This study explored the utilisation of GeoGebra as a modelling tool to develop undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers. This mission was accomplished by implementing GeoGebra-enriched activities, which provided carefully designed representational support to mediate between students’ initially developed conceptual and procedural knowledge gains. The rectangular and polar forms of the complex number were connected and merged using GeoGebra’s computer algebra systems and dynamic geometric systems platforms. Despite the centrality of complex numbers to the undergraduate mathematics curriculum, students tend to experience conceptual and procedural obstacles in mathematics-dependent physics engineering topics such as mechanical vector analysis and electric-circuit theory. The study adopted an exploratory sequential mixed methods design and involved purposively selected first-year engineering mathematics students at a South African university. The constructivist approach and Realistic Mathematical Education underpinned the empirical investigation. Data were collected from students’ scripts. Implementing GeoGebra-enriched activities and providing carefully designed representational support sought to enhance students’ conceptual and procedural knowledge of complex numbers and problem representational competence. The intervention additionally helped students to conceptualise and visualise a complex rectangular number. Implications for technology-enhanced pedagogy are discussed.

The article provides exploratory insights into the development of undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers using GeoGebra as a dynamic digital tool. Key findings from the study demonstrated that GeoGebra appears to be an effective modelling tool that can be harnessed to demystify the complexity of mathematics students’ conceptual and procedural knowledge of complex numbers.

The complex number system (^{2} + 1 = 0. However, they are now essential in learning post-school mathematics-dependent topics in engineering physics courses (Smith et al., ^{2} = −1). While many mathematicians, GeoGebra, and researchers use

The link between a complex number representation’s rectangular and polar forms.

A complex number can also be expressed in exponential form ^{jθ}

Complex numbers’ multiple forms and representations are equally essential and intertwine algebraic, trigonometric, and exponential functions. Therefore, students must understand how to navigate within and among the three forms. Lesh et al. (

Furthermore, many studies at various educational levels reveal that students need help with conceptual and procedural knowledge of complex numbers (Ahmad & Shahrill, ^{1}

Despite the centrality of complex numbers to post-school mathematics-dependent topics in engineering physics courses (Smith et al.,

Many students need clarification on different forms and representations of complex numbers (Hui & Lam,

Given the articulated research problem, the study is accordingly premised on the following research questions: (1)

The study explored the affordances of GeoGebra as a modelling tool to develop undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers. The following objectives underpinned the empirical investigation:

To develop undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers.

To develop undergraduate engineering mathematics students’ problem representational competence in complex numbers.

To explore the effectiveness of GeoGebra-enriched activities on undergraduate engineering mathematics students’ academic achievement in complex numbers tasks.

There is a paucity of studies on the teaching and learning of complex numbers; therefore, there is a need for more studies on them. Norlander and Norlander (

Ramaila and Seloane (

The assertion by Veith and Bitzenbauer (

This empirical study focuses on the efficacy of GeoGebra as a mathematical teaching tool, focusing on: (1) the enhancement of students’ conceptual knowledge by connecting, linking, reifying, and merging the different forms of complex numbers and (2) the improvement of students’ overall achievements on complex numbers tasks. It is for this reason that this study lessened the abstraction of complex numbers by exploring the visualisation and multi-representational affordances of GeoGebra as a modelling tool.

Hiebert and Lefevre (

In this study, students shall be deemed to have attained conceptual and procedural understanding of a complex number if they: (1) implicitly show an understanding that a complex number is one coherent mathematical entity expressible in three different representations, (2) know how the forms and representations are interconnected and can further reconstruct each, and (3) can convert or move seamlessly within and between the forms of representation where appropriate. There are many and sometimes polarised theoretical viewpoints on the existence or non-existence of the relationship between conceptual and procedural knowledge which one must be taught first. Gelman and Williams (

Iterative model for the development of conceptual and procedural knowledge.

Rittle-Johnson et al. (

Hallet et al. (

On the basis of these findings, Hallet et al. (

This study focused on the effect of GeoGebra-enriched activities on undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers. Research at various educational levels found GeoGebra to be effective in the development of students’ conceptual and procedural knowledge of many mathematics topics such as calculus (Oscal,

The study adopted a concurrent mixed methods design (Creswell,

The study involved 48 volunteering first-year mechanical and industrial, and electrical engineering mathematics students at a South African university. Twenty-four students from the Mechanical and Industrial Engineering groups constituted the control group (CG), while the other 24 were from the electrical group and formed the experimental group (EG). The Mechanical and Industrial, and electrical groups didn’t attend modules together and hence were purposively selected, minimising potential contamination. The test was administered to the two groups as a pre-test and post-test.

The lecturer-researcher taught both groups complex numbers simultaneously for two weeks. The CG intervention was conducted using the conventional marker-and-whiteboard, pen-and-paper traditional approach. As for the EG, the lecturer was the only one using GeoGebra in class. However, he assisted students in downloading the software and encouraged them to take activities beyond the mathematics classroom. Hence, some activities were given to students as ‘homework’, and feedback was provided in the next session.

The lecturer-researcher used interactive teaching and learning facilitated by implementing the GeoGebra-enabled instructional intervention and traditional approach. GeoGebra was used to prepare the teaching instructions (which interacted with a whiteboard and pen-and-paper) for demonstration, explanation, and visualisation during the intervention. Hohenwarter and Fuchs (

Summary of Hypothetical Learning Trajectory for the teaching of complex numbers.

Phase | Designed activities | Theories addressed |
---|---|---|

1 | Solutions of linear, quadratic (real and non-real roots) equations | Constructivism RME: reality, activity |

2 | Introduction of a complex number in rectangular form; modelling the sum and difference of two complex numbers in rectangular form | Constructivism RME: activity, level Guided reinvention Instrumentation: explain-the-screen, discuss-the-screen |

3 | Simplifying expressions involving sums/differences/products/quotients of complex numbers in rectangular form | Constructivism RME: activity, level Guided reinvention Instrumentation: discuss-the-screen, work-and-walk |

4 | Introduction of polar and exponential forms of complex numbers. Linking polar and rectangular forms. Modelling the product/quotient of complex polar numbers. The conversions between the different forms of complex numbers | Constructivism RME: activity, level Guided reinvention, intertwinement Instrumentation: explain-the-screen, discuss-the-screen, work-and-walk |

5 | Complex expressions involving powers and quotients of complex numbers | Constructivism RME: activity, Guided reinvention, intertwinement Instrumentation: discuss-the-screen, work-and-walk |

6 | Roots of complex numbers | Constructivism RME: activity, Guided reinvention, intertwinement Instrumentation: discuss-the-screen, work-and-walk |

RME, Realistic Mathematical Education.

Phase 1, a precursor for introducing a complex number in phase 2, was conventionally underpinned by constructivism and the reality and activity approaches of RME. The introduction of complex numbers in phase 2 was further driven by Driver and Tarran’s (

Two complex numbers, _{1} = 2 – _{2} = 1 + 2_{3} = _{1} + _{2} = 3 + _{1} vertically in the graphic/geometrical window. Through the interactions, guided reinvention, and level principles of RME, the lecturer-researcher guided students in linking the visible geometrical and algebraic changes of the real parts of _{1} to the changes in the dependent _{3}. The most prominent lecturer-researcher orchestrations were explain-the-screen and discuss-the-screen. Students engaged, self-discovered, and correctly linked the material numerical changes of the imaginary parts of _{3} in the algebraic and graphic windows when _{1} was dragged slowly horizontally in the geometrical window. These processes and procedures are impossible in the traditional pen-and-paper approaches. The GeoGebra-facilitated activities provided students with ample opportunities for meaningful explorations, engagements, physical interactions, self-discovery, and visualisations of the sum modelling of the rectangular form of the complex numbers, thereby enhancing the attainment of conceptual knowledge (Baroody et al., _{1} = _{1} + _{1} and _{2} = _{2} + _{2}, the algebraic sum formula is _{3} = (_{1} + _{2}) + _{1} + _{2}). This is like Sfard (_{3} = (_{1} + _{2}) + _{1} + _{2}).

Exploration of GeoGebra’s enablement and visualisation modelling of the sum of two rectangular complex numbers: (a) _{3} = (2 − _{3} = (2 + _{3} = (2 − 2_{3} = (1 − _{3} = (3 − _{3} = (1 + 2

Phase 4 was driven through constructivism and the activity, interaction, level, guided reinvention, and intertwinement principles of RME to represent, connect, link, and merge complex numbers’ different representations (rectangular and polar) to enhance conceptual understanding of complex numbers further (Haapasalo,

How GeoGebra’s visual affordance is explored to show the link between the rectangular and polar forms of complex numbers: (a) _{1} = 2 + 2_{1} = 3 + 3_{1} = 4 + 4_{1} = 2_{1} = −2 + 2_{1} = 5 + 0

Quantitative and qualitative data were collected through a complex analysis test (CAT), administered as a pre-test and post-test to both EG and CG. Students’ scripts (pre-test and post-test) were analysed qualitatively for conceptual and procedural knowledge and quantitatively to determine general academic achievements and improvements in tasks involving complex numbers. To ensure the validity and reliability of the CAT, students’ scripts were reviewed and moderated by two lecturers (who were mathematics experts and lecturing the other first-year engineering mathematics groups).

Students’ scripts were analysed qualitatively to determine the effectiveness of GeoGebra in developing undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers.

Students’ scripts were also analysed quantitatively using both descriptive and inferential statistics guided by the following four null hypotheses to gauge the effectiveness of GeoGebra-enriched activities on undergraduate engineering mathematics students’ overall academic achievement in tasks involving complex numbers:

_{01}

_{02}

_{03}

_{04}

Appropriately paired (dependent) and independent t-tests were conducted using Statistical Package for Social Sciences (SPSS) version 25 to test the hypotheses. This indicated whether there was any statistically significant difference between or within the means of the two groups. Cohen’s

Although the enablement and visual affordances of GeoGebra-facilitated intervention underpinned by the six principles of RME and carefully designed problem representation support improve students’ problem representation and enhanced conceptual and procedural knowledge of some students, the CG students also benefitted from the conventional approach.

Below are EG and CG vignette samples tracing conceptual and procedural understanding progress (or lack thereof) of two students’ pre-test and post-test.

Vignette of EG student showing initial knowledge during the pre-test.

Vignette of EG student showing post-intervention knowledge during post-test.

Vignette of CG student’s pre-test showing initial knowledge.

Vignette of CG student’s post-test script showing post-intervention knowledge.

Students from both groups benefitted from their respective interventions for the problem representational support. Although both sampled students improved their marks for these two problem representations support test items, there are some conceptual and procedural concerns. The most appropriate approach for test item one is as follows:

However, the EG student converted the argument, 30°, from degrees into radians first before the calculations, which is a long and unnecessary approach. Maybe the student confused the polar form (expressible in degrees or radians) with an exponential form, where the argument must be in radians only.

The most appropriate approach for test item two is as follows:

Although the CG student got the correct answer in both tests, the concern is the change in the student’s correct procedural approach used during the pre-test.

The student’s mark improved from 50% (2 out of 4: see

Students’ total test item marks after problem representation support (

Group | Pre-test | Post-test |
---|---|---|

Experimental group | 31/(24×4) = 31/96 = 32.29% | 86/(24×4) = 86/96 = 89.58% |

Control group | 43/(24×4) = 43/96 = 44.79% | 66/(24×4) = 66/96 = 68.75% |

Furthermore, the vignettes shown in

EG student’s initial geometrical interpretation of the sum of rectangular complex numbers.

CG student’s initial geometrical interpretation of the sum of rectangular complex numbers.

EG student’s post-intervention geometrical interpretation of the sum of rectangular complex numbers.

CG student’s post-intervention geometrical interpretation of the sum of rectangular complex numbers.

The lecturer-researcher guided the EG and CG students in the geometric interpretation meaning of the sum of rectangular complex numbers. Subsequently, the two sample students used different approaches. However, what is worth mentioning is the students’ preferences in the post-test, the noticeable power and influence of GeoGebra’s visualisation, and enablement affordance in EG students. The EG student’s approach (see

Students’ total marks for the sum of complex numbers on the Argand diagram (

Group | Pre-test | Post-test |
---|---|---|

Experimental | 13/(24×3) = 13/72 = 18.06% | 62/(24×3) = 62/72 = 86.11% |

Control | 11/(24×3) = 11/72 = 15.28% | 46/(24×3) = 46/72 = 63.89% |

Although a few CG students showed conceptual and procedural improvements (see ^{3}, (3 < −25.78°)^{3} and

CG student’s post-intervention De Moivre’s theorem.

CG student’s post-intervention conceptual and procedural knowledge.

EG student’s post-intervention De Moivre’s theorem.

EG student’s post-intervention conceptual and procedural knowledge.

EG student’s pre-test conceptual and procedural knowledge.

Students’ total marks for the three ‘expert-like behaviours’ test items (

Group | Pre-test | Post-test |
---|---|---|

Experimental | 27/(24×9) = 27/216 = 12.5% | 184/(24×9) = 184/216 = 85.19% |

Control | 36/(24×9) = 36/216 = 16.7% | 121/(24×9) = 121/216 = 56.02% |

Although GeoGebra-enriched intervention helped students to conceptualise and visualise a complex rectangular number, including the role and meaning of the symbol

Basic algebraic and exponential misconceptions, errors, and role of

Some students, in both the EG and the CG, displayed fragmented basic algebraic and exponential knowledge and misconceptions before and after the intervention (see ^{4} + 3 ^{2})^{2} + 3 and correctly substituted ^{2})^{2} + 3 to get (–1)^{2} + 3 in the pre-test (see ^{2} = –1, resulting in a drop of 33.33% marks for this test item. The same student wrote ^{7} – 4 = (^{2})^{5} – 4 = (–1)^{5} – 4 = –5 in the pre-test (see ^{4} + 3 and improved the marks for this test item from 66.67% (2 out of 3) to 100% (3 out of 3), the marks dropped by 33.33% in the other test item in the post-test (see ^{7} –4 = –1 – 4 = –3. The student’s marks for these two items remained at 50% (3 out of 6) in the tests, and it could have been better in both tests had it not been for prevailing misconceptions and weak algebraic knowledge, as highlighted above.

Weak algebraic and exponential displayed pre intervention.

Weak algebraic and exponential knowledge that persisted post intervention.

As indicated earlier, both groups benefitted from their respective interventions, and their overall academic achievements on complex numbers tasks improved and are reflected in

Independent

Test | Group | Mean | Standard deviation | Cohen’s |
Conclusion ( |
||
---|---|---|---|---|---|---|---|

Pre-test | Control | 13.58 | 4.52 | −0.149 | 0.8820 | - | < 0.05 |

Experimental | 13.75 | 3.11 | - | - | - | - | |

Post-test | Control | 19.50 | 4.66 | −3.880 | 0.0003 | 1.12 | < 0.05 |

Experimental | 24.21 | 3.69 | - | - | - | - |

, Not significant;

, significant.

Dependent samples t-test results for the pre-test and post-test for both groups (

Group | Test | Mean | Standard deviation | Cohen’s |
Conclusion ( |
||
---|---|---|---|---|---|---|---|

Control | Pre-test | 5.92 | 3.73 | 7.773 | < 0.00001 | 1.59 | < 0.05 |

Post-test | - | - | - | - | - | - | |

Experimental | Pre-test | 10.46 | 3.60 | 14.235 | < 0.00001 | 2.91 | < 0.05 |

Post-test | - | - | - | - | - | - |

, Significant.

The CG pre-test mean score (

The CG post-test mean score (

The

The CG mean score difference between the pre-test and the post-test (

The EG mean difference score between the pre-test and the post-test (

After providing problem representation support, EG students’ total marks for the two problem representational support test items improved by 177.42% to an average of 3,58 compared to 53.49% recorded by the CG to 2.75. The EG showed significant improvement, which can be attributed to the effectiveness of GeoGebra-enriched activities: (1) modelling the sum and difference of rectangular complex numbers to give it a geometrical meaning, and (2) linking the rectangular and polar forms of complex numbers. The activities, according to Haapasalo (

The EG’s total marks for representing the sum of complex numbers on the Argand diagram improved by 376.92% compared to 318.18% recorded by the CG. The average mark of the EG was 86.11% (62 out of 72) compared to the CG’s 62.89% (46 out of 72) on the post-test. The significance testing for the total marks for this test item yielded a chi-square statistic of 0.0841 and a

Students’ total marks for the three ‘expert-like behaviours’ test items showed that the EG’s total marks for the pre-test conceptual and procedural test items were 33.33% less than the CG’s. However, this changed in the post-test as the EG total marks were 52.07% more than the CG. It is worth mentioning that the EG’s average mark for the two items in the post-test was a remarkable 85.19% (184 out of 216) compared to the CG’s 56.02% (121 out of 216). The significance testing for the total marks for these two conceptual and procedural test items yielded a chi-square statistic of 6.5152 and a

As indicated earlier, both groups benefitted from their respective interventions. At the post-test, (1) the CG’s pre-post-test mean score (

However, what is noticeable is the independent

Although most students displayed what Smith et al. (

The study’s key findings strongly suggest that implementing GeoGebra-enriched activities is promising for developing students’ conceptual and procedural knowledge of complex numbers. The provision of carefully designed representation enhanced students’ problem representational competence linking their initial conceptual and procedural knowledge gains. These gains translated into improved achievement in tasks involving complex numbers. The prevalence of misconceptions hurt students’ learning of complex numbers. There is a crucial need to examine the pedagogical affordances of modelling tools such as GeoGebra to enhance students’ conceptual and procedural knowledge of mathematics topics. Similarly, more complex numbers topics and subtopics could be included in the study, and GeoGebra’s effect could further be investigated over a more extended period. According to Chan and Leung (

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

S.M.P. conceptualised the study and was involved in methodology, formal analysis, investigation, resources and writing, reviewing and editing the article. S.R. and M.N. assisted with validation, formal analysis, investigation, and writing, reviewing and editing the final article.

The Research Ethics Committee of the Faculty of Education at the University of Johannesburg granted permission to conduct research (ethical clearance number Sem. 1-2022-026).

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

The authors confirm that the data supporting the findings of this study are available within the article.

The views and opinions expressed in this article are those of the author and do not reflect the official policy or position of the World Health Organization.

GeoGebra has statistical capabilities as well.