# How Learning to Speak the Language of a Computer-Based Digital Environment Can Plant Seeds of Algebraic Generalisation: The Case of a 12-Year-Old Student and eXpresser

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction: Students’ Learning in Digital Interactive Mathematics Learning Environments

## 2. Our Perspective on Algebraic Generalisation, Elaboration of the Theoretical Lens and Introduction of the DIMLE

#### 2.1. Digital Technology as a Gateway to Overcome Difficulties with Algebraic Generalisation

#### 2.2. The Theory of Instrumental Genesis

#### 2.3. The Microworld eXpresser and Three Key Ways of Thinking for Algebraic Generalisation

- AWOT.1—perceiving structure and exploiting its power: recognising the constituent elements of a complex structure, and using them to build the structure—both physically and mentally;
- AWOT.2—seeing the general in the particular: identifying variants and invariants, manipulating a special and familiar case in order to get a sense of what stays the same and what changes;
- AWOT.3—recognising and articulating generalisations, including expressing them symbolically: describing structure by using variables (symbols) and expressing symbolically their relationships to structure and generality; that is, building an algebraic rule.

^{th}term in a figural pattern sequence. Students had to perceive the structure in the given pattern or what was repeated and what remained the same in any instantiation of the pattern (AWOT1); then they had to explore what stayed the same and what varied and consider constants and variables (AWOT2); before taking the final step of articulating the “rule” that gave the total number in any pattern instantiation writing it using the eXpresser language (AWOT3). In other words, students are asked to create a pattern on a grid using a set of coloured tiles and repeat it with a specific regularity. Below, we present an example to further showcase eXpresser’s design and the specific tasks it poses to students who interact with it (Figure 1).

#### 2.4. Our Specific Research Question and Broader Aim

## 3. Capturing a Student’s Learning Journey in eXpresser

#### 3.1. Participants and Data Collected

#### 3.2. Re-Enactments of the Observed-Activity Sessions

- The specific task Molly is solving;
- The artefacts she uses (a visible part of how Molly interacts with the DIMLE);
- Her instrumentation schemes, through the techniques and the technical elements (the visible parts of her becoming fluent in the language of the DIMLE), as well as the conceptual elements (that we will associate to the emergence of the expected algebraic ways of thinking).

## 4. Results

#### 4.1. Vignette 1

#### 4.2. Vignette 2

- Researcher:
- Alright let’s write it down here with the expression. So […], over here, you take the 6 and then … what are you going to do?

- Molly:
- [she drags out the 6] and then I think I have to [she moves the mouse on the screen but does not find what she seems to be expecting.]

- Researcher:
- How do you put them together, if you want? You drag it on top of the other one right?

- Molly:
- oh yeah. I think it’s times…

- Researcher:
- okay and then what do you need to do?

- Molly:
- calculate it and that’s 36. So, I think you have to do, drag the 36 there.

#### 4.3. Vignette 3

#### 4.4. Vignette 4

#### 4.5. Looking Back at Molly’s Scheme Development

- A phase of guided sense-making that coincided with the use of her “compute and answer” scheme, and in which she tries to make sense of the task and of the artefacts in eXpresser she chooses to use, supporting her activity by recalling previous techniques;
- A phase of mental recollection, in which she mentally recalls previous actions associated with her task. The sense-making component is weak in this phase, and the scheme appears to be unstable, as Molly easily falls back to previous schemes (especially to her calculate value of expression scheme or calculate value of expression with unlocked number scheme) and seeks support from eXpresser or a nearby researcher;
- A phase of adjustment and stabilisation of the scheme “answer with general rule”, during which the instrumented techniques in eXpresser become condensed and are applied in a more automatic manner. In this phase, Molly’s techno-mathematical fluency becomes more apparent as she becomes more and more experienced with how to interact with eXpresser to achieve what she wants;
- A phase of generalisation, though still quite situated, in which Molly explains to her classmate how to “answer with general rule” in a more flexible (possibly more general) way, in the sense that she separates what needs to be done from what “you can do”.

## 5. Discussion and Conclusions

#### 5.1. A Broader Glance at Planting Seeds of Algebraic Generalisation in eXpresser

#### 5.2. Generalisability of Our Method

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Papert, S. Mindstorms: Children, Computers, and Powerful Ideas; Harvester Press: London, UK, 1980. [Google Scholar]
- Karadag, Z.; Martinovic, D.; Freiman, V. Dynamic and Interactive Mathematics Learning Environments (DIMLE). In Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV, USA, 1–4 October 2011. [Google Scholar]
- Leung, A.; Baccaglini-Frank, A. Digital Technologies in Designing Mathematics Education Tasks; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Ye, H.; Liang, B.; Ng, O.-L.; Chai, C.S. Integration of computational thinking in K-12 mathematics education: A systematic review on CT-based mathematics instruction and student learning. Int. J. STEM Educ.
**2023**, 10, 1–26. [Google Scholar] [CrossRef] - Hoyles, C.; Noss, R.; Kent, P.; Bakker, A. Improving Mathematics at Work: The Need for Techno-Mathematical Literacies; Routledge: New York, NY, USA, 2010. [Google Scholar]
- Lew, H.-C.; Baccaglini-Frank, A. Creating constructive interference between the 4th Industrial Revolution (+ COVID 19) and the teaching and learning of mathematics. In Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education, Khon Kaen, Thailand, 19–22 July 2021; Volume 1, pp. 76–84. [Google Scholar]
- Tamborg, A.L.; Elicer, R.; Brating, K.; Geraniou, E.; Jankvist, U.T.; Misfeldt, M. The politics of computational thinking and programming in mathematics education: Comparing curricula and resources in England, Sweden, and Denmark. In Handbook of Digital Resources in Mathematics Education; Pepin, B., Gueudet, G., Choppin, J., Eds.; Living Edition; Springer International Handbooks of Education: Cham, Switzerland, 2023. [Google Scholar]
- Wing, J. Research notebook: Computational thinking—What and why. Link Mag.
**2011**, 6, 20–23. [Google Scholar] - Ng, O.; Cui, Z. Examining primary students’ mathematical problem-solving in a programming context: Towards computationally enhanced mathematics education. ZDM Math. Educ.
**2021**, 53, 847–860. [Google Scholar] [CrossRef] - Pérez, A. A framework for computational thinking dispositions in mathematics education. J. Res. Math. Educ.
**2018**, 49, 424–461. [Google Scholar] [CrossRef] - Noss, R.; Hoyles, C. Windows on Mathematical Meanings: Learning Cultures and Computers; Kluwer Academic Publishers: London, UK, 1996. [Google Scholar]
- Noss, R.; Hoyles, C. Constructionism and Microworlds. In Technology Enhanced Learning; Duval, E., Sharples, M., Sutherland, R., Eds.; Springer: Cham, Switzerland, 2017; pp. 29–35. [Google Scholar]
- Clark-Wilson, A.; Robutti, O.; Sinclair, N. The Mathematics Teacher in the Digital Era. International Research on Professional Learning and Practice; Springer: Cham, Switzerland, 2022. [Google Scholar]
- Vérillon, P.; Rabardel, P. Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. Eur. J. Psychol. Educ.
**1995**, 10, 77–101. [Google Scholar] [CrossRef] - Artigue, M. Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. Int. J. Comput. Math. Learn.
**2002**, 7, 245–274. [Google Scholar] [CrossRef] - Trouche, L. Managing complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. Int. J. Comput. Math. Learn.
**2004**, 9, 281–307. [Google Scholar] [CrossRef] - Drijvers, P.; Godino, J.D.; Font, V.; Trouche, L. One episode, two lenses: A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educ. Stud. Math.
**2013**, 82, 23–49. [Google Scholar] [CrossRef] - Arcavi, A. Symbol sense: Informal sense-making in formal mathematics. Learn. Math.
**1994**, 14, 24–35. [Google Scholar] - Küchemann, D. Looking for Structure: A Report of the Proof Materials Project; Dexter Graphics: London, UK, 2008. [Google Scholar]
- Arcavi, A.; Drijvers, P.; Stacey, K. Learning and Teaching of Algebra: Ideas, Insights and Activities; Routledge: New York, NY, USA, 2016. [Google Scholar]
- Küchemann, D. Algebra. In Children’s Understanding of Mathematics; Hart, K., Ed.; Antony Rowe Publishing Services: London, UK, 1981; pp. 102–119. [Google Scholar]
- Küchemann, D.; Hoyles, C. From empirical to structural reasoning: Tracking changes over time. In Teaching and Learning Proof across the Grades; Blanton, N., Stylianou, M., Knuth, D., Eds.; Lawrence Erlbaum Associates: Hillsdale, MI, USA, 2009; pp. 171–190. [Google Scholar]
- Ellis, A.B. Algebra in the middle school: Developing functional relationships through quantitative reasoning. In Early Algebraization a Global Dialogue from Multiple Perspectives; Cai, J., Knuth, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 215–238. [Google Scholar]
- Knuth, E.J.; Alibali, M.W.; McNeil, N.M.; Weinberg, A.; Stephens, A.C. Middle school students’ understanding of core algebraic concepts: Equivalence & variable. In Early Algebraization A Global Dialogue from Multiple Perspectives; Cai, J., Knuth, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 259–276. [Google Scholar]
- Kieran, C.; Drijvers, P. The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of Cas use in secondary school algebra. Int. J. Comput. Math. Learn.
**2006**, 11, 205–263. [Google Scholar] [CrossRef] - Hoyles, C.; Healy, L. Visual and symbolic reasoning in mathematics: Making connections with computers? Math. Think. Learn.
**1999**, 1, 59–84. [Google Scholar] - Presmeg, N.C. Visualisation in high school mathematics. Learn. Math.
**1986**, 6, 42–46. [Google Scholar] - Duval, R. Understanding the Mathematical Way of Thinking—The Registers of Semiotic Representations; Springer International Publishing: Cham, Switzerland, 2017. [Google Scholar]
- Kaput, J. Technology and mathematics education. In Handbook on Research in Mathematics Teaching and Learning; Grouws, D., Ed.; Macmillan: New York, NY, USA, 1992; pp. 515–556. [Google Scholar]
- Mason, J. Developing Thinking in Algebra; Sage: London, UK, 2005. [Google Scholar]
- Noss, R.; Hoyles, C.; Mavrikis, M.; Geraniou, E.; Gutierrez-Santos, S.; Pearce, D. Broadening the sense of “dynamic”: A microworld to support students’ mathematical generalisation. ZDM Math. Educ.
**2009**, 41, 493–503. [Google Scholar] [CrossRef] - Benton, L.; Saunders, P.; Kalas, I.; Hoyles, C.; Noss, R. Designing for learning mathematics through programming: A case study of pupils engaging with place value. Int. J. Child.-Comput. Interact.
**2018**, 16, 68–76. [Google Scholar] [CrossRef] - Benton, L.; Hoyles, C.; Kalas, I.; Noss, R. Bridging primary programming and mathematics: Some findings of design research in England. Digit. Exp. Math. Educ.
**2017**, 3, 115–138. [Google Scholar] [CrossRef] - Jacinto, H.; Carreira, S. Mathematical problem solving with technology: The techno-mathematical fluency of a Student-with-GeoGebra. Int. J. Sci. Math. Educ.
**2017**, 15, 1115–1136. [Google Scholar] [CrossRef] - Monaghan, J.; Trouche, L.; Borwein, J.M. Tools and Mathematics: Instruments for Learning; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Vergnaud, G. The theory of conceptual fields. Hum. Dev.
**2009**, 52, 83–94. [Google Scholar] [CrossRef] - Roorda, G.; Ros, P.; Drijvers, P.; Goedhart, M. Solving Rate of Change Tasks with a Graphing Calculator: A Case Study on Instrumental Genesis. Digit. Exp. Math. Educ.
**2016**, 2, 228–252. [Google Scholar] [CrossRef] - Gregersen, R.; Baccaglini-Frank, A. Lower secondary students reasoning competency in a digital environment: The case of instrumented justification. In Mathematical Competencies in the Digital Era; Jankvist, U., Geraniou, E., Eds.; Springer: Cham, Switzerland, 2022; pp. 119–138. [Google Scholar]
- Gregersen, R.M. Analysing instrumented justification: Unveiling student’s tool use and conceptual understanding in the prediction and justification of dynamic behaviours. Digit. Exp. Math. Educ.
**2024**, 10, 47–75. [Google Scholar] [CrossRef] - Mavrikis, M.; Noss, R.; Hoyles, C.; Geraniou, E. Sowing the seeds of algebraic generalization: Designing epistemic affordances for an intelligent microworld. J. Comput. Assist. Learn.
**2013**, 29, 68–84. [Google Scholar] [CrossRef] - Sinclair, N. Knowing as remembering: Methodological experiments in embodied experiences of number. Digit. Exp. Math. Educ.
**2023**, 10, 29–46. [Google Scholar] [CrossRef] - Güss, C.D. What is going through your mind? Thinking aloud as a method in cross-cultural psychology. Front. Psychol.
**2018**, 9, 1292. [Google Scholar] [CrossRef] [PubMed] - Lewins, A.; Silver, C. Using Software in Qualitative Research: A Step-by-Step Guide; Sage: London, UK, 2007. [Google Scholar]
- Mavrikis, M.; Geraniou, E. Using Qualitative Data Analysis Software to analyse students’ computer-mediated interactions: The case of MiGen and Transana. Int. J. Soc. Res. Methodol.
**2010**, 14, 245–252. [Google Scholar] [CrossRef] - Hiebert, J. Conceptual and Procedural Knowledge: The Case of Mathematics; Lawrence Erlbaum Associates, Inc.: Hillsdale, NJ, USA, 1986. [Google Scholar]
- Kieran, C. The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In Vital Directions for Mathematics Education Research; Leatham, K.R., Ed.; Springer: New York, NY, USA, 2013; pp. 153–171. [Google Scholar]
- Baccaglini-Frank, A.; Maracci, M. Multi-touch technology and preschoolers’ development of number-sense. Digit. Exp. Math. Educ.
**2015**, 1, 7–27. [Google Scholar] [CrossRef] - Jupri, A.; Drijvers, P.; Van den Heuvel-Panhuizen, M. An instrumentation theory view on students’ use of an applet for algebraic substitution. Int. J. Technol. Math. Educ.
**2016**, 23, 63–80. [Google Scholar] - Baccaglini-Frank, A.; Carotenuto, G.; Sinclair, N. Eliciting preschoolers’ number abilities using open, interactive environments. ZDM Math. Educ.
**2020**, 52, 779–791. [Google Scholar] [CrossRef] - Geraniou, E.; Jankvist, U.T. Towards a definition of “mathematical digital competency”. Educ. Stud. Math.
**2019**, 102, 29–45. [Google Scholar] [CrossRef]

**Figure 1.**Screenshot of eXpresser, in which a “crosses” pattern has been constructed by repeating a building block made of 4 blue tiles and a green tile. Rules for the numbers of tiles of each colour in the pattern have been constructed using the “unlocked” number (which is 3 in this instance) named “z”.

**Figure 2.**(

**a**,

**b**). The bridges model and the fence model as presented on the screen in activities (

**a**) 1 and (

**b**) 2.

**Figure 3.**Molly successfully coloured the green tiles for the 6 repetitions using the number generator, as highlighted by the red circles added to the screenshot.

**Figure 4.**Molly writes a correct rule for the red tiles (5 × 6), calculates its value (30) and uses this calculated value to replace her previous answer (which she first tried to fix by unlocking it), highlighted by the red circles added to the screenshot.

Specific Task | Artefacts Used | Instrumentation Scheme: Compute and Answer |
---|---|---|

colour the green tiles (see Figure 3) | number generator; properties window | technique: compute the number of tiles mentally; answer with a number |

conceptual elements: perceiving structure visually (as per AWOT.1) and describing it arithmetically: seeing 12 as 6 blocks of 2 | ||

technical elements: use the number generator to write the number of green tiles; drag this into the “?” under “How many tiles?” in the properties window |

Specific Task | Artefacts Used | Instrumentation Scheme: Calculate Value of Expression |
---|---|---|

colour the red tiles (see Figure 3) | geometric representation of the pattern; expression -blocks; properties window; calculate value | technique: make an expression for the number of tiles and calculate its value; answer with a number |

conceptual elements: a numerical calculation can be represented symbolically; a number must be used to answer, “How many red tiles?”; perceiving structure, and recognizing and articulating generalisations expressing them symbolically through the expression-blocks (as per AWOT.3) | ||

technical elements: drag onto the canvas a number from the building block properties window; drag onto the canvas another number; multiply the two numbers on the canvas; calculate the value; use this number to answer, “How many tiles?” |

Specific Task | Artefacts Used | Instrumentation Scheme: Calculate Value of Expression with Unlocked Number |
---|---|---|

colour the red tiles (see Figure 4) | geometric representation of the pattern; unlocked number; pattern animation; expression blocks; properties windows; calculate value | technique: make an expression for the number of tiles, using the unlocked number of building blocks, and calculate its value; answer with a number |

conceptual elements: a numerical calculation can be represented symbolically; a numerical calculation can be related to a geometrical structure (working towards AWOT.1); there are special numbers that “can change” (working towards AWOT.2); a number must be used to answer, “How many red tiles?” (working towards AWOT.3) | ||

technical elements: drag out the number of tiles of the desired colour in a building block properties window; drag out the unlocked number of repetitions of the building block; multiply these two numbers; calculate the value; use the calculated value to answer “How many tiles?” |

**Table 4.**Molly’s instantiation 10 (the final one) of her instrumentation scheme for the colouring task.

Specific Task | Artefacts Used | Instrumentation Scheme: Answer with General Rule |
---|---|---|

colour the tiles of a selected colour | geometric representation of the pattern; expression-blocks; number generator; calculate value; properties window; unlocked number | technique: make an expression for the number of tiles; use it to answer |

conceptual elements: a numerical calculation can be represented symbolically; a numerical calculation can represent properties of a geometrical structure; algebraic rules can answer questions by asking “how many…?” (but it is better if you calculate the value in your head ahead of time); perceiving structure, seeing the general in the particular and articulating generalisations (working towards AWOT.1, AWOT.2, AWOT.3) | ||

technical elements: calculate the number of tiles and type that number into the number generator; drag out the number of tiles of the desired colour in a building block properties window; unlock the number of repetitions of the building block; drag out the number of repetitions; multiply the two numbers; use the rule in the answer to answer the “How many tiles?” question |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baccaglini-Frank, A.E.; Geraniou, E.; Hoyles, C.; Noss, R.
How Learning to Speak the Language of a Computer-Based Digital Environment Can Plant Seeds of Algebraic Generalisation: The Case of a 12-Year-Old Student and eXpresser. *Educ. Sci.* **2024**, *14*, 409.
https://doi.org/10.3390/educsci14040409

**AMA Style**

Baccaglini-Frank AE, Geraniou E, Hoyles C, Noss R.
How Learning to Speak the Language of a Computer-Based Digital Environment Can Plant Seeds of Algebraic Generalisation: The Case of a 12-Year-Old Student and eXpresser. *Education Sciences*. 2024; 14(4):409.
https://doi.org/10.3390/educsci14040409

**Chicago/Turabian Style**

Baccaglini-Frank, Anna E., Eirini Geraniou, Celia Hoyles, and Richard Noss.
2024. "How Learning to Speak the Language of a Computer-Based Digital Environment Can Plant Seeds of Algebraic Generalisation: The Case of a 12-Year-Old Student and eXpresser" *Education Sciences* 14, no. 4: 409.
https://doi.org/10.3390/educsci14040409