open trapezium (Figure 7). Something went wrong (Y.1.66). Again, the student turned to the 'what-if-not' approach to elaborate further the situation by choosing to reduce the number of variables. What if there were not so many variables? This led him to 'see' separately the sides of the trapezium and the existing relationships between them. Variable 'c' is now seen in the light of its dependence from variable 'a'. The student managed to work with only two variables making clear the role each one of them plays in the construction of the shape (Figure 8). Variable 'a' is the length of one side but determines also the length of the distance between the two parallels (:a/sqrt 2). Variable 'b' is the small base but at the same time determines, in conjunction with variable 'a', the length of the big base (:b +: a/sqrt 2).</p> <p>Having finalized the shape of the wing it was then time to shift his focus to the construction of the fan. The crucial element was now the number of the wings that should be separated by equal angles. One of his first steps was to make a wrong generalization (Υ.1.88): "If the number of the wings is 'n' the left turn after completing each wing should be 225/:n." Since the outcome is not examined in the light of its mathematical structure (How is the size of the angle determined? How is it connected with the number of the wings?) the only way to proceed was by substituting random values in the variable 'n' which in turn contributed to identifying the source of his error (Y.1.92) but not the way to overcome this. Although easy to change specific numerical data into another one, much attention must be given to the mathematical meaning of this change, in the sense of the way such changes have an impact on the problem's solution. The students' problem-solving process employed an embodied spatial articulation to express the execution of each line of the Logo-code (Abrahamson [<reflink idref="bib2" id="ref51">2</reflink>]). He used his figures to follow on the screen the path of the bird vis-à-vis the lines of the Logo-code. Again, working concurrently in two representations of the same mathematical situation can be considered an instance of conceptual understanding. His mathematical understanding was enhanced and now he focused on the source of his error: How to organize the last part of his code to prepare the entity (e.g., the bird) to correctly construct the next wing of the fan.</p> <p>Therefore, in order to find the drawback of his thinking he decided to check the reasonableness of his arguments step-by-step (Y.1.103). Checking his argumentation can be considered an important act that contributed to his mathematical meaning-making. Students who are engaged in mathematical argumentation and check its reasonableness demonstrate greater understanding of mathematical concepts (Cross [<reflink idref="bib10" id="ref52">10</reflink>]; Schwarz, Hershkowitz, and Prusak [<reflink idref="bib38" id="ref53">38</reflink>]). This thorough examination of his arguments made him to admit that his choice of 3 wings was a special case, but since "The number of wings will not always be 3" (Y.1.103) he should try to get the general case.</p> <p>The student distinguished two different parts in the Logo-code that had an impact on the wrong shape of the fan: (i) the size of the left turn of the bird and (ii) the orientation of the bird after completing a wing. This was the new problem posed by himself. His problem-solving effort included to follow again step-by-step the movement of the bird on the screen vis-à-vis the lines of the code. Thus, he managed to reach the correct final position of the bird through its left turn of 180 degrees and then its right turn of (360–45*:n)/:n (Y.1.118).</p> <p>The solution of this sub-problem signaled the posing of the next one: whether any value for the 'n' variable can be chosen. Some random choices (Y.1.134) resulted in overlapping wings, a result that was not considered correct according to the students' image of the Greek windmills. To overcome this difficulty, he used the sliders he accidentally discovered which helped him to reach the general solution of the task. This process of the general solution negotiates the minimum and maximum value for the 'n' variable. This effort was supported by discriminating the behavior of the code for specific cases (e.g. excluding the case <emph>n</emph> = 8 since the wings will be adjacent with no space between them). He finally realized that the interval [<reflink idref="bib3" id="ref54">3</reflink>, 7] provided the values that would satisfy the conditions for a proper fan.</p> <p>The whole work of the student was mainly around the use of variable in a constructionist problem-solving environment and the meaning-making process included an iterative review—revise—solve cycle (Figure 14). This whole process can be seen as a continuous interplay between problem solving (<emph>solve in the sense of writing the aimed code</emph>) and constructionist activities that take the form of problem posing (<emph>when the student reviews-revises his code</emph>) (Papadopoulos et al. [<reflink idref="bib32" id="ref55">32</reflink>]).</p> <p>PHOTO (COLOR): Figure 14. The review–revise–solve cycle.</p> <p>Each <emph>review</emph> was provoked by a <emph>what-if-not</emph> question of the student which negates one of the features of the given situation (e.g., the number or the shape of the wings). This resulted to the reformulation of the initial problem (<emph>posing - revise</emph>). According to his response to the previous negation, the student comes up with a new problem. The next step was to <emph>solve</emph> the new problem. Then, after solving it, a new <emph>what-if-not</emph> signaled the beginning of another cycle and so on. For example, the first <emph>what-if-not</emph> question challenges the case of the isosceles triangle for the wing's shape (<emph>review</emph>). This resulted to the revision of the problem. The new one was to construct a wing having the shape of an equilateral triangle (<emph>revise</emph>), which then was solved by the student (<emph>solve</emph>). This was later reviewed again by another <emph>what-if-not</emph> question that resulted to a wing having the shape of a right trapezium and so on.</p> <p>This interplay between problem solving and constructionist activities (revise the code) seems to serve as enabler of meaning-making in mathematics. The student built a progressive understanding of the concept of variable. He managed to develop and exhibit an understanding of the concept of variable as a general number. The translation from problem situation to symbolic algebra is difficult and linking the problematic situation with mathematics symbolism requires students to invest on abstract thinking including interpretation and construction. This was implied by the ability of the student to interpret a symbol as representing a general, indeterminate entity that can assume any value and manipulate (simplify, develop) the symbolic variable (Trigueros et al. [<reflink idref="bib46" id="ref56">46</reflink>]). This is in alignment with Cañadas, Molina, and del Río ([<reflink idref="bib7" id="ref57">7</reflink>]) who consider algebraic symbolism and verbal representations as representation systems. According to them, translation from verbal representations to algebraic symbolism is part of the problem-solving process whereas the opposite (e.g., from algebraic symbolism to verbal representation) is part of the problem-posing process.</p> <p>Moreover, the student's ignorance of the existence of the sliders made him to handle mathematically their absence. For example, he could use the sliders to correct the case of the open trapezium (Figure 7). Instead, he had to work mathematically to 'close' the trapezium. So, he first decided to reduce the number of the variables involved in the procedure (Y.1.67). Then, he handled variables 'a' and 'c' as covarying entities (Y.1.69, Y.1.74), exhibiting thus an advanced understanding of the concept of variable (Thompson and Carlson [<reflink idref="bib45" id="ref58">45</reflink>]).</p> <p>Apart from noticing the student building an understanding of the concept of variable, more instances of meaning making can be identified in his work. The affordances of the media allowed the student to understand several mathematical ideas that were involved. For example, the student was able to understand that when constructing a triangle, the right turn does not refer to the triangle's angle but to its supplementary angle. For the isosceles triangle he realized that the size of the angle opposite to its base determines the number of wings and there is an inverse relationship between them. He grasped the significance of the right turns after completing a wing. He polished his general solution by determining that the range of values for the number of the wings is from 3 to 7.</p> <p>Although the previous research has emphasized the significance of the use of such half-baked microworlds for meaning making in mathematics, the relevant studies were investigating the case of separately solving given problems (Hoyles and Noss [<reflink idref="bib17" id="ref59">17</reflink>]; Lantz-Andersson, Linderoth, and Säljö [<reflink idref="bib26" id="ref60">26</reflink>]) or de-bugging artifacts with bugs (Kynigos [<reflink idref="bib22" id="ref61">22</reflink>], [<reflink idref="bib24" id="ref62">24</reflink>]). However, it seems that meaning making can occur also in digital environments that allow the student to move successively from solving a problem to review and revise it. In this paper, an interplay between these two processes was used in a constructionist environment exemplifying how meaning making was constructed around (but not limited to) the concept of variable.</p> <hd id="AN0187842866-7">Conclusions</hd> <p>In this paper a half-baked microworld (part of an e-book) was used by a 10<sups>th</sups> grader. The findings give evidence that the e-book environment by its affordances contributed to generating mathematical meaning around the concept of variable. The continuous interplay between problem solving and constructionist activities (in a way that resembles the interplay between problem solving and problem posing through the what-if-not questions) facilitated the student to exploit his mathematical knowledge to achieve his goals. The student constructed mathematical meaning through discriminating and using parts of the Logo-codes, generalizing and debugging efforts to achieve new purposes, and synthesizing procedural computer experiences with 'other' mathematical work. Inviting students to work in such constructionist environments provides them the opportunity to think in terms of posing new inquiries asking what-if-not questions. This engages students in an interplay between problem-solving and problem-posing processes that can facilitate the mathematical meaning making. A special affordance of this environment, as Kynigos and Diamantidis ([<reflink idref="bib25" id="ref63">25</reflink>]) emphasize, is "a kind of pedagogical engineering of agency, giving students space and legitimacy for coming up with and owning their problems" (p. 161). There is also another affordance valuable for the classroom. 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National Association for Research in Science Teaching.</bibtext> </blist> </ref> <ref id="AN0187842866-10"> <title> Footnotes </title> <blist> <bibtext> MALT2 − 'Machine Lab Turtlesphere' was conceived at Educational Technology Lab (NKUA) and can be accessed at http://etl. ppp. uoa. gr/malt2/.</bibtext> </blist> </ref> <aug> <p>By Ioannis Papadopoulos</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib29" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib41" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib28" firstref="ref5"></nolink> <nolink nlid="nl4" bibid="bib18" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib48" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib21" firstref="ref8"></nolink> <nolink nlid="nl7" bibid="bib39" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib27" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib13" firstref="ref12"></nolink> <nolink nlid="nl10" bibid="bib36" firstref="ref13"></nolink> <nolink nlid="nl11" bibid="bib17" firstref="ref16"></nolink> <nolink nlid="nl12" bibid="bib22" firstref="ref17"></nolink> <nolink nlid="nl13" bibid="bib24" firstref="ref18"></nolink> <nolink nlid="nl14" bibid="bib42" firstref="ref19"></nolink> <nolink nlid="nl15" bibid="bib40" firstref="ref20"></nolink> <nolink nlid="nl16" bibid="bib16" firstref="ref21"></nolink> <nolink nlid="nl17" bibid="bib11" firstref="ref22"></nolink> <nolink nlid="nl18" bibid="bib31" firstref="ref23"></nolink> <nolink nlid="nl19" bibid="bib47" firstref="ref24"></nolink> <nolink nlid="nl20" bibid="bib14" firstref="ref25"></nolink> <nolink nlid="nl21" bibid="bib12" firstref="ref26"></nolink> <nolink nlid="nl22" bibid="bib26" firstref="ref27"></nolink> <nolink nlid="nl23" bibid="bib19" firstref="ref28"></nolink> <nolink nlid="nl24" bibid="bib23" firstref="ref30"></nolink> <nolink nlid="nl25" bibid="bib34" firstref="ref31"></nolink> <nolink nlid="nl26" bibid="bib33" firstref="ref32"></nolink> <nolink nlid="nl27" bibid="bib15" firstref="ref35"></nolink> <nolink nlid="nl28" bibid="bib37" firstref="ref36"></nolink> <nolink nlid="nl29" bibid="bib20" firstref="ref39"></nolink> <nolink nlid="nl30" bibid="bib32" firstref="ref41"></nolink> <nolink nlid="nl31" bibid="bib30" firstref="ref43"></nolink> <nolink nlid="nl32" bibid="bib43" firstref="ref44"></nolink> <nolink nlid="nl33" bibid="bib35" firstref="ref45"></nolink> <nolink nlid="nl34" bibid="bib44" firstref="ref50"></nolink> <nolink nlid="nl35" bibid="bib10" firstref="ref52"></nolink> <nolink nlid="nl36" bibid="bib38" firstref="ref53"></nolink> <nolink nlid="nl37" bibid="bib46" firstref="ref56"></nolink> <nolink nlid="nl38" bibid="bib45" firstref="ref58"></nolink> <nolink nlid="nl39" bibid="bib25" firstref="ref63"></nolink>
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Meaning Generation in the Interplay between Problem Solving and Posing in a Constructionist Environment
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  Data: English
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Ioannis+Papadopoulos%22">Ioannis Papadopoulos</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-6548-1499">0000-0001-6548-1499</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Computers+in+the+Schools%22"><i>Computers in the Schools</i></searchLink>. 2025 42(3):233-256.
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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– Name: Pages
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  Data: 24
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  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
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  Label: Education Level
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  Data: <searchLink fieldCode="EL" term="%22Grade+10%22">Grade 10</searchLink><br /><searchLink fieldCode="EL" term="%22High+Schools%22">High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink>
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  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Problem+Solving%22">Problem Solving</searchLink><br /><searchLink fieldCode="DE" term="%22Concept+Formation%22">Concept Formation</searchLink><br /><searchLink fieldCode="DE" term="%22Comprehension%22">Comprehension</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+10%22">Grade 10</searchLink><br /><searchLink fieldCode="DE" term="%22High+School+Students%22">High School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Constructivism+%28Learning%29%22">Constructivism (Learning)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Software%22">Computer Software</searchLink><br /><searchLink fieldCode="DE" term="%22Troubleshooting%22">Troubleshooting</searchLink>
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  Data: 10.1080/07380569.2024.2412300
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  Label: ISSN
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  Data: 0738-0569<br />1528-7033
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: In this paper the potentiality of meaning generation in a constructionist environment through an interplay between problem solving and problem posing is examined. The meaning-making process is examined by following how a Grade-10 student is engaged in an iterative cycle of reformulating and solving the initial problem within a Logo-based microworld. The findings show that the contribution of this iterative cycle is in favor of the generation of mathematical meaning. The shift from reformulating to solving and vice versa supported the student to initially use his mathematical knowledge for achieving certain goals, then to discriminate and use parts of his Logo program through generalizing and debugging efforts to achieve new purposes.
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      – Text: English
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        PageCount: 24
        StartPage: 233
    Subjects:
      – SubjectFull: Problem Solving
        Type: general
      – SubjectFull: Concept Formation
        Type: general
      – SubjectFull: Comprehension
        Type: general
      – SubjectFull: Grade 10
        Type: general
      – SubjectFull: High School Students
        Type: general
      – SubjectFull: Constructivism (Learning)
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: Computer Software
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      – SubjectFull: Troubleshooting
        Type: general
    Titles:
      – TitleFull: Meaning Generation in the Interplay between Problem Solving and Posing in a Constructionist Environment
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              Type: published
              Y: 2025
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            – TitleFull: Computers in the Schools
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