Artificial intelligence, computational thinking, and mathematics education

Author(s):

George Gadanidis, (Western University, London, Canada)

Single Sentence Summary:

An examination of the intersection of artificial intelligence (AI), computational thinking (CT), and mathematics education (ME) for young students reveals that there is a fruitful overlap between AI, CT and ME that is of value to consider in mathematics education.

Abstract:

Purpose

The purpose of this paper is to examine the intersection of artificial intelligence (AI), computational thinking (CT), and mathematics education (ME) for young students (K-8). Specifically, it focuses on three key elements that are common to AI, CT and ME: agency, modeling of phenomena and abstracting concepts beyond specific instances.

Design/methodology/approach

The theoretical framework of this paper adopts a sociocultural perspective where knowledge is constructed in interactions with others (Vygotsky, 1978). Others also refers to the multiplicity of technologies that surround us, including both the digital artefacts of our new media world, and the human methods and specialized processes acting in the world. Technology is not simply a tool for human intention. It is an actor in the cognitive ecology of immersive humans-with-technology environments (Levy, 1993, 1998) that supports but also disrupts and reorganizes human thinking (Borba and Villarreal, 2005).

Findings

There is fruitful overlap between AI, CT and ME that is of value to consider in mathematics education.

Originality/value

Seeing ME through the lenses of other disciplines and recognizing that there is a significant overlap of key elements reinforces the importance of agency, modeling and abstraction in ME and provides new contexts and tools for incorporating them in classroom practice.

Keywords:

Artificial intelligence, Mathematics education, Computational thinking

Type:

Conceptual paper

Publisher:

Emerald Publishing Limited

Received:

17 September 2016

Revised:

29 October 2016

Accepted:

29 November 2016

Copyright:

© Emerald Publishing Limited 2017
Published by Emerald Publishing Limited
Licensed re-use rights only

Citation:

George Gadanidis, (2017) "Artificial intelligence, computational thinking, and mathematics education", The International Journal of Information and Learning Technology, Vol. 34 Issue: 2, pp.133-139, https://doi.org/10.1108/IJILT-09-2016-0048

Downloads:

The fulltext of this document has been downloaded 1883 times since 2017

Article

Introduction

Section:ChooseTop of pageIntroduction <<AICTAI∩CT∩MEConcluding remarksReferencesCorresponding Author

This paper examines the intersection of artificial intelligence (AI), computational thinking (CT) and mathematics education (ME) for young students (K-8). Specifically, it focuses on three key elements that are common to AI, CT and ME: agency, modeling of phenomena and abstracting concepts beyond specific instances (see Figure 1).

As is the case with a lot of the author’s work, the theoretical framework of this paper adopts a sociocultural perspective where knowledge is constructed in interactions with others (Vygotsky, 1978). Others also refers to the multiplicity of technologies that surround us, including both the digital artefacts of our new media world, and the human methods and specialized processes acting in the world. Technology is not simply a tool for human intention. It is an actor in the cognitive ecology of immersive humans-with-technology environments (Levy, 1993, 1998) that supports but also disrupts and reorganizes human thinking (Borba and Villarreal, 2005). Actor-network theory (Latour, 2005) emphasizes the reciprocal relationship between the “actor” and technology, where we are both acting and acted upon (Thumlert et al., 2014). In this examination of the overlap of AI, CT and ME, I identify and explore key elements of CT as actors we (can) think-with in the learning and teaching process.

The first two sections below introduce AI and CT. The third section discusses how agency, modeling and abstraction may be seen as three common key elements of AI, CT and ME.

AI

Section:ChooseTop of pageIntroductionAI <<CTAI∩CT∩MEConcluding remarksReferencesCorresponding Author

AI is the intelligence evident in machines or software:

It is also the name of the academic field of study which studies how to create computers and computer software that are capable of intelligent behavior. Major AI researchers and textbooks define this field as “the study and design of intelligent agents,” in which an intelligent agent is a system that perceives its environment and takes actions that maximize its chances of success

(“Artificial Intelligence,” n.d., para. 1).

Today, AI is increasingly pursued in a variety of ways by industry, such as seen in the development of self-driving cars by Google and cognitive systems like Watson by IBM.

AI singularity

Some experts estimate that we are 20-50 years away from an AI singularity, where machines capable of recursive self-learning surpass human intellectual capacity and control.

AI machines that match and surpass human intelligence may be seen as leading to positive technological advances, such as eliminating aging and disease or enhanced space travel (Bostrom and Yudkowsky, 2014). At the same time, an AI singularity may prove disastrous. Stephen Hawking told the BBC (Cellan-Jones, 2014), “The development of full AI could spell the end of the human race.” Hawking (2014, para. 7) wrote:

If a superior alien civilisation sent us a message saying, “We’ll arrive in a few decades,” would we just reply, “OK, call us when you get here – we’ll leave the lights on”? Probably not – but this is more or less what is happening with AI. Although we are facing potentially the best or worst thing to happen to humanity in history, little serious research is devoted to these issues [...]. All of us should ask ourselves what we can do now to improve the chances of reaping the benefits and avoiding the risks.

AI in education

AI in education has historically focused on the design of digital tutors that not only provide exposition of concepts to be learned, but also have the intelligence to respond meaningfully to student behavior, such as providing adaptive support (Gilbert et al., 2015), addressing student learning styles (Dorca, 2015) or providing culturally appropriate communication (Blanchard, 2015). Historically, these tutors were embedded in software packages designed for specific content areas, such as mathematics.

Today, especially in higher grades and in post-secondary settings, with student learning increasingly occurring in online settings, there is a focus on web-based intelligent agents that may act as content tutors or as online discussion facilitators (Adamson et al., 2014; Tegos et al., 2014). AI support of online learning is especially important with the growth of Massive Open Online Courses (MOOCs), where enrollment in the most popular MOOC platforms averages over 40,000 students (Ferenstein, 2014). AI can play a role in organizing and supporting online collaboration and in assessing student learning.

Another form of educational AI, which most of us take for granted, is online search engines coupled with the tremendous amount of freely accessible online information. If we need a definition, the knowledge to complete a task, or help to understand a concept, a quick search of available online knowledge will identify a variety of text and multimedia resources to assist us.

CT

Section:ChooseTop of pageIntroductionAICT <<AI∩CT∩MEConcluding remarksReferencesCorresponding Author

CT in education has three instances: screen-based coding, digital tangibles (such as programmable robots and circuits), and off-screen algorithms or pseudocode. The term CT was popularized by Wing’s (2006) advocacy, “To reading, writing, and arithmetic, we should add CT to every child’s analytical ability” (p. 33).

Currently CT in education is more as its own, isolated curriculum objective, rather than integrated with, and enriching, existing subject areas (Gadanidis, 2014). However, there is a natural connection between CT and mathematics – such as in the logical structure or in the ability to model mathematical relationships (Wing, 2008).

AI∩CT∩ME

Section:ChooseTop of pageIntroductionAICTAI∩CT∩ME <<Concluding remarksReferencesCorresponding Author

Let us now turn to the intersection of AI, CT and ME and explore their common focus on agency, modeling and abstraction.

Agency

AI

Agency and the associated features of self-regulation and self-learning are key aspects of AI. Let’s take self-driving cars as an example, where a core problem is the analysis of sensor and image data. For instance, what kind of object is in front of the car, and how should the car respond?

It examines the images and guesses the kind of object in each image. Initially most of its guesses will be wrong. Therefore, the algorithm modifies internal parameters or parts of its structure somewhat and tries again. This process continues, discarding changes that reduce the algorithm’s accuracy, keeping changes that increase the accuracy, until it correctly classifies all images. Afterward, when entirely new images are presented to the algorithm it will classify them with high accuracy. The algorithm has learned!

(Top misconceptions of autonomous cars and self-driving vehicles, 2015, para. 29).

The team of programmers designing the self-driving car could attempt to anticipate every obstacle or situation, but variations are too numerous. The car-in-action has to be able to learn from its experience and to make decisions based on that self-learning. What is also interesting is that once one car learns something from a situation, its knowledge can be immediately shared with all other cars, so that all cars learn.

CT

Student agency is a key feature of education-oriented CT environments. Building on Papert’s (1980) work with Logo programming, several programming languages are available today (e.g. Scratch, available: at https://scratch.mit.edu/), that offer a low floor, enabling even young children to engage with little prerequisite knowledge, and a high ceiling, providing opportunities to explore more complex relationships. As elaborated in greater detail in Gadanidis (in press), this environment offers students opportunities to abstract, automate and dynamically model concepts, to explore their relationships and to experience conceptual surprise and insight, not only by implementing pre-programmed simulations, but also by creating and editing their own, thus experiencing CT and mathematics as producers as well as consumers. For example, Figure 2 shows the Scratch code for drawing a set of circles, rotated about a point. Young students can drag and drop code blocks that snap together to model various of mathematical concepts. In such computer coding experiences, students are in control, writing personally meaningful code and exploring related problems and extensions.

ME

Students’ agency is also a key feature of ME theory. Burton (1999) suggests that agentic control makes a substantial difference in mathematics attitude and achievement. Schoenfeld (1987) suggests, “Many students come to believe that school mathematics consists of mastering formal procedures that are completely divorced from real life, from discovery, and from problem solving” (p. 197). Papert (1993, p. 25) adds, “I am convinced that the best learning takes place when the learner takes charge.”

Modeling

AI

Developing a self-driving car involves conceptualizing models of how other cars move and react and how pedestrians interact with vehicles, to give two examples. Similarly, designing intelligent agents in education contexts, such as tutoring or online learning facilitation, requires the development of models of the subject matter and of the learners. This model-creation and the associated model-testing and -refinement is an integral component of AI development.

CT

CT is an approach to problem solving that focuses on the logic and design of computational algorithms, or sequences of steps that can be implemented using a computer (Aho, 2012; Wing, 2006, 2008, 2011). The power of CT modeling is its dynamic nature: making a change in the computer code shows the mathematical reaction immediately. For example, changing the values of parameters in Figure 2 can cause the program to draw fewer circles or different shapes.

ME

Dynamic modeling allows students to “play” with mathematics and helps bring to life the concepts students are studying. Play naturally engages children with creative problem solving (Ginsburg, 2006) and has historically been valued in early childhood learning (Perry and Dockett, 2002; Duncan and Lockwood, 2008).

Abstraction

AI

Abstraction “plays a key role in representing knowledge and in reasoning” (Saitta and Zucker, 2013, p. 2), and is an integral component of AI development. For example, in the case of the self-driving car, creating a model of “pedestrian” abstracts key attributes.

CT

Yadav et al. (2014) note that abstraction is a key element of CT. Wing (2008, p. 3717) states, “In computing, we abstract notions beyond the physical dimensions of time and space. Our abstractions are extremely general because they are symbolic, where numeric abstractions are just a special case.” This process of abstraction can be seen in Figure 1, where the code used represents a variety of related cases at once.

ME

Abstraction is at the heart of mathematics. Abstraction, in the everyday sense of the word, is also a natural human activity. For example, very young children easily abstract beyond specific instances of objects and develop mental models of classes of objects, such as “cat,” despite the many different sizes, colors and behaviors of cat instances.

However, as I have argued in Gadanidis (2014, 2015) the idea of engaging young students with abstraction is not widely accepted in education, primarily due to the widespread acceptance of Piaget’s stages of development. Egan (2002) notes that “Piaget’s ideas and overall approach absolutely dominate in education” (p. 105). Papert (1980), Egan (1997), Fernandez-Armesto (1997) and Schmittau (2005) challenge Piaget’s notion that young children are not capable of abstract thinking, which Egan identifies as integral to language development. Abstraction helps students conceptualize and engage with complex problems and relationships by reducing information and detail. Wing (2011) notes that we use abstraction to better manage complexity.

Concluding remarks

Section:ChooseTop of pageIntroductionAICTAI∩CT∩MEConcluding remarks <<ReferencesCorresponding Author

There is tremendous interest and enthusiasm today for CT in education. In Canada, for example, in January 2016 Prime Minister Justin Trudeau said, “We need to do a lot better job of getting young people to understand what coding is and how it’s important, how to program, how to problem solve, how to create the most elegant algorithm possible” (Kitchener Post, 2016). Around the same time, the provinces of British Columbia and Nova Scotia announced that computer coding will be added to all grades of the K-12 curriculum. Internationally, as one example, England in 2014 mandated a coding curriculum for all K-12 students. At the same time, as discussed above, there is a growing industry focus on AI. These phenomena are not distinct or separate. As the ancient Greek playwright Sophocles suggested, there are few plots in life. To better understand the evolving phenomena around us it is important to examine how they overlap and to see each through the lens of another.

This paper offers a nascent exploration of the intersection of AI, CT and ME, highlighting three of their common elements: agency, modeling and abstraction. This seeing of ME through the lenses of other disciplines helps us recognize that there is a significant overlap of key elements, draws our pedagogical attention to the importance of agency, modeling and abstraction in ME, and provides new contexts and tools for incorporating them in classroom practice.

Figure 1 Three elements common to AI, CT and ME

Figure 2 Creating a circles pattern in Scratch

References

1.

Adamson, D., Dyke, G., Jang, H. and Rose, C.P. (2014), “Towards an agile approach to adapting dynamic collaboration support to student needs”, International Journal of Artificial Intelligence in Education, Vol. 24, pp. 2-6. [Crossref], [Google Scholar]

2.

Aho, A.V. (2012), “Computation and computational thinking”, Computer Journal, Vol. 55, pp. 832-835. [Crossref], [Google Scholar]

3.

Artificial Intelligence (n.d.), “In Wikipedia”, available at: https://en.wikipedia.org/wiki/Artificial_intelligence (accessed March 15, 2016). [Google Scholar]

4.

Blanchard, E.G. (2015), “Socio-cultural imbalances in AIED research: investigations, implications and opportunities”, International Journal of Artificial Intelligence in Education, Vol. 25, pp. 204-228. [Crossref], [Google Scholar]

5.

Borba, M.C. and Villarreal, M.E. (2005), Humans-With-Media and Reorganization of Mathematical Thinking: Information and Communication Technologies, Modeling, Experimentation and Visualization, Springer, New York, NY. [Google Scholar]

6.

Bostrom, N. and Yudkowsky, E. (2014), “The ethics of artificial intelligence”, in Frankish, K. and Ramsey, W. (Eds), The Cambridge Handbook of Artificial Intelligence, Cambridge University Press, Cambridge, pp. 316-334. [Google Scholar]

7.

Burton, L. (1999), “The practices of mathematicians: what do they tell us about coming to know mathematics?”, Educational Studies in Mathematics, Vol. 37, pp. 121-143. [Crossref], [Google Scholar]

8.

Cellan-Jones, R. (2014), “Stephen hawking warns artificial intelligence could end mankind”, British Broadcasting Corporation, December 2, available at: www.bbc.com/news/technology-30290540 (accessed February 23, 2017). [Google Scholar]

9.

Dorca, F. (2015), “Implementation and use of simulated students for test and validation of new adaptive educational systems: a practical insight”, International Journal of Artificial Intelligence in Education, Vol. 25, pp. 319-345. [Crossref], [Google Scholar]

10.

Duncan, J. and Lockwood, M. (2008), Learning through Play: A Work-Based Approach for the Early Years, Continuum International Publishing Group, New York, NY. [Google Scholar]

11.

Egan, K. (1997), The Educated Mind: How Cognitive Tools Shape our Understanding, University of Chicago Press, Chicago, IL. [Crossref], [Google Scholar]

12.

Egan, K. (2002), Getting it Wrong from the Beginning: Our Progressive Inheritance from Herbert Spencer, John Dewey, and Jean Piaget, Yale University Press, New Haven, CT. [Google Scholar]

13.

Ferenstein, G. (2014), “Study: massive online courses enroll an average of 43,000 students, 10% completion”, Tech Crunch, available at: http://techcrunch.com/2014/03/03/study-massive-online-courses-enroll-an-average-of-43000-students-10-completion/ (accessed February 23, 2017). [Google Scholar]

14.

Fernandez-Armesto, F. (1997), Truth: A History and a Guide for the Perplexed, Bartam, London. [Google Scholar]

15.

Gadanidis, G. (2014), “Young children, mathematics and coding: a low floor, high ceiling, wide walls learning environment”, in Polly, D. (Ed.), Cases on Technology Integration in Mathematics Education, IGI Global, Hershey, PA, pp. 312-344. [Google Scholar]

16.

Gadanidis, G. (2015), “Coding as a trojan horse for mathematics education reform”, Journal of Computers in Mathematics and Science Teaching, Vol. 34 No. 2, pp. 155-173. [Google Scholar] [Infotrieve]

17.

Gadanidis, G., Hughes, J., Minniti, L. and White, B. (in press), “Computational thinking, grade 1 students and the binomial theorem”, Digital Experience in Mathematics Education. [Google Scholar]

18.

Gilbert, S.B., Blessing, S.B. and Guo, E. (2015), “Authoring effective embedded tutors: an overview of the extensible problem specific tutor (xPST) system”, International Journal of Artificial Intelligence in Education, Vol. 25, pp. 428-454. [Crossref], [Google Scholar]

19.

Ginsburg, H.P. (2006), “Mathematical play and playful mathematics: a guide for early education”, in Golinkoff, R.M., Hirsh-Pasek, K. and Singer, D. (Eds), Play=learning, Oxford University Press, New York, NY, pp. 145-165. [Google Scholar]

20.

Hawking, S. (2014), “Transcendence looks at the implications of artificial intelligence – but are we taking AI seriously enough?”, Independent, May 1, available at: www.independent.co.uk/news/science/stephen-hawking-transcendence-looks-at-the-implications-of-artificial-intelligence-but-are-we-taking-9313474.html (accessed February 23, 2017). [Google Scholar]

21.

Kitchener Post (2016), “Google opens its doors on Breithaurt; PM Trudeau takes part”, available at: www.kitchenerpost.ca/news-story/6240341-google-opens-its-doors-on-breithaupt-pm-trudeau-takes-part/ (accessed June 20, 2016). [Google Scholar]

22.

Latour, B. (2005), Reassembling the Social: An Introduction to Actor-Network-Theory, Oxford University Press, Oxford. [Google Scholar]

23.

Levy, P. (1993), Tecnologias da Inteligência: O futuro do pensamento na era da informática [Technologies of Intelligence: the future of thinking in the informatics era], Editora 34, Rio de Janeiro. [Google Scholar]

24.

Levy, P. (1998), Becoming Virtual: Reality in the Digital Age, Plenum Press, New York, NY. [Google Scholar]

25.

Papert, S. (1980), Mindstorms: Children, Computers, and Powerful Ideas, Basic Books, New York, NY. [Google Scholar]

26.

Papert, S. (1993), The Children’s Machine. Rethinking School in the Age of the Computer, Basic Books, New York, NY. [Google Scholar]

27.

Perry, B. and Dockett, S. (2002), “Young children’s access to powerful mathematical ideas”, in English, L.D. (Ed.), Handbook of International Research in Mathematics Education: Directions for the 21st Century, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 81-111. [Google Scholar]

28.

Saitta, L. and Zucker, J.D. (2013), Abstraction in Artificial Intelligence and Complex Systems, Springer, New York, NY. [Crossref], [Google Scholar]

29.

Schmittau, J. (2005), “The development of algebraic thinking – a Vygotskian perspective”, ZDM, Vol. 37 No. 1, pp. 16-22. [Google Scholar] [Infotrieve]

30.

Schoenfeld, A.H. (1987), “What’s all the fuss about metacognition?”, in Schoenfeld, A.H. (Ed.), Cognitive Science and Mathematics Education, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 189-215. [Google Scholar]

31.

Tegos, S., Demetriadis, S. and Tsiatsos, T. (2014), “A configurable conversational agent to trigger students’ productive dialogue: a pilot study in the CALL domain”, International Journal of Artificial Intelligence in Education, Vol. 24, pp. 62-91. [Crossref], [Google Scholar]

32.

Thumlert, J., de Castell, S. and Jenson, J. (2014), “Short cuts and extended techniques: rethinking relations between technology and educational theory”, Educational Philosophy and Theory, Vol. 47 No. 8, pp. 786-803. [Crossref], [Google Scholar] [Infotrieve]

33.

Top misconceptions of autonomous cars and self-driving vehicles (2015), “In driverless car market watch”, available at: www.driverless-future.com/?page_id=774 (accessed February 23, 2017). [Google Scholar]

34.

Vygotsky, L.S. (1978), Mind in Society, Harvard University Press, Cambridge, MA. [Google Scholar]

35.

Wing, J. (2011), “Research notebook: computational thinking – what and why?”, The Link Magazine, Spring, Carnegie Mellon University, Pittsburgh, PA, available at: http://link.cs.cmu.edu/article.php?a=600 [Google Scholar]

36.

Wing, J.M. (2006), “Computational thinking”, Communications of the ACM, Vol. 49 No. 3, pp. 33-35. [Crossref], [Google Scholar] [Infotrieve]

37.

Wing, J.M. (2008), “Computational thinking and thinking about computing”, Philosophical Transactions of the Royal Society A, Vol. 366 No. 1881, pp. 3717-3725. [Crossref], [Google Scholar] [Infotrieve]

38.

Yadav, A., Mayfield, C., Zhou, N., Hambrusch, S. and Korb, J.T. (2014), “Computational thinking in elementary and secondary teacher education”, ACM Transactions on Computing Education, Vol. 14 No. 1, pp. 1-5, 16. [Crossref], [Google Scholar] [Infotrieve]

Corresponding Author

Section:ChooseTop of pageIntroductionAICTAI∩CT∩MEConcluding remarksReferencesCorresponding Author <<

George Gadanidis can be contacted at: ggadanid@uwo.ca