# What do we know about Computing Education in Africa?
A Systematic Review of Computing Education Research Literature

Ismaila Temitayo Sanusi<EMAIL_ADDRESS>0000-0002-5705-6684 School of
Computing, University of Eastern FinlandP.O.Box 111, 80101Joensuu80101Finland
and  Fitsum Gizachew Deriba<EMAIL_ADDRESS>0000-0002-0699-0274 School of
Computing, University of Eastern FinlandP.O.Box 111, 80101Joensuu80101Finland

###### Abstract.

Noticeably, Africa is underrepresented in the computing education research
(CER) community. However, there has been some effort from the researchers in
the region to contribute to the growing need of computing for all. In order to
understand the body of works that emerged from the global south region and
their area of focus in computing education, we conducted a systematic review
of the literature. This research investigates the prominent CER journals and
conferences to discern the kind of research that has been published and how
much contribution they have made to the growing field. Of the 68 selected
studies, 45 papers were from South Africa. The prominent aspect of computing
in the literature is programming, which accounts for 43%. We identified open
areas for research in the context and discussed the implication of our
findings for the development of CER in Africa.

Computing education research, broadening participation, inclusion, diversity,
Africa

††ccs: Social and professional topics Computer Science Education††ccs:
Computing methodologies K-12 Education††ccs: Computing methodologies Higher
Education Institutions Figure 1. Landscape of Computing Education Research in
Africa

##  1\. Introduction

Researchers claimed that the ”CS for All movement is in full swing”
(payton2019reality, ). Meanwhile, little evidence exist to show how the
initiative is promoted in the world’s second-largest and second-most populous
continent, Africa. For over a decade, research within the computer science
education (CSEd) community has decry poor geographic representation and the
adverse effect it may have on computing education (chvalovsky1978computer, ;
wollowski2009expanding, ; becker2021expanding, ). The underrepesentation of
the region was also reported in a recent study by Brecker et al.
(becker2021expanding, ) who noted that one percent of ACM membership and less
than one percent of SIGSCE membership are from Africa. While being a member of
these professional bodies is not a prerequisite to publish in CSEd publication
outlet, there is a report of significantly low participation and
representation in computing education in Africa (becker2021expanding, ).

Increasingly, broadening participation, inclusion, and diversity has become
common terms in computing education. However, the African region remains
underresearched and less represented in the discussion regarding computing
education and its development across all levels from K-12 to higher education
institutions (HEIs) (agbo2023computing, ). Addressing this obvious gap will
contribute to high-quality research and equitable research that meets the
teaching of all students computing. According to Jurado de Los Santos et al.
(jurado2020term, ) equity is related to the need established by many
countries. As a result, it is imperative to explore learners with various
needs and identify opportunities to support them within their contexts to
participate in learning computing (gizachew2022design, ). To provide evidence
for education policymakers and relevant stakeholders on how to achieve
equitable access, participation, and learning experience in computing among
all students, researchers has call for equitable enabling research
(mcgill2023conducting, ; fletcher2021cape, ). In achieving equitable computing
education, it is important to understand across different contexts and regions
how learners, educators, education policymakers among other relevant
stakeholders can be supported to successfully implement computing education.

While it has been established that Africa has been greatly underrepresented in
the global contribution to computing education (mcgill2023meeting, ), there
are still some significant contributions to the field from the region. To
understand the existing research that have been conducted on computing
education research in Africa, we investigated major international computing
education outlets. This paper provides insights from a systematic literature
review, specifically on CER in Africa over the years, making at least the
following contributions:

  * •

Identifying the aspects of computing education that have been the focus of CER
in Africa

  * •

Summarising the computing education research practices within the African
community

  * •

Discussing the implication of the findings for development of CER in Africa

  * •

Identifying how the results of this exposition will advance contributions of
CER in Africa

  * •

Discussing broadening of participation in computing education through the
inclusion of understudied region to encapsulate holistically what equity-
enabling research entails

  * •

Research gaps in computing education in the African context.

Figure 1 presents the landscape of CER in Africa based on the result of our
analysis.

##  2\. Methodology

This study conducted a systematic literature review by following the
guidelines proposed by Kitchenham (kitchenham2004procedures, ). As adopted by
prior studies in computing education research (luxton2018introductory, ), this
review, adopts a structured process which includes: Identifying research
questions, Performing databases search, Selecting studies, Filtering the
studies by evaluating their pertinence, Extracting data, Synthesizing the
results and Writing the review report.

###  2.1. Identifying research questions

Based on the study aim, this study identified three research questions to
guide the study.

RQ1.:

What are the characteristics of CER papers from Africa?

RQ2.:

What aspects of computing education have been the focus of literature from
Africa?

RQ3.:

What open research areas need exploration in computing education in Africa?

###  2.2. Performing databases search

The literature search was performed in July 2023. Since this research focuses
on understanding the research that emanates from Africa, this research adopts
“Africa” as the sole search term within the whole document or field. The use
of Africa as the only search term is justifiable since the databases searched
focus only on computing education research. This study is not interested in
specific fields of interest such as programming, or artificial intelligence
(AI), etc. Hence, the authors of this research believe it is sufficient to
proceed with a single search term.

We used 10 different databases as shown in Table 1. These include the
proceedings of the ACM Conference on Innovation and Technology in Computer
Science Education (ITiCSE), ACM Conference on International Computing
Education Research (ICER), ACM Technical Symposium on Computer Science
Education(SIGCSE TS), Koli Calling International Conference on Computing
Education Research (KOLI), WiPSCE Conference on Primary and Secondary
Computing Education Research (WiPSCE), ACM Global Computing Education
Conference (CompEd), Australasian Computing Education Conference (AUS-CE),
Computer Science Education Research Conference (CSERC) and journal articles
from ACM Transactions on Computing Education (TOCE) and Computer Science
Education (CSE).

CSERC is mostly hosted in Europe but has been hosted once in South Africa in
2016. Till date, none of the other listed conferences has been hosted in
Africa. With CompEd created to be typically hosted outside North America or
Europe, it may be hosted in Africa in the coming years. While other CSEd
publication forums exist, the 10 outlets we consider in this study have been
described as central forums in the CSEd field. Past research has utilized some
of these publication forums (lunn2021computer, ; hao2019systematic, ) to
understand the development of computing education.

Table 1. Database search

ID | Publication outlet | Search result | Possible relevant results | Relevant papers
---|---|---|---|---
1 | SIGCSE TS | 357 | 15 | 9
2 | ITiSCE | 123 | 32 | 17
3 | ICER | 46 | 0 | 0
4 | KOLI | 27 | 12 | 10
5 | WiPSCE | 11 | 2 | 2
6 | CompED | 3 | 0 | 0
7 | CSE | 61 | 6 | 6
8 | ACM TOCE | 79 | 4 | 2
9 | ACE [AUS-CE] | 8 | 0 | 0
10 | CSERC | 27 | 26 | 22
| Total | 742 | 97 | 68

###  2.3. Selecting studies

Based on the identified search terms and publication outlets, the first author
searched and retrieved a total of 742 studies. Most of our searches returned
articles that reported African-Americans as part of their participants. We did
not consider a few studies that combined data from another continent outside
Africa, such as (buchele2013two, ) which utilized samples from Ghana and the
USA. While we recognize that such collaboration broadens and encourages
participation in the CSEd community as opined by (becker2021expanding, ), this
study is interested in studies focusing on subjects within the African
context. Future studies may explore studies that combine participants across
continents or regions.

###  2.4. Inclusion and exclusion criteria

Before proceeding with the selection process for study inclusion, we
established specific inclusion and exclusion criteria to identify potential
papers.

####  2.4.1. Inclusion criteria

All articles to be included should be primary studies conducted with samples
or participants within the African continent and written in the English
language. Note that some papers consider different countries in Africa and
they were included in our analysis (e.g. (tshukudu2023investigating, )). We
included these papers because our interest is to understand works specifically
focused on African participants.

####  2.4.2. Exclusion criteria

Secondary studies, studies conducted in non-English languages, studies
comparing Western and African populations, and studies that combine data from
populations other than those in Africa are excluded from the study.

###  2.5. Filtering the studies by evaluating their pertinence

After potential papers were identified (n=742), all articles were exported to
Rayyan systematic tools for further identification of relevant studies
(johnson2018rayyan, ). Two authors independently scanned the first 100 papers
and categorized them as include, exclude, or maybe, by reading the abstract
and title with the help of a filtering tool. The agreement between raters
demonstrated high reliability according to Cohen’s Kappa test ($\kappa=0.89$),
which results in near perfect according to (belur2021interrater, ). Any
disagreements were addressed through discussion and resolution; in instances
where the two authors held differing opinions on paper classification, they
engaged in discussion until reaching a consensus. Finally, the author
specifically selected papers that reported findings based on participants
within the African context, which yielded 97 papers.

###  2.6. Full-text reading and Data extraction

After identifying potentially relevant papers, pertinent data was extracted
into a spreadsheet by reading the entire document. Following, we perform
coding to each study (wubineh2023exploring, ). This process involved
retrieving details such as study objectives, study characteristics, and
computing aspects as outlined in the selected articles. At this stage, 29
studies were further excluded because they did not meet the inclusion
criteria; for example, studies that reported teaching computing without
empirical evidence were not considered (e.g., (anyanwu1978computer, ;
bishop2004developing, )). Finally, 68 studies were included in this study,
which are eligible for our analysis.

###  2.7. Synthesizing the results

To synthesize the results of our review, we evaluate the individual findings
of the 68 studies on spreadsheets and combine them based on their similar
themes. Subsequently, we analyze and assess the reviewed studies to determine
the outcomes of the research question (pati2018write, ). Finally, drawing upon
the synthesized data, we generate insights from the findings and provide
implications of the study.

###  2.8. Writing the review report

Lastly, we present our findings based on the evidence derived from the
synthesized results of CER included in the study. We succinctly summarize the
CER findings within the African context, interpret their implications, and
draw conclusions aimed at enriching the comprehension of the research
landscape in Africa.

##  3\. Result

###  3.1. Characteristics of the Selected Papers

Figure 2. selected studies

####  3.1.1. Year of Publication and Submission categories

As shown in Figure 2, the 68 papers we identified were published over 3
decades from 1993 – 2023. The highest number of articles were published in
2012 and 2022 respectively with 7 papers. There appears to be increased
submissions from 2012 even though they are not so significant. The total
number of accepted papers per year in computing education publication outlets
currently seems to be about 4 on average.

With respect to the submission categories, most of the articles were published
as full papers (n=50), followed by posters (n=11) and short papers (n=7).

####  3.1.2. Education Level and Context of Study Setting

As shown in Figure 3, 76.5% (n=52) of the studies were from Higher Education
and 16.2% (n=11) were from K-12 education out of which 8 studies focused on
high school students. The remaining 7.5% (n=5) of the studies did not specify
the education level.

Our analysis also shows that only 9 papers stated the geographic area (Urban
vs Rural) where the study were conducted. Out of the 9 papers, 6 were in rural
areas and 3 were conducted in rural and urban areas without comparing samples
in the two distinct location.

####  3.1.3. Research Design

As illustrated in Figure 3, this study indicate that the selected studies uses
different research approaches. The majority (31%; n=22) of the articles used
mixed methods, followed by qualitative methods (25%; n=17), design-based (19%;
n=13), and quantitative methods (16%, n=11). The remaining 5 studies (7%) do
not explicitly state the approaches adopted.

Figure 3. Research method, Education level and Learning setting

###  3.2. Aspects of Computing Education

As shown in Figure 4, this study identified eleven aspects of computing
education that have been the focus of literature from Africa. The eleven areas
of focus are: programming, general CS concepts, software engineering, general
IT concepts, computational thinking, machine learning for K-12, data structure
and algorithm, database design course, data warehouse, cloud computing, and
human-computer interaction (HCI).

####  3.2.1. Programming

Thirty-one studies (43%) were programming-related. From them, seven studies
focused on tool development such as E-tutoring (horner2016tutoring, ),
E-learning (messer2011use, ), Saucode (boateng2021suacode, ; boateng2019keep,
) , Autograd (annor2021autograd, ), Jenuity (van2008jenuity, ), and Recursive
game generator (anyango2021supporting, ). Eight studies were on assessment and
pedagogical strategies to programming (chetty2015towards, ;
pieterse2014managing, ; pieterse2017automatic, ; pieterse2013automated, ;
apiola2012new, ; sanders2006mental, ; scholtz2010mental, ; gotschi2003mental,
). Thirteen studies discussed different aspects and approaches to learning
programming, including game-based learning (oyelere2017integrating, ) and the
use of blended learning (bati2014blended, ); experience, impact, and
cooperativeness in pair programming (benade2017pair, ; mentz2008effect, ;
liebenberg2012pair, ); effectiveness of scaffolding learning (mbogo2016design,
), the impact of iconic programming notation (cilliers2005effect, ),
interactive learning in discussion forums (pieterse2011student, ), and
describing of genetic programming (pillay2004first, ). The other category of
programming-related discussed topics were student perception and experience
(govender2012students, ), workshop experiences (tshukudu2022broadening, ),
misconceptions and challenges in programming (anyango2018teaching, ;
pieterse2018triviality, ; motara2020obstacles, ); the impact of parent
technology adoption on students’ education (breedt2012student, ).

####  3.2.2. General CS Concepts

Thirteen studies (22.4%) discussed general computer science concepts,
including teaching tools (e.g., MobileEdu (oyelere2016evaluating, ), Automatic
grading (klein2011automated, )), 4 studies on professional development such as
teachers and students preconceptions (tshukudu2023investigating, ;
sanders2000fundamentals, ), preparedness (mbogo2019structured, ), and capacity
building (apiola2015building, ). Three studies discussed learning styles and
settings, such as interactive learning (kotze1998hypermedia, ), informal
learning (klomsri2013social, ), and learning styles (galpin2007learning, ).
Another three studies related to curriculum development
(joseph2021ithinksmart, ; motara2020obstacles, ; pieterse2012participation, ),
and one study about student attitude on ethics (boateng2019keep, ).

####  3.2.3. Software Engineering

Five studies (7.3%) were related to software engineering courses: three
discussed student involvement, aspiration, and teamwork in software
development (marshall2017student, ; liebenberg2016career, ;
pieterse2012participation, ); two on the impact of experiential learning
(marshall2016exploration, ), and experiences (pieterse2012intensive, )
respectively.

####  3.2.4. General IT Concepts

five studies (7.2%) were focused on general IT-related concepts. From them,
two (tedre2008implementing, ; mentz2013empowering, ) on professional
development; two (laine2011refreshing, ; marais2016towards, ) on
contextualizing and developing curriculum; one on perception
(govender2007challenges, ).

####  3.2.5. Computational Thinking (CT)

In this aspect, only three studies (4.4%), focus on computational thinking
(integration of ICT in curriculum (kassa2022computational, ), evaluation of
framework (gouws2013computational, ), and education with Virtual reality
(joseph2021ithinksmart, )).

####  3.2.6. Machine Learning for K-12

Two studies on machine learning education focus on how to teach machine
learning to young learners (sanusi2021intercontinental, ; sanusi2023learning,
) within the K-12 levels.

####  3.2.7. Data Structure and Algorithm

Data structure and algorithm with three studies [4.4%] in different concepts
such as the influence of other languages and other subjects in algorithm
courses (rauchas2006language, ), collaborative learning in algorithm courses
(nazir2019teaching, ), and teaching approaches (sanders2002teaching, ).

####  3.2.8. Database Design Course

Database design course (on self-directed learning (zyl2022developing, )). The
study aims to enhance SDL by identifying factors (e.g., considering various
perspectives of peers when solving problems in cooperative learning groups)
that emerged from the data.

####  3.2.9. Data warehouse

Data warehouse (on student view (goede2016listening, )). The author assessed
the perceptions of students who have chosen a data warehousing module using
critical systems heuristics, to improve the module’s design and ensure long-
term benefits for all stakeholders.

####  3.2.10. Cloud Computing

Cloud computing (student engagement (nazir2021enhancing, )). The study
explored and implemented effective approaches for maintaining student
engagement in an online and distance learning environment during the COVID-19
pandemic, focusing on the Cloud Platform Development coursework.

####  3.2.11. Human-Computer Interaction (HCI)

HCI (theoretical knowledge on group project (clarke1998teaching, )). The study
aimed to stress the value of hands-on experience with empirical methods in HCI
education, enabling students to better apply theoretical knowledge to user-
interface design.

Figure 4. Aspects of Computing Education Research in Africa

###  3.3. Open Research Areas for Computing Education Research

Based on the RQ2 findings regarding the aspects of computing education
researchers focused on, we identify some open areas of possible research for
scholars in the context.

####  3.3.1. Emerging Technologies

Our findings indicate that research on understanding how students learn and
educators teach emerging technologies such as artificial intelligence (AI)
(including generative & regenerative AI), machine learning (ML), and quantum
computing is limited. As a result of these findings, research focusing on
teaching and learning these rapidly-emerging technologies needs to be fully
explored including policies that supports their implementation
(temitayo2021teaching, ).

####  3.3.2. Diversity Dimensions

According to McGill et al. (mcgill2023conducting, ), ”diversity dimensions
include various characteristics that have been historically used to
differentiate groups, such as ethnicity, gender, religious beliefs, or socio-
economic status.” Africa is a continent with a very high linguistic diversity
with an estimated 1500-2000 African languages (Oneworld, ). With these
statistics, it is important to explore the complications that may arise based
on language as past research has identified the links between language and
learning computing among other diversity measures (villegas2023effects, ). Our
findings showed that little or no report was given regarding the vulnerable
groups, learning disability (e.g., neurodiversity), language complexity, and
demographic influence in learning computing.

####  3.3.3. Ethics

Only a single paper focused on ethics in the retrieved articles. This finding
is an indication that research on ethical implications has not been a subject
of scrutiny. According to Sætra & Danaher (saetra2022each, ), ethics plays a
key role in the normative analysis of the impacts of technologies of different
types, e.g., computer ethics, AI ethics, data ethics among others. Based on
the proliferation of technology with specific impacts, ethical implications of
these technologies should be understood by exploring several stakeholders in
the region(sanusi2022insight, ).

####  3.3.4. Educational Level

Our results suggest that majority of the studies were conducted in HEIs and
only eleven (16.2%) considered K-12 population. To this end, more research is
required to explore various level and specific grade band from early childhood
education through high school. Each educational stage such as primary, middle
and high school have their own peculiarities which demands to be investigated
to support the students within the cohort. Adult education is another domain
to be examined.

####  3.3.5. Learning Context

Research has shown that learning context has implications for students
learning of computing. For example, studies (dunton2018examining, ;
ma2023developing, ; sanusi2023machine, ) have shown that out-of-school
programs are an essential tool for supporting and expanding K-12 computer
science education. Hence, it is imperative to examine learning opportunities
in-and-out of school system to support all learners within the developing
region.

####  3.3.6. Use of Innovative Approaches

Pedagogy is not a static concept; rather, it evolves over time (kunnari2019co,
). There is a need to always rethink our pedagogy, particularly in a region
with limited evidence of resources and evidence of effective pedagogical
approaches that support students learning of computing. Researching into the
use of novel approaches is in tandem with the global call for development of
innovative ways to prepare learners with adequate knowledge and skills. For
instance, in the ever-changing world, co-design provides an effective approach
to foster engagement in learning(sanusi2023preparing, ), hence teachers must
be willing to co-design with relevant stakeholders (sanusi2022exploring, ).
Globally, there is a need for innovative ways to teach and introduce topics,
learning materials, and ideas to learners regardless of educational level.

####  3.3.7. Teacher Education

Since computer science teacher preparation is critical to the development of
CS education (gal2010computer, ), it is important to investigate this domain.
With the continuous call for computing education for all learners, there
should be increasing discussion about the educators who are to teach the
learners. It is necessary to understand CS teacher education programs and
professional learning opportunities available within the region.

####  3.3.8. Conceptions of Computing

While studies (xu2021exploring, ; grosse2019s, ) have investigated conceptions
of various education stakeholders (e.g., students and educators) in relation
to learning computing, there is a dearth of such research in the developing
region of Africa. To this end, in relation to computing, extensive research
that considers the perspectives of children, parents, education policymakers
among other relevant stakeholders should be conducted to ensure the effective
implementation of computing education initiatives.

##  4\. Discussion

###  4.1. Insights generated from the findings

The number of relevant studies we identified in our search of computing
education publication outlets for our investigation supports the claim that
there is a worrying lack of published work in international venues about CS
education in Africa (mcgill2023meeting, ). The focus of the included papers on
HEIs mostly suggests that less attention is paid to teaching and learning of
CS education at k-12 levels. There is also a clear indication that
geographical location of the participants is not considered important enough
to be reported.

Regarding the aspect of computing that has been the focus of literature from
Africa, we identified programming as the prominent topic. However, these
studies on programming are HEI focused and none of them were conducted in the
context of K-12. This finding shows that there is limited initiative on young
learners and programming education which is consistent with existing research
(tshukudu2022broadening, ). Other computing courses such as ML, CT and
software engineering as shown in Subsection 3.2 were barely researched. The
limited research focus on ML education within the K-12 levels is consistent
with reports in past research (sanusi2021intercontinental, ). However, no
report of a study conducted on AI or ML in HEIs was found.

This study provides an overview of the state of CSEd research in Africa. We
specifically identified the aspects of computing education that have been the
focus within the region. Our findings further provides insight into the areas
of computing that needs exploration. These findings will contribute to
broadening participation in computing. For instance, addressing the open areas
identified in literature will contribute to meeting the needs of all learners
to learn computing across educational level and learning settings.

###  4.2. Limitation and Future research

While this research attempts to be exhaustive by investigating prominent
computing education conferences and journals, Other computer science and
education databases such as ACM, IEEE, and ERIC which may return more data
were not considered. To broaden the search beyond ”Africa,” we recommend
adding specific African country names and employing snowballing for enhanced
study retrieval. Even though this limitation is recognized, our approach is
not inadequate since most the retrieved studies mention the word ”Africa”
despite their focus on specific regions in Africa.

To strengthen computing education in the UK and Ireland, UKICER was initiated.
This is an indication that a CER conference hosted only in African countries
be may be instituted. Future work could investigate computing education
researchers in Africa and Where they publish. In addition, future research
should explore why there is low research output in computing education
research in Africa (Sanusi2024exploring, ).

##  5\. Conclusion

This research employed a systematic review approach to understand the CER
landscape in Africa with a focus on computing education literature. While it
is recognized that some of the research conducted on computing education
within the African context can be published in other forums (e.g. national or
regional publication venues), this study holds the view that the CER-focused
conferences and journal need to be investigated to get an overview of
submissions from the continent. There could be a follow-up study to understand
why conferences such as ICER has no record of contributions from Africa. While
we identified panels have focused on enhancing computer and programming
knowledge in developing countries including Africa (khailany1982enhancing, ;
sterling1978computer, ; adams2006approaches, ), there is no record of subject
matters focusing specifically on CER in Africa. We did not also identify a
report of Working Groups, organised by ITiSCE or CompEd focusing on CER in
Africa. This paper fills a gap in the literature providing a picture of the
state of the published literature about computing education research in
Africa.

###  5.1. Some Guiding Questions for Future Research

Based on the eight open areas we identified for future research, researchers
interested in the context may explore the following questions.

####  5.1.1. Emerging Technologies

How can we design models for young children’s learning of programming?; How
does generative AI supports learning of programming? (deriba2023enhancing, ;
aruleba2023integrating, ); What opportunities exist for teaching quantum
computing concepts in k-12 education in Africa?; How is cybersecurity
education implemented in schools across the region of Africa?; How is AI
curriculum implemented in schools in the developing regions of Africa? What
are teachers practices and disposition and students learning experiences with
respect to AI? How might we develop a framework for teaching generative AI
(thomas2024artificial, )?

####  5.1.2. Diversity Dimensions

What are the implications of language complexity for explaining computing
concepts for CSEd in Africa?; How can learners with varying disabilities
(including neurodiversity) types be supported to learn computing?; How much do
we know about demographic influence in learning computing?; Are there ways to
the address urban versus rural and public versus private schools dichotomy in
computing learning?; What initiatives can instituted to assist street-
connected children and youth in computing education?

####  5.1.3. Ethics

What do children know about computer ethics (including emerging technology)
ethics in Africa?; Is computer ethics education integrated into school
curricula?; How do teachers perceive and teach computer ethics in schools?;
What are the students’ conceptions and attitudes toward computer ethics?

####  5.1.4. Educational Level

Does participating in K-12 computing courses have an impact on students’
future career choices in computing?; What do we know about learning computing
skills in vocational schools and colleges(Oladele2024What, )?

####  5.1.5. Learning Context

What are the out-of-school initiatives that currently support computing
education in Africa?; How can computing education be further promoted in
schools?

####  5.1.6. Use of Innovative Approaches

Does co-design pedagogy support learning of computing?; How can contextualized
resources be employed to support learning of computing ?; How much of support
does unplugged activities (as a low-cost solution) foster for computing
skills?

####  5.1.7. Teacher Education

Where is the computing education teacher program happening in Africa?; Why
there are Low Research Output in Computing Education Research in Africa?
(Sanusi2024exploring, ).

####  5.1.8. Conceptions of Computing

What are young children’s conceptions of computing in African settings?
(oyedoyin2024young, ). What do we know about the perspectives of various
education stakeholders (e.g. parents, students, teachers, and policymakers)
with respect to computing in Africa?

## References

  * (1) Elizabeth Adams, Doug Baldwin, Judith Bishop, John English, Pamela Lawhead, and Daniel Stevenson.  Approaches to teaching the programming languages course: A potpourri.  In Proceedings of the 11th annual SIGCSE conference on Innovation and technology in computer science education, pages 299–300, Bologna, Italy, 2006. ACM Press.
  * (2) Friday Joseph Agbo, Maria Ntinda, Sonsoles López-Pernas, Mohammed Saqr, and Mikko Apiola.  Computing education research in the global south.  In Past, Present and Future of Computing Education Research: A Global Perspective, pages 311–333. Springer, Willamette University, USA, 2023.
  * (3) Prince Steven Annor, Edwin Kayang, Samuel Boateng, and George Boateng.  Autograd: Automated grading software for mobile game assignments in suacode courses.  In Proceedings of the 10th Computer Science Education Research Conference, pages 79–85, ACM New York, NY, USA, 2021. Association for Computing Machinery.
  * (4) Jecton Tocho Anyango and Hussein Suleman.  Teaching programming in kenya and south africa: What is difficult and is it universal?  In Proceedings of the 18th Koli Calling International Conference on Computing Education Research, pages 1–2, 2018.
  * (5) Jecton Tocho Anyango and Hussein Suleman.  Supporting cs1 instructors: Design and evaluation of a game generator.  In Proceedings of the 26th ACM Conference on Innovation and Technology in Computer Science Education V. 1, pages 115–121, ACM New York, NY, USA, 2021. Association for Computing Machinery.
  * (6) JA Anyanwu.  Object-orientation in java for scientific programmers.  In Papers of the SIGCSE/CSA technical symposium on Computer science education, pages 37–40, 1978.
  * (7) Mikko Apiola, Jarkko Suhonen, Abbi Nangawe, and Erkki Sutinen.  Building cs research capacity in sub-saharan africa by implementing a doctoral training program.  In Proceedings of the 46th ACM Technical Symposium on Computer Science Education, pages 633–638, 2015.
  * (8) Mikko Apiola and Matti Tedre.  New perspectives on the pedagogy of programming in a developing country context.  Computer Science Education, 22(3):285–313, 2012.
  * (9) Kehinde Aruleba, Ismaila Temitayo Sanusi, George Obaido, and Blessing Ogbuokiri.  Integrating chatgpt in a computer science course: Students perceptions and suggestions.  arXiv preprint arXiv:2402.01640, 2023.
  * (10) Tesfaye Bayu Bati, Helene Gelderblom, and Judy Van Biljon.  A blended learning approach for teaching computer programming: design for large classes in sub-saharan africa.  Computer Science Education, 24(1):71–99, 2014.
  * (11) Brett A Becker, Amber Settle, Andrew Luxton-Reilly, Briana B Morrison, and Cary Laxer.  Expanding opportunities: Assessing and addressing geographic diversity at the sigcse technical symposium.  In Proceedings of the 52nd ACM Technical Symposium on Computer Science Education, pages 281–287, 2021.
  * (12) Jyoti Belur, Lisa Tompson, Amy Thornton, and Miranda Simon.  Interrater reliability in systematic review methodology: exploring variation in coder decision-making.  Sociological methods & research, 50(2):837–865, 2021.
  * (13) Trudie Benadé and Janet Liebenberg.  Pair programming as a learning method beyond the context of programming.  In Proceedings of the 6th Computer Science Education Research Conference, pages 48–55, 2017.
  * (14) Judith Bishop and Nigel Horspool.  Developing principles of gui programming using views.  In Proceedings of the 35th SIGCSE technical symposium on Computer science education, pages 373–377, 2004.
  * (15) George Boateng, Prince Steven Annor, and Victor Wumbor-Apin Kumbol.  Suacode africa: Teaching coding online to africans using smartphones.  In Proceedings of the 10th Computer Science Education Research Conference, pages 14–20, 2021.
  * (16) George Boateng, Victor Wumbor-Apin Kumbol, and Prince Steven Annor.  Keep calm and code on your phone: A pilot of suacode, an online smartphone-based coding course.  In Proceedings of the 8th Computer Science Education Research Conference, pages 9–14, 2019.
  * (17) Hugo Breedt and Vreda Pieterse.  Student confidence in using computers: the influence of parental adoption of technology.  In Proceedings of Second Computer Science Education Research Conference, pages 17–21, 2012.
  * (18) Suzanne Fox Buchele.  Two models of a cryptography and computer security class in a liberal arts context.  In Proceeding of the 44th ACM technical symposium on Computer science education, pages 543–548, 2013.
  * (19) Jacqui Chetty and Duan van der Westhuizen.  Towards a pedagogical design for teaching novice programmers: design-based research as an empirical determinant for success.  In Proceedings of the 15th Koli Calling Conference on Computing Education Research, pages 5–12, 2015.
  * (20) Thomas KF Chiu, Zubair Ahmad, Murod Ismailov, and Ismaila Temitayo Sanusi.  What are artificial intelligence literacy and competency? a comprehensive framework to support them.  Computers and Education Open, page 100171, 2024.
  * (21) Vaclav Chvalovsky.  Computer science education at universities: the case of developing countries.  In Papers of the SIGCSE/CSA technical symposium on Computer science education, pages 41–47, 1978.
  * (22) Charmain Cilliers, André Calitz, and Jéan Greyling.  The effect of integrating an iconic programming notation into cs1.  In Proceedings of the 10th annual SIGCSE conference on Innovation and technology in computer science education, pages 108–112, 2005.
  * (23) Matthew C Clarke.  Teaching the empirical approach to designing human-computer interaction via an experiential group project.  In Proceedings of the twenty-ninth SIGCSE technical symposium on Computer science education, pages 198–201, 1998.
  * (24) Fitsum Gizachew Deriba, Ismaila Temitayo Sanusi, and Amos Oyelere Sunday.  Enhancing computer programming education using chatgpt-a mini review.  In Proceedings of the 23rd Koli Calling International Conference on Computing Education Research, pages 1–2, 2023.
  * (25) Sarah T Dunton and Stephanie Rodriguez.  Examining the role of informal education in k-12 computing pathways & cs education reform efforts.  In Proceedings of the 49th ACM Technical Symposium on Computer Science Education, pages 1064–1064, 2018.
  * (26) Carol L Fletcher and Jayce R Warner.  Cape: A framework for assessing equity throughout the computer science education ecosystem.  Communications of the ACM, 64(2):23–25, 2021.
  * (27) Judith Gal-Ezer and Chris Stephenson.  Computer science teacher preparation is critical.  ACM Inroads, 1(1):61–66, 2010.
  * (28) Vashti C Galpin, Ian D Sanders, and Pei-yu Chen.  Learning styles and personality types of computer science students at a south african university.  ACM SIGCSE Bulletin, 39(3):201–205, 2007.
  * (29) Fitsum Gizachew.  Design and implementation of amharic-based ide with high-level programming language.  EAI Endorsed Transactions on Creative Technologies, 9(30):e2–e2, 2022.
  * (30) Roelien Goede.  Listening to the affected: Student views after starting a 4th year module in data warehousing.  In Proceedings of the Computer Science Education Research Conference 2016, pages 12–21, 2016.
  * (31) Tina Götschi, Ian Sanders, and Vashti Galpin.  Mental models of recursion.  In Proceedings of the 34th SIGCSE technical symposium on Computer science education, pages 346–350, 2003.
  * (32) Lindsey Ann Gouws, Karen Bradshaw, and Peter Wentworth.  Computational thinking in educational activities: an evaluation of the educational game light-bot.  In Proceedings of the 18th ACM conference on Innovation and technology in computer science education, pages 10–15, 2013.
  * (33) Desmond Wesley Govender and Irene Govender.  Are students learning object oriented programming in an object oriented programming course? student voices.  In Proceedings of the 17th ACM annual conference on Innovation and technology in computer science education, pages 395–395, 2012.
  * (34) Desmond Wesley Govender and Manoj Maharaj.  Challenges with respect to the e-readiness of secondary school teachers in kwazulu-natal, south africa.  In Proceedings of the 12th annual SIGCSE conference on Innovation and technology in computer science education, pages 191–195, 2007.
  * (35) Gregor Große-Bölting, Yannick Schneider, and Andreas Mühling.  It’s like computers speak a different language: Beginning students’ conceptions of computer science.  In Proceedings of the 19th Koli Calling International Conference on Computing Education Research, pages 1–5, 2019.
  * (36) Qiang Hao, David H Smith IV, Naitra Iriumi, Michail Tsikerdekis, and Amy J Ko.  A systematic investigation of replications in computing education research.  ACM Transactions on Computing Education (TOCE), 19(4):1–18, 2019.
  * (37) V Horner and Patricia Gouws.  E-tutoring support for undergraduate students learning computer programming at the university of south africa.  In Proceedings of the Computer Science Education Research Conference 2016, pages 29–36, 2016.
  * (38) Nastasha Johnson and Margaret Phillips.  Rayyan for systematic reviews.  Journal of Electronic Resources Librarianship, 30(1):46–48, 2018.
  * (39) Friday Joseph Agbo, Solomon Sunday Oyelere, Jarkko Suhonen, and Markku Tukiainen.  ithinksmart: Immersive virtual reality mini games to facilitate students’ computational thinking skills.  In Proceedings of the 21st Koli Calling International Conference on Computing Education Research, pages 1–3, 2021.
  * (40) Pedro Jurado de Los Santos, Antonio-José Moreno-Guerrero, José-Antonio Marín-Marín, and Rebeca Soler Costa.  The term equity in education: A literature review with scientific mapping in web of science.  International Journal of Environmental Research and Public Health, 17(10):3526, 2020.
  * (41) Ermias Abebe Kassa and Enguday Ademe Mekonnen.  Computational thinking in the ethiopian secondary school ict curriculum.  Computer Science Education, 32(4):502–531, 2022.
  * (42) Asad Khailany.  Enhancing computer knowledge in less developed countries(panel discussion).  ACM SIGCSE Bulletin, 14(1):260, 1982.
  * (43) Barbara Kitchenham.  Procedures for performing systematic reviews.  Keele, UK, Keele University, 33(2004):1–26, 2004.
  * (44) Richard Klein, Angelo Kyrilov, and Mayya Tokman.  Automated assessment of short free-text responses in computer science using latent semantic analysis.  In Proceedings of the 16th annual joint conference on Innovation and technology in computer science education, pages 158–162, 2011.
  * (45) Tina Klomsri, Linn Grebäck, and Matti Tedre.  Social media in everyday learning: How facebook supports informal learning among young adults in south africa.  In Proceedings of the 13th Koli Calling International Conference on Computing Education Research, pages 135–144, 2013.
  * (46) Paula Kotzé.  Why the hypermedia model is inadequate for computer-based instruction.  ACM SIGCSE Bulletin, 30(3):148–152, 1998.
  * (47) Irma Kunnari, Jari Jussila, Vesa Tuomela, and Jukka Raitanen.  Co-creation pedagogy from cschool towards hamk design factory.  2019\.
  * (48) Teemu H Laine and Erkki Sutinen.  Refreshing contextualised it curriculum with a pervasive game project in tanzania.  In Proceedings of the 11th Koli Calling International Conference on Computing Education Research, pages 66–75, 2011.
  * (49) Janet Liebenberg, Elsa Mentz, and Betty Breed.  Pair programming and secondary school girls’ enjoyment of programming and the subject information technology (it).  Computer Science Education, 22(3):219–236, 2012.
  * (50) Janet Liebenberg and Vreda Pieterse.  Career goals of software development professionals and software development students.  In Proceedings of the Computer Science Education Research Conference 2016, pages 22–28, 2016.
  * (51) Stephanie Lunn, Maíra Marques Samary, and Alan Peterfreund.  Where is computer science education research happening?  In Proceedings of the 52nd ACM Technical Symposium on Computer Science Education, pages 288–294, 2021.
  * (52) Andrew Luxton-Reilly, Simon, Ibrahim Albluwi, Brett A Becker, Michail Giannakos, Amruth N Kumar, Linda Ott, James Paterson, Michael James Scott, Judy Sheard, et al.  Introductory programming: a systematic literature review.  In Proceedings companion of the 23rd annual ACM conference on innovation and technology in computer science education, pages 55–106, 2018.
  * (53) Ruizhe Ma, Ismaila Temitayo Sanusi, Vaishali Mahipal, Joseph E Gonzales, and Fred G Martin.  Developing machine learning algorithm literacy with novel plugged and unplugged approaches.  In Proceedings of the 54th ACM Technical Symposium on Computer Science Education V. 1, pages 298–304, 2023.
  * (54) Craig Marais and Karen Bradshaw.  Towards a technical skills curriculum to supplement traditional computer science teaching.  In Proceedings of the 2016 ACM Conference on Innovation and Technology in Computer Science Education, pages 338–343, 2016.
  * (55) Linda Marshall and Janet Liebenberg.  Student curriculum development buy-in: A study from an educational software development module.  In Proceedings of the 6th Computer Science Education Research Conference, pages 66–72, 2017.
  * (56) Linda Marshall, Vreda Pieterse, Lisa Thompson, and Dina M Venter.  Exploration of participation in student software engineering teams.  ACM Transactions on Computing Education (TOCE), 16(2):1–38, 2016.
  * (57) Chao Mbogo.  A structured mentorship model for computer science university students in kenya.  In Proceedings of the 50th ACM Technical Symposium on Computer Science Education, pages 1109–1115, 2019.
  * (58) Chao Mbogo, Edwin Blake, and Hussein Suleman.  Design and use of static scaffolding techniques to support java programming on a mobile phone.  In Proceedings of the 2016 ACM Conference on Innovation and Technology in Computer Science Education, pages 314–319, 2016.
  * (59) Monica M McGill, Leigh Ann DeLyser, Ismaila Temitayo Sanusi, and Selina Marianna Shah.  Meeting the needs of all learners through high quality k-12 computing education research.  In Proceedings of the ACM Conference on Global Computing Education Vol 2, pages 185–186, 2023.
  * (60) Monica M McGill, Sarah Heckman, Christos Chytas, Michael Liut, Vera Kazakova, Ismaila Temitayo Sanusi, Selina Marianna Shah, and Claudia Szabo.  Conducting sound, equity-enabling computing education research.  In Proceedings of the 2023 Working Group Reports on Innovation and Technology in Computer Science Education, pages 30–56. 2023.
  * (61) Elsa Mentz, Roxanne Bailey, Betty Breed, and Marietjie Havenga.  Empowering information technology teachers through professional development: an evaluation.  In Proceedings of the 8th Workshop in Primary and Secondary Computing Education, pages 37–38, 2013.
  * (62) Elsa Mentz, Johannes L van der Walt, and Leila Goosen.  The effect of incorporating cooperative learning principles in pair programming for student teachers.  Computer science education, 18(4):247–260, 2008.
  * (63) Orry M Messer and Angelo Kyrilov.  The use of mediating artifacts in embedding problem solving processes in an e-learning environment.  In Proceedings of the 16th annual joint conference on Innovation and technology in computer science education, pages 390–390, 2011.
  * (64) Yusuf Moosa Motara.  Obstacles when teaching functional programming.  In Proceedings of the 9th Computer Science Education Research Conference, pages 1–2, 2020.
  * (65) Sajid Nazir, Stephen Naicken, and James H Paterson.  Teaching data structures through group based collaborative peer interactions.  In Proceedings of the 8th Computer Science Education Research Conference, pages 98–103, 2019.
  * (66) Sajid Nazir, Yovin Poorun, James Paterson, and Brian Hainey.  Enhancing student engagement through cloud computing coursework: Challenges and opportunities in the time of covid-19.  In Proceedings of the 10th Computer Science Education Research Conference, pages 110–112, 2021.
  * (67) Campbell O. Oladele, Ismaila Temitayo Sanusi, and Harrison I. Atagana.  From palette to program: Exploring the impact of visual art backgrounds on programming achievement in college scratch classes.  In Proceedings of the 2024 Innovation and Technology in Computer Science Education. 2024.
  * (68) One World Nations Online.  List of official, national and spoken languages of africa., 2023.
  * (69) Mayowa Oyedoyin, Ismaila Temitayo Sanusi, and Musa Adekunle Ayanwale.  Young children’s conceptions of computing in an african setting.  Computer Science Education, pages 1–36, 2024.
  * (70) Solomon S Oyelere, Jarkko Suhonen, Greg M Wajiga, and Erkki Sutinen.  Evaluating mobileedu: third-year undergraduate computer science students’ mobile learning achievements.  In Proceedings of the 16th Koli Calling International Conference on Computing Education Research, pages 176–177, 2016.
  * (71) Solomon Sunday Oyelere, Jarkko Suhonen, and Teemu H Laine.  Integrating parson’s programming puzzles into a game-based mobile learning application.  In Proceedings of the 17th Koli Calling International Conference on Computing Education Research, pages 158–162, 2017.
  * (72) Debajyoti Pati and Lesa N Lorusso.  How to write a systematic review of the literature.  HERD: Health Environments Research & Design Journal, 11(1):15–30, 2018.
  * (73) Jamie Payton, Jamika D Burge, and Jill Denner.  The reality of inclusion: The role of relationships, identity, and academic culture in inclusive and equitable practices for broadening participation in computing education.  In Proceedings of the 50th ACM Technical Symposium on Computer Science Education, pages 494–495, 2019.
  * (74) Vreda Pieterse.  Automated assessment of programming assignments.  CSERC, 13:4–5, 2013.
  * (75) Vreda Pieterse and Janet Liebenberg.  Automatic vs manual assessment of programming tasks.  In Proceedings of the 17th Koli Calling International Conference on Computing Education Research, pages 193–194, 2017.
  * (76) Vreda Pieterse and Christoph Stallmann.  Managing a large tertiary computer science class.  In Proceedings of the Computer Science Education Research Conference, pages 79–90, 2014.
  * (77) Vreda Pieterse and Estelle Taylor.  On the triviality of the assignment statement.  In Proceedings of the 7th Computer Science Education Research Conference, pages 51–57, 2018.
  * (78) Vreda Pieterse, Lisa Thompson, Linda Marshall, and Dina M Venter.  An intensive software engineering learning experience.  In Proceedings of Second Computer Science Education Research Conference, pages 47–54, 2012.
  * (79) Vreda Pieterse, Lisa Thompson, Linda Marshall, and Dina M Venter.  Participation patterns in student teams.  In Proceedings of the 43rd ACM technical symposium on Computer Science Education, pages 265–270, 2012.
  * (80) Vreda Pieterse and Isabel J Van Rooyen.  Student discussion forums: what is in it for them?  In Computer Science Education Research Conference, pages 59–70, 2011.
  * (81) Nelishia Pillay.  A first course in genetic programming.  ACM SIGCSE Bulletin, 36(4):93–96, 2004.
  * (82) Sarah Rauchas, Benjamin Rosman, George Konidaris, and Ian Sanders.  Language performance at high school and success in first year computer science.  ACM SIGCSE Bulletin, 38(1):398–402, 2006.
  * (83) Henrik Skaug Sætra and John Danaher.  To each technology its own ethics: The problem of ethical proliferation.  Philosophy & Technology, 35(4):93, 2022.
  * (84) Ian Sanders.  Teaching empirical analysis of algorithms.  ACM SIGCSE Bulletin, 34(1):321–325, 2002.
  * (85) Ian Sanders, Vashti Galpin, and Tina Götschi.  Mental models of recursion revisited.  In Proceedings of the 11th annual sigcse conference on innovation and technology in computer science education, pages 138–142, 2006.
  * (86) Ian Sanders and Conrad Mueller.  A fundamentals-based curriculum for first year computer science.  ACM SIGCSE Bulletin, 32(1):227–231, 2000.
  * (87) Ismaila Temitayo Sanusi.  Intercontinental evidence on learners’ differentials in sense-making of machine learning in schools.  In Proceedings of the 21st Koli Calling International Conference on Computing Education Research, pages 1–2, 2021.
  * (88) Ismaila Temitayo Sanusi.  Machine Learning Education in the K–12 Context.  PhD thesis, Itä-Suomen yliopisto, 2023.
  * (89) Ismaila Temitayo Sanusi and Sunday Adewale Olaleye.  An insight into cultural competence and ethics in k-12 artificial intelligence education.  In 2022 IEEE global engineering education conference (EDUCON), pages 790–794. IEEE, 2022.
  * (90) Ismaila Temitayo Sanusi, Joseph Olamide Omidiora, Solomon Sunday Oyelere, Henriikka Vartiainen, Jarkko Suhonen, and Markku Tukiainen.  Preparing middle schoolers for a machine learning–enabled future through design-oriented pedagogy.  IEEE Access, 2023.
  * (91) Ismaila Temitayo Sanusi, Solomon Sunday Oyelere, and Joseph Olamide Omidiora.  Exploring teachers’ preconceptions of teaching machine learning in high school: A preliminary insight from africa.  Computers and Education Open, 3:100072, 2022.
  * (92) Ismaila Temitayo Sanusi, Kissinger Sunday, Solomon Sunday Oyelere, Jarkko Suhonen, Henriikka Vartiainen, and Markku Tukiainen.  Learning machine learning with young children: exploring informal settings in an african context.  Computer Science Education, pages 1–32, 2023.
  * (93) Ismaila Temitayo Sanusi and Ethel Tshukudu.  Exploring barriers and strategies to boost scientific output in computing education in africa: Early insights.  In Proceedings of the 2024 Innovation and Technology in Computer Science Education. 2024.
  * (94) Tamarisk Lurlyn Scholtz and Ian Sanders.  Mental models of recursion: investigating students’ understanding of recursion.  In Proceedings of the fifteenth annual conference on Innovation and technology in computer science education, pages 103–107, 2010.
  * (95) Theodor Sterling, FK Allotey, Asad Khailany, Maria Lucia Blanck Lisboa, Ignacio Mijares, and Edward Robertson.  Computer science education in developing countries.  In Papers of the SIGCSE/CSA technical symposium on Computer science education, pages 179–179, 1978.
  * (96) Matti Tedre, Fredrick D Ngumbuke, Nicholas Bangu, and Erkki Sutinen.  Implementing a contextualized it curriculum: Ambitions and ambiguities.  In Proceedings of the 8th International Conference on Computing Education Research, pages 51–61, 2008.
  * (97) Ismaila Temitayo Sanusi.  Teaching machine learning in k-12 education.  In Proceedings of the 17th ACM conference on international computing education research, pages 395–397, 2021.
  * (98) Ethel Tshukudu, Sofiat Olaosebikan, Kenechi Omeke, Alexandrina Pancheva, Stephen McQuistin, Lydia John Jilantikiri, and Maha Al-Anqoudi.  Broadening participation in computing: Experiences of an online programming workshop for african students.  In Proceedings of the 27th ACM Conference on on Innovation and Technology in Computer Science Education Vol. 1, pages 393–399, 2022.
  * (99) Ethel Tshukudu, Sue Sentance, Oluwatoyin Adelakun-Adeyemo, Brenda Nyaringita, Keith Quille, and Ziling Zhong.  Investigating k-12 computing education in four african countries (botswana, kenya, nigeria, and uganda).  ACM Transactions on Computing Education, 23(1):1–29, 2023.
  * (100) Martin van Tonder, Kevin Naude, and Charmain Cilliers.  Jenuity: a lightweight development environment for intermediate level programming courses.  ACM SIGCSE Bulletin, 40(3):58–62, 2008.
  * (101) Ismael Villegas Molina, Adrian Salguero, Shera Zhong, and Adalbert Gerald Soosai Raj.  The effects of spanish-english bilingual instruction in a cs0 course for high school students.  In Proceedings of the 2023 Conference on Innovation and Technology in Computer Science Education V. 1, pages 75–81, 2023.
  * (102) Michael Wollowski, M Daniels, and Å Cajander.  Expanding opportunities: Assessing and addressing geographic diversity at the sigcse technical symposium.  2009\.
  * (103) Betelhem Zewdu Wubineh, Fitsum Gizachew Deriba, and Michael Melese Woldeyohannis.  Exploring the opportunities and challenges of implementing artificial intelligence in healthcare: A systematic literature review.  In Urologic Oncology: Seminars and Original Investigations. Elsevier, 2023.
  * (104) Zhen Xu, Albert D Ritzhaupt, Karthikeyan Umapathy, Yang Ning, and Chin-Chung Tsai.  Exploring college students’ conceptions of learning computer science: A draw-a-picture technique study.  Computer Science Education, 31(1):60–82, 2021.
  * (105) Sukie Van Zyl and Elsa Mentz.  Developing deeper self-directed learning in database design: Factors that influence knowledge transfer.  In Proceedings of the 17th Workshop in Primary and Secondary Computing Education, pages 1–2, 2022.


On the twofold Moutard transformation

of the stationary Schrödinger equation

with axial symmetry

A. G. Kudryavtsev

Institute of Applied Mechanics,

Russian Academy of Sciences, Moscow 125040, Russia

<EMAIL_ADDRESS>
###### Abstract

The generalized Moutard transformation of the stationary axially symmetric
Schrödinger equation is considered. It is shown that a superposition of two
Moutard transformations can provide new potentials for the eigenvalue problem.
Examples of two - dimensional potentials and exact solutions for the
stationary axially symmetric Schrödinger equation are obtained as an
application of the twofold Moutard transformation.

PACS: 02.30.Jr, 02.30.Ik, 03.65.Ge

##  1 Introduction

The stationary Schrödinger equation $\left(\Delta-{\it
u}\left(x,y,z\right)\right)Y\left(x,y,z\right)=0$, where $\Delta$ is Laplace
operator, is of greate interest because it describes different physical
phenomena. In the case ${\it u}=-E+v\left(x,y,z\right)$ this equation
describes nonrelativistic quantum system with energy $E$ [1], in the case
${\it u}=-{\omega}^{2}/{c\left(x,y,z\right)}^{2}$ equation describes an
acoustic wave with temporal frequency $\omega$ in inhomogeneous media with
sound velocity $c$ [2]. A lot of physical environments have axial symmetry. In
the case of axial symmetry the stationary Schrödinger equation in spherical
coordinates has the form

$\left({\frac{{\frac{\partial}{\partial
r}}\left({r}^{2}{\frac{\partial}{\partial
r}}\right)}{{r}^{2}}}+{\frac{{\frac{\partial}{\partial\theta}}\left(\sin\left(\theta\right){\frac{\partial}{\partial\theta}}\right)}{{r}^{2}\sin\left(\theta\right)}}-u\left(r,\theta\right)\right)Y\left(r,\theta\right)=0$
(1)

The importance of the existence of exactly solvable problems of quantum
mechanics is indisputable. An exactly solvable model is understood as a model
that allows one to construct exact solutions in explicit form. Such models are
important not only for describing physical systems, but also serve as reliable
initial approximations when constructing perturbation theory, and are also
useful for testing numerical algorithms. Therefore, the search for new exact
solutions of the Schrödingerr equation is an actual theme of scientific
research, and various methods of solution are used [3] \- [9].

The useful tool for one - dimensional Schrödinger equation is the Darboux
transformation [10]. The Moutard transformation [11], [12] is the
generalization of the Darboux transformation for the flat two - dimensional
Schrödinger equation. In the papers [13], [14] the nonlocal Darboux
transformation of the two - dimensional stationary Schrödinger equation in
cartesian coordinates was considered and its relation to the Moutard
transformation was established. In the paper [15] the nonlocal Darboux
transformation of the stationary axially symmetric Schrödinger equation is
considered and it is shown that a special case of the nonlocal Darboux
transformation provides the generalization of the Moutard transformation. In
the present paper eigenvalue problem for the axially symmetric Schrödinger
equation is considered and new two - dimensional potentials and exact
solutions are obtained as an application of the generalized Moutard
transformation.

##  2 Generalized Moutard transformation in spherical coordinates

As a result of the transition from cylindrical to spherical coordinates, the
formulas for the generalized Moutard transformation from work [15] take the
form

${\frac{\partial}{\partial\theta}}\left({\it\tilde{Y}}\left(r,\theta\right){\it
Y_{0}}\left(r,\theta\right)\sin\left(\theta\right)\right)-\sin\left(\theta\right)r\left({\it
Y_{0}}\left(r,\theta\right)\right)^{2}{\frac{\partial}{\partial
r}}\left({\frac{Y\left(r,\theta\right)}{{\it
Y_{0}}\left(r,\theta\right)}}\right)=0$ (2) ${\frac{\partial}{\partial
r}}\left({\it\tilde{Y}}\left(r,\theta\right){\it
Y_{0}}\left(r,\theta\right)\sin\left(\theta\right)r\right)+\sin\left(\theta\right)\left({\it
Y_{0}}\left(r,\theta\right)\right)^{2}{\frac{\partial}{\partial\theta}}\left({\frac{Y\left(r,\theta\right)}{{\it
Y_{0}}\left(r,\theta\right)}}\right)=0$ (3)
$\tilde{u}\left(r,\theta\right)=u\left(r,\theta\right)-{\frac{\partial^{2}}{\partial{r}^{2}}}\ln\left(\sin\left(\theta\right)\left({\it
Y_{0}}\left(r,\theta\right)\right)^{2}\right)\\\
-{\frac{{\frac{\partial}{\partial
r}}\ln\left(\sin\left(\theta\right)\left({\it
Y_{0}}\left(r,\theta\right)\right)^{2}\right)}{r}}-{\frac{{\frac{\partial^{2}}{\partial{\theta}^{2}}}\ln\left(\sin\left(\theta\right)\left({\it
Y_{0}}\left(r,\theta\right)\right)^{2}\right)}{{r}^{2}}}$ (4)

Here, $Y_{0}$ and $Y$ are the solutions of equation (1) with the initial
potential $u$. The function $\tilde{Y}$ defined as a solution of a consistent
system of equations (2) and (3) is a solution of equation (1) with the new
potential $\tilde{u}$.

Note that $Y=Y_{0},\tilde{Y}={\frac{1}{{\it Y_{0}}\sin\left(\theta\right)r}}$
is a simple example of solution of equations (2) and (3).

In the papers [15], [16] some examples of two-dimensional potentials and exact
solutions of the time-independent axially symmetric Schrödinger equation have
been obtained on the basis of the generalized Moutard transformation formulas.
This examples illustrate that repeated application of the generalized Moutard
transformation can lead to the potentials that are more interesting for
physical interpretation than the potentials obtained by the first
transformation. In particular, repeated application of the generalized Moutard
transformation is effective for obtaining potentials that do not have a
singularity. Likewise for the two-dimensional flat Schrödinger equation in
cartesian coordinates, it was shown in the paper [17] that the twofold
application of the classical Moutard transformation is efficient for obtaining
nonsingular potentials. Similarly, nonsingular potentials in cylindrical
coordinates can be effectively obtained using the twofold generalized Moutard
transformation [15], [16] .

When studying the Moutard transformations of the Schrödinger equation with
axial symmetry, another interesting property of twofold transformations was
discovered [15], namely the ability to obtain the same transformation for
eigenfunctions with different eigenvalues. The fact is that for the eigenvalue
problem $u=-k^{2}+v\left(r,\theta\right)$ and with a single application of the
Moutard transformation (4) using the selected eigenfunction
$Y_{0}\left(r,\theta,k\right)$ corresponding to the eigenvalue $k$, the new
potential has the form
${\tilde{u}}=-k^{2}+{\tilde{v}}\left(r,\theta,k\right)$. Due to ${\tilde{v}}$
contains a dependence on k, one obtains different transformations for
eigenfunctions with different eigenvalues. Similarly, the classical Moutard
transformation in the two-dimensional flat case leads to different
transformations for eigenfunctions with different eigenvalues. Unlike the
Moutard transformation, the Darboux transformation gives us the
transformations of all eigenfunctions. This difference between the two-
dimensional case and the one-dimensional case is apparently reflected in the
well-known fact about integrability only at a selected energy level in the
two-dimensional case (see, for example, [18], [19]).

Let us consider $u=-k^{2}$ and
$Y_{0}=\sin\left(kr\cos\left(\theta\right)\right)$ as a selected
eigenfunction. From (4) we obtain

$\tilde{u}=-{k}^{2}+2\,{\frac{{k}^{2}}{\left(\sin\left(kr\cos\left(\theta\right)\right)\right)^{2}}}+{\frac{1}{{r}^{2}\left(\sin\left(\theta\right)\right)^{2}}}$
(5)

This is an example of ${\tilde{v}}$ dependence on $k$. To avoid this
dependence we use twofold Moutard transformation. As a result of the
transition from cylindrical to spherical coordinates, the formulas for the
superposition of two generalized Moutard transformations from work [15] take
the form

${\tilde{\tilde{u}}}=u\left(r,\theta\right)-2\,{\frac{\partial^{2}}{\partial{r}^{2}}}\ln\left(F\left(r,\theta\right)\right)-2\,{\frac{{\frac{\partial}{\partial
r}}\ln\left(F\left(r,\theta\right)\right)}{r}}-2\,{\frac{{\frac{\partial^{2}}{\partial{\theta}^{2}}}\ln\left(F\left(r,\theta\right)\right)}{{r}^{2}}}$
(6) ${\frac{\partial}{\partial\theta}}F\left(r,\theta\right)=\\\
\sin\left(\theta\right){r}^{2}\left(\left({\frac{\partial}{\partial r}}{\it
Y_{2}}\left(r,\theta\right)\right){\it
Y_{1}}\left(r,\theta\right)-\left({\frac{\partial}{\partial r}}{\it
Y_{1}}\left(r,\theta\right)\right){\it Y_{2}}\left(r,\theta\right)\right)$ (7)
${\frac{\partial}{\partial r}}F\left(r,\theta\right)=\\\
-\sin\left(\theta\right)\left(\left({\frac{\partial}{\partial\theta}}{\it
Y_{2}}\left(r,\theta\right)\right){\it
Y_{1}}\left(r,\theta\right)-\left({\frac{\partial}{\partial\theta}}{\it
Y_{1}}\left(r,\theta\right)\right){\it Y_{2}}\left(r,\theta\right)\right)$ (8)

Here, $Y_{1}$ and $Y_{2}$ are the solutions of equation (1) with the initial
potential $u$.

Equations (7), (8) are invariant under the substitution ${\it
Y_{1}}\rightarrow{\it Y_{2}},{\it Y_{2}}\rightarrow{\it
Y_{1}},F\rightarrow-F$, which represents the commutativity of the generalized
Moutard transformations. The result does not depend on the order of choice of
the functions ${\it Y_{1}},{\it Y_{2}}$ for the transformation. We also note
that $F$ is determined up to multiplication by a constant. Simple examples of
solutions for the equation (1) with potential (6) can be obtained by the
formulas ${\tilde{\tilde{Y}}}_{1}={\it
Y_{1}}F^{-1},\,{\tilde{\tilde{Y}}}_{2}={\it Y_{2}}F^{-1}$.

##  3 Examples of potentials and exact solutions

Let us consider
$u=-k^{2},Y_{1}=\sin\left(kr\cos\left(\theta\right)\right),Y_{2}=\cos\left(kr\cos\left(\theta\right)\right)$.
From equations (7), (8) we obtain
$F={r}^{2}\left(\sin\left(\theta\right)\right)^{2}+C$ where $C$ is an
arbitrary constant. From formula (6) we get new potential

${\tilde{\tilde{u}}}=-{k}^{2}+4\,{\frac{{r}^{2}\left(\sin\left(\theta\right)\right)^{2}-C}{\left({r}^{2}\left(\sin\left(\theta\right)\right)^{2}+C\right)^{2}}}$
(9)

Here ${\tilde{\tilde{v}}}$ does not depend on $k$ and we have obtained new
eigenvalue problem. As a solution to the original eigenvalue problem with
$u=-k^{2}$ we will consider a plane wave ${{\rm
e}^{ikr\cos\left(\theta\right)}}$. Carrying out a twofold Moutard
transformation or using a simple formula ${\it Y_{2}}F^{-1}+i{\it
Y_{1}}F^{-1}$ we obtain the following solution to equation (1) with potential
(9)

${\frac{{{\rm
e}^{ikr\cos\left(\theta\right)}}}{{r}^{2}\left(\sin\left(\theta\right)\right)^{2}+C}}$
(10)

As another example, consider two solutions to equation (1) with potential
$u=-k^{2}$

$Y_{1}={\frac{{{\sl J}_{p+1/2}\left(kr\right)}{\it
P}\left(p,\cos\left(\theta\right)\right)}{\sqrt{r}}}$ (11) $Y_{2}={\frac{{{\sl
Y}_{p+1/2}\left(kr\right)}{\it
P}\left(p,\cos\left(\theta\right)\right)}{\sqrt{r}}}$ (12)

where $p$ is a parameter, ${\sl J}_{p+1/2},{\sl Y}_{p+1/2}$ are Bessel
functions of the first and second kind, ${\it P}$ is Legendre function of the
first kind. From equations (7), (8) we obtain $F$ depending only on $\theta$

$F_{p}\left(\theta\right)=-\int\\!\sin\left(\theta\right)\left({\it
P}\left(p,\cos\left(\theta\right)\right)\right)^{2}\,{\rm d}\theta$ (13)

From formula (6) we get new potential

${\tilde{\tilde{u_{p}}}}=-k^{2}-2\,{\frac{{\frac{\partial^{2}}{\partial{\theta}^{2}}}\ln\left(F_{p}\left(\theta\right)\right)}{{r}^{2}}}$
(14)

where ${\tilde{\tilde{v}}}$ does not depend on $k$. Note that
${\tilde{\tilde{v}}}={\frac{f\left(\theta\right)}{{r}^{2}}}$ has the form of a
Calogero-Moser potential. The problem of integrability of Calogero-Moser type
potentials continues to attract the attention of mathematicians and physicists
[20].

For non-negative integer values of the parameter $p$, the function $F$ can be
obtained explicitly. So for p=0,1,2 we have

$F_{0}=\cos\left(\theta\right)+C,F_{1}=\left(\cos\left(\theta\right)\right)^{3}+C,$

$F_{2}=\left(9\,\left(\cos\left(\theta\right)\right)^{4}-10\,\left(\cos\left(\theta\right)\right)^{2}+5\right)\cos\left(\theta\right)+C$

where arbitrary constants $C$ for each $F_{p}$ can of course be chosen
independently.

From (14) we obtain the corresponding potentials

${\tilde{\tilde{u}}}_{0}=-{k}^{2}+2\,{\frac{\cos\left(\theta\right)C+1}{\left(\cos\left(\theta\right)+C\right)^{2}{r}^{2}}}$
(15)
${\tilde{\tilde{u}}}_{1}=-{k}^{2}+6\,{\frac{\cos\left(\theta\right)\left(\left(3\,\left(\cos\left(\theta\right)\right)^{2}-2\right)C+\left(\cos\left(\theta\right)\right)^{3}\right)}{\left(\left(\cos\left(\theta\right)\right)^{3}+C\right)^{2}{r}^{2}}}$
(16)
${\tilde{\tilde{u}}}_{2}=-{k}^{2}+10\,{\frac{N\left(\theta\right)}{\left(\left(9\,\left(\cos\left(\theta\right)\right)^{4}-10\,\left(\cos\left(\theta\right)\right)^{2}+5\right)\cos\left(\theta\right)+C\right)^{2}{r}^{2}}}$
(17)

where

$N\left(\theta\right)=\left(45\,\left(\cos\left(\theta\right)\right)^{4}-54\,\left(\cos\left(\theta\right)\right)^{2}+13\right)\cos\left(\theta\right)C\\\
+\left(9\,\left(\cos\left(\theta\right)\right)^{4}+72\,\left(\cos\left(\theta\right)\right)^{2}-70\right)\left(\cos\left(\theta\right)\right)^{4}+5$

As examples of exact solutions to the scattering problem with potentials (15)
- (17), we again consider a plane wave ${{\rm
e}^{ikr\cos\left(\theta\right)}}$ for the original $u=-k^{2}$ and carry out a
twofold Moutard transformation with the corresponding $Y_{1}$ and $Y_{2}$ from
formulas (11) and (12). As a result, we obtain the following exact solutions

${\tilde{\tilde{Y}}}_{0}={{\rm
e}^{ikr\cos\left(\theta\right)}}\left(1+{\frac{i}{kr\left(\cos\left(\theta\right)+C\right)}}\right)$
(18) ${\tilde{\tilde{Y}}}_{1}={{\rm
e}^{ikr\cos\left(\theta\right)}}\left(1+3\,{\frac{\cos\left(\theta\right)\left(ikr\cos\left(\theta\right)-1\right)}{{k}^{2}{r}^{2}\left(\left(\cos\left(\theta\right)\right)^{3}+C\right)}}\right)$
(19) ${\tilde{\tilde{Y}}}_{2}=\\\ {{\rm
e}^{ikr\cos\left(\theta\right)}}\left(1+{\frac{M\left(r,\theta,k\right)}{{r}^{3}{k}^{3}\left(\left(9\,\left(\cos\left(\theta\right)\right)^{4}-10\,\left(\cos\left(\theta\right)\right)^{2}+5\right)\cos\left(\theta\right)+C\right)}}\right)$
(20)

where

$M\left(r,\theta,k\right)=5\,i\left(3\,\left(\cos\left(\theta\right)\right)^{2}-1\right)\left(\left(3\,\left(\cos\left(\theta\right)\right)^{2}-1\right){r}^{2}{k}^{2}+6\,ikr\cos\left(\theta\right)-6\right).$

##  4 Results and discussion

The eigenfunction of the eigenvalue problem of the two-dimensional Schrödinger
operator allows us to obtain the Moutard transformation for the potential.
However, the resulting potential depends on the eigenvalue corresponding to
the eigenfunction. The article shows that in a number of cases, applying the
Moutard transformation twice makes it possible to obtain a potential that does
not depend on the eigenvalues. It is important to note that the Moutard
transformation to obtain new solutions requires quadratures, so obtaining
eigenfunctions in explicit analytical form for a new potential is not always
possible. If for the new potential the eigenfunctions are obtained in explicit
form, then we can talk about a new solvable two-dimensional eigenvalue
problem. Here we say that a problem is solvable if it allows obtaining of
solutions in an explicit functional form. This article provides examples of
new solvable eigenvalue problems in the case of the Schrödinger operator with
axial symmetry. For a new solvable problem, one can try again to construct a
twofold Moutard transformation. Finding all solvable eigenvalue problems of
the Schrödinger operator that can be obtained using the twofold Moutard
transformation remains an open problem.

## References

  * [1] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, 1977.
  * [2] P.M. Morse and K.U. Ingard, Theoretical Acoustics, New York, NY: McGraw- Hill, 1968.
  * [3] L. Infeld, T.E. Hull, The factorization method, Rev. Modern Physics, v. 23, 1951, p, 21–68.
  * [4] Nikiforov, A. F. and Uvarov, V. B., Special Functions of Mathematical Physics. A Unified Introduction With Applications, Birkhauser Verlag, Basel-Boston, 1988.
  * [5] Fred Cooper, Avinash Khare, Uday Sukhatme, Supersymmetry and quantum mechanics, Physics Reports, Volume 251, 1995, Pages 267-385.
  * [6] A A Andrianov and M V Ioffe, Nonlinear supersymmetric quantum mechanics: concepts and realizations, 2012, J. Phys. A: Math. Theor., v, 45, 503001
  * [7] D.J. Fernandez, Trends in Supersymmetric Quantum Mechanics. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham, 2019.
  * [8] Sanjana Bhatia, C.N. Kumar, Ajay Nath, Investigation of one-dimensional quantum droplets in a temporally perturbed external harmonic trap, Physics Letters A, Volume 492, 2023, 129228.
  * [9] Guillermo Gordillo-Núñez, Renato Alvarez-Nodarse, Niurka R. Quintero, The complete solution of the Schrodinger equation with the Rosen–Morse type potential via the Nikiforov–Uvarov method, Physica D: Nonlinear Phenomena, Volume 458, 2024, 134008.
  * [10] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, 1991.
  * [11] T. Moutard, J. Ecole Polyt., 45 (1878), 1-11.
  * [12] C. Athorne and J. J. C. Nimmo, On the Moutard transformation for integrable partial differential equations, Inverse Problems, 7 (1991), p. 809–826.
  * [13] A.G. Kudryavtsev, Exactly solvable two - dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation, Phys. Lett. A, 377 (2013) 2477-2480.
  * [14] A.G. Kudryavtsev, Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation, Theoretical and Mathematical Physics, 187(1) (2016) 455-462.
  * [15] A.G. Kudryavtsev, Exact solutions of the time-independent axially symmetric Schrödinger equation, JETP Letters, 2020, Vol. 111, No. 2 (2020), pp. 126–128.
  * [16] A. G. Kudryavtsev, On the Nonlocal Darboux Transformation for Time-Independent Axially Symmetric Schrodinger and Helmholtz Equations, JETP Letters, 2021, Vol. 113, No. 6, pp. 409–412.
  * [17] I.A. Taimanov, S.P. Tsarev, Two-dimensional rational solitons and their blowup via the moutard transformation, Theoretical and Mathematical Physics, November 2008, Volume 157, Issue 2, pp. 1525-1541.
  * [18] A.P. Veselov and S.P. Novikov, Finite-zone two-dimensional Schroedinger operators. Potential operators, Dokl. Akad. Nauk SSSR, 1984, v. 279, N 4, 784-788 ( English transl., Soviet Math. Dokl. 30:3 (1984), 705-708).
  * [19] A.V. Ilina, I.M. Krichever, N.A. Nekrasov, Two-Dimensional Periodic Schrödinger Operators Integrable at an Energy Eigenlevel. Funct Anal Its Appl 53, 23–36 (2019).
  * [20] Berest, Y., Chalykh, O. Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras. Commun. Math. Phys. 400, 133–178 (2023).