Elly Kang, Sean Sehoon Kim, Sarah Porter, Taylor Torres

Abstract

There has been widespread enthusiasm for artificial intelligence and machine learning (AIML) curricula and instruction. Yet, integrating these fields into schools remains challenging. One underexplored avenue, presented in this paper, involves integrating AIML curricula with mathematics. To explore this approach, we conducted interviews to illustrate how high school students of various mathematics backgrounds explained how AIML and mathematics work within FaceID, a well-known technology. Interviews were analyzed with the Knowledge in Pieces (KiP) framework and compared to AIML experts’ responses, who were asked the same questions. The findings showcase where students’ primitive responses took on characteristics of experts’ and where they diverged. Given our results, we highlight potential starting points for AIML curricula to be integrated with mathematics concepts.

Purpose of Study

Within the last half-decade, there have been a number of calls for youth to learn about artificial intelligence and machine learning (AIML) (Chiu & Chai, 2020; Touretsky et al., 2019). However, much remains unknown about how concepts from AIML can be operationalized in pre-collegiate classrooms. There are already several empirically-backed studies that provide examples of how students can learn about artificial intelligence successfully. Some of these curricula approach AIML with an ethical lens (Williams, Kaputsos, & Breazeal, 2021) while others build from a computer science standpoint, encouraging students to explore AIML concepts through relevant coding projects (Estevez, Garate & Graña, 2019).

Our interest in distributing AIML knowledge to youth lies in examining its mathematical underpinnings. While many mathematics concepts involved in AIML applications are beyond the scope of most pre-collegiate students’ mathematical knowledge (e.g., multivariate calculus), others are not (e.g., probability and geometry). Our overarching aim is to provide AIML curricula for students that are integrated within Common Core State Standards for Mathematics (CCSSM), so that students from diverse backgrounds may learn how AIML is supported in part by mathematics available to them.

To progress toward a “mathematics of AI” curriculum, a crucial first step is to understand what intuitions, ideas, and conjectures students offer when asked to explain how AI works. To address this aim, our research team asked students and experts to predict how FaceID, a common AIML application, operates, and what math concepts might be included. Through this study, we address where students’ conceptions mirrored those of experts’ and where they diverged. By doing so, we wish to identify common ground between experts’ and novices’ explanations of AI systems and their mathematics that may be used in service of broader AI curricula creation efforts. To that end we ask, how do novices conceive of AI and its relationship to mathematics, and, to what degree do novices’ conceptions parallel those of AI experts?

Theoretical Perspectives

Interest in studying human expertise arose from developments in artificial intelligence itself (Glaser, Chi, & Farr, 2014). Characteristics of expert thinkers were explored in cognitive psychology (Newell & Simon, 1972) and further defined throughout the 1970s-80s. Glaser (1987) noted that experts: excel and perceive large, meaningful patterns in their domains, solve domain problems quickly, have superior memories, represent domain-specific problems at deep levels compared to novices, qualitatively analyze problems for long periods of time, and have strong self-monitoring skills. For the present study, we will focus only on experts’ recognition of patterns, analysis of systems, and domain-specific representations.

Novices construct explanations for novel phenomena based on superficial, everyday experiences. Their knowledge of novel phenomena is diverse and dynamically cued, as characterized by diSessa’s Knowledge in Pieces (KiP) framework (diSessa, 1993). Elements in a KiP system have multiple forms and levels of complexity. The most basic unit, coined phenomenological primitives (p-prims), describes novices’ fragmented knowledge and causal explanations that often appear to be self-evident when evoked. As novices build expertise in a domain through carefully designed instruction, they learn to cue p-prims more productively and build expert-like explanations (diSessa, Gillespie, & Esterly, 2004), which can take the form of explanatory primitives (e-prims), offering more detailed explanations (Kapon & diSessa, 2012). Our work at present seeks to identify p-prims and e-prims in novices’ explanations of AIML.

Although the KiP framework was first constructed to examine expert-novice conceptions of physics (diSessa, 1993), p-prims have been applied elsewhere. For example, Southerland et al. (2001) used p-prims to investigate students’ tentative, shifting descriptions of biological phenomena, concluding that p-prims were useful in characterizing students’ formations of scientific ideas. Likewise, Iszak applied KiP to both students and pre-service teachers studying proportion, functions, and other multiplicative relationships in mathematics (Iszak, 2005; Izsak & Jacobson, 2017), and suggested that the KiP framework was productive for suggesting improvements for mathematics curriculum and instruction design.

To the best of our knowledge, KiP has not yet been used to describe students’ conceptions of AIML. Yet, it shares many commonalities with physics. AIML is a domain that is positioned as difficult for non-experts to learn (diSessa, 1996) and difficult to access without a statistics or computing background (Sulmont et al., 2017). Like physics, AIML is omnipresent in the lives of 21st century citizens. We therefore hypothesize that novice students will have informal conceptions of AIML based on their repeated interactions with technologies that rely upon it.

Methods

Participants and Procedures

We recruited 7 experts and 36 high school students via snowball sampling on their conceptions of AIML and its mathematics. Students participated from 20 schools in eight states and territories from all regions of the U.S. They gave demographic information and their most recent math course completed, which varied from Algebra 1 to advanced courses such as Differential Equations. Although not explicitly asked, some students described their experiences in computer science courses such as AP Computer Science A. Experts were identified based on their position in AIML fields, such as data scientist or computer science professor. Rather than inquiring about prior coursework, experts were asked how they used mathematics in their current work.

We conducted semi-structured interviews designed to elicit explanations of how FaceID, an AIML application that uses face recognition to unlock mobile phones, works. Interviews took place on Zoom and lasted 30-45 minutes. To ensure that all participants understood what we meant by FaceID, we showed a video segment of a user setting up his iPhone by capturing photos of his face from different perspectives, then using it to successfully unlock his phone. After the clip, the team asked follow-up questions to determine what each respondent noticed, how they believed the technology worked, and what mathematics concepts might be involved in FaceID. Probing questions were asked until the interviewer felt confident that they understood the hypothesis presented by each participant.

Data Analysis

After the interview phase, each member of the research team independently coded all transcripts for similarities in participants’ explanations (Saldaña, 2014). Researchers attended to subject domains that participants drew upon when explaining AIML concepts (e.g., references to computing, human cognition), which mathematics topics participants connected to FaceID, and how participants explained the role of mathematics in the technology. For students’ descriptions, researchers noted patterns in explanations that reflected primitiveness (e.g., “AI is a smart robot”) and patterns that suggested formal theories (e.g., “A computer program that makes decisions from the data.”). After data were independently coded, the research team discussed which students’ explanations contained similar elements and should be classified together. The meeting continued until all disagreements were resolved, and a categorization hierarchy that was motivated by Southerland et al. (2001) was created to group similar student explanations (see Table 1). A second coding pass of the data was conducted to ensure that all responses fit within the coding rubric.

Results

Experts’ Descriptions of AI Systems

The seven experts initially described AI with two standout features. First, all but one foregrounded AI’s historical roots in cognitive psychology rather than its presence in modern computational systems. They connected AI with the humanistic notions of decision-making, intelligence, and behaviors of humans, such as using a priori past observations to make future judgments. Second, experts supplanted explanations with domain-specific examples. For instance, Expert 7, a native of California, explained how machine learning systems draw upon past climate data to make predictions about present wildfire frequencies. In doing so, experts demonstrated focused attention to multiple facets of AI as well as multiple real-world uses of it.

When asked to describe FaceID and its associated mathematics concepts, all but one expert gave both high-level descriptions and esoteric details. All experts described FaceID as a large network that took image data as input and processed it with a machine learning algorithm through pixel comparisons. Some experts elaborated further, explaining that convolutional neural networks extract relevant features from images, such as edges, facial features, and ratios between facial features, to create a distinguishable profile of the user to be used during face recognition. Mathematically, experts agreed that calculus and linear algebra comprised the core mathematics domains used in FaceID, which would have been out of scope for many high school students. However, they observed that knowing topics prerequisite for calculus and linear algebra, such as matrices, matrix operations, probability, general functions, trigonometric functions, and inequalities, also supported understanding FaceID’s operations.

Students’ Descriptions of AI Systems

A high-level summary of each student groups’ hypotheses and examples is shown in Table 2. Students with anthropomorphic knowledge constructions attributed FaceID’s operations to human-like characteristics. Students with teleological knowledge constructions attributed FaceID’s recognition abilities to iPhone’s camera. The only mathematics topics identified were geometric knowledge of angles and taking measurements. We suspect those topics may have been cued by the FaceID video in the interview protocol. A few students did not believe that mathematics was involved. Although some students’ explanations mirrored experts’ attention to humanistic features, students did not connect their conjectures to computing.

Students with mechanistic proximate or mechanistic anthropomorphic knowledge constructions connected aspects of computing and mathematics to their explanations of FaceID. They mentioned computer programs, tasks, and data in their hypotheses, although they stopped short of explaining how those elements were coordinated to accomplish AIML. Mechanistic anthropomorphic students additionally attributed humanistic properties to AIML systems. Mathematically, students’ explanations varied. Some did not know how math was involved, some identified the same mathematics concepts as students in the first level, and others drew inferences about invisible mathematics, such as probability. Student 29, for example, stated, “They must use math to analyze how well the system is working. Probability to ensure that…it is getting the right face, geometry to map out the dimensions of your face and looking for color proportions, and maybe intervals to calculate a certain area of your face.” Although a primitive explanation, this student identified many pre-linear algebra math topics identified by experts.

Students with mechanistic ultimate knowledge constructions explained AIML as a system of interconnected computing-based components that were coordinated to perform humanistic tasks. These students offered cause-and-effect inferences that explained how FaceID’s infrastructure relied on bigger ideas from AIML (e.g., training data) and what mechanism from AIML permitted FaceID to make decisions. Geometry topics were still the most prominently cited as involved with AIML. Surprisingly, their levels of explanatory detail attributed to mathematics varied greatly. Student 12, for example, explained that FaceID involved, “an absolute or local minimum…so when the algorithm keeps tuning itself, it’s hoping to move down,” whereas Student 20 offered, “I believe that math has a place in this. I can’t tell you what specific mathematical function, but it’s definitely determining values of…shapes and stuff.” While these students’ overall explanations of FaceID showed some characteristics of e-prims, their connections to mathematics sounded more primitive than we expected.

Discussions and Implications

In aggregate, students’ conceptions tended to contain one or more the following p-prims: AI as a humanistic agent, AI as a machine that executes tasks, and AI as a data-driven system. Some students offered more explanatory power to p-prims into what could be considered e-prims, for example, AI as a machine that executes tasks by combining data from humans with a learning algorithm. Though some students’ explanations paralleled experts’ descriptions in highlighting humanistic aspects of AIML or by offering detail on how machines are trained, not even the most detailed hypotheses mirrored experts in mathematical precision.

In some ways, this finding is not surprising. Druga, Otero, and Ko (2022) reviewed over 50 AI curricula, where only three connected AIML to mathematics. A majority of the curricula focused on overarching concepts in AIML while abstracting away mathematics. One hypothesis is that many students who gave mechanistic ultimate knowledge constructions may have received prior AIML instruction in one such curricula, yet were never implored to consider mathematical connections.

However, a vast majority of students offered conjectures about mathematics’ involvement in AIML’s infrastructure even without formal knowledge. In our view, this suggests that students could learn about the mathematics of AIML, albeit at a basic level. We suggest that future curricula strongly consider designs that explicitly bridge AIML with its prerequisite mathematics.

References

Chi, M. T., Glaser, R., & Farr, M. J. (2014). The nature of expertise. Psychology Press.

Chiu, T. K., & Chai, C. S. (2020). Sustainable curriculum planning for artificial intelligence education: A self-determination theory perspective. Sustainability, 12(14), 5568.

diSessa, A. A., Gillespie, N. M., & Esterly, J. B. (2004). Coherence versus fragmentation in the development of the concept of force. Cognitive science, 28(6), 843-900.

diSessa, A. A. (1996). What do “just plain folk” know about physics. The handbook of education and human development, 709-730.

diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and instruction, 10(2-3), 105-225.

Druga, S., Otero, N., & Ko, A. J. (2022). The Landscape of Teaching Resources for AI Education.

Estevez, J., Garate, G., & Graña, M. (2019). Gentle introduction to artificial intelligence for high-school students using scratch. IEEE access, 7, 179027-179036.

Glaser, R. (1987). Thoughts on Expertise in C. Schooler e K.. W. Schaie (dir.), Cognitive Functioning and Social Structure over the Life Course (p. 81-94), Norwood.

Izsak, A. (2005). ” You have to count the squares”: applying knowledge in pieces to learning rectangular area. The Journal of the Learning Sciences, 14(3), 361-403.

Izsák, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300-339.

Kapon, S., & diSessa, A. A. (2012). Reasoning through instructional analogies. Cognition and Instruction, 30(3), 261-310.

Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104, No. 9). Englewood Cliffs, NJ: Prentice-hall.

Saldaña, J. (2014). Coding and analysis strategies. The Oxford handbook of qualitative research, 581-605.

Southerland, S. A., Abrams, E., Cummins, C. L., & Anzelmo, J. (2001). Understanding students’ explanations of biological phenomena: Conceptual frameworks or p‐prims?. Science Education, 85(4), 328-348.

Sulmont, E., Patitsas, E., & Cooperstock, J. R. (2019). What is hard about teaching machine learning to non-majors? Insights from classifying instructors’ learning goals. ACM Transactions on Computing Education (TOCE), 19(4), 1-16.

Touretzky, D., Gardner-McCune, C., Martin, F., & Seehorn, D. (2019, July). Envisioning AI for K-12: What should every child know about AI?. In Proceedings of the AAAI conference on artificial intelligence (Vol. 33, No. 01, pp. 9795-9799).

Williams, R., Kaputsos, S. P., & Breazeal, C. (2021, May). Teacher Perspectives on How To Train Your Robot: A Middle School AI and Ethics Curriculum. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 35, No. 17, pp. 15678-15686).

Novices Students’ Explanations of AI	Description	Illustrative Quote
Anthropomorphic or Teleological	A: An explanation based on the use of human attributes as the causal agent for AI. T: An explanation where the technological ends are considered as the agent that explains AI.	“AI is computers that have brains and are running everything.” —Student 9 “AI is smart technology.” —Student 19
Mechanistic Proximate	An explanation in which an aspect of computing, such as programming, is identified as the underlying determinant of AI systems.	“Technology that is programmed to respond like a human.” —Student 17
Mechanistic Anthropomorphic	An explanation in which an aspect of computing, such as programming is identified and connected to a human attribute of AI, such as behaving autonomously, reasoning, or making a decision.	“Computers that are coded to think on their own, that are coded to generate answers or do tasks that are learned by themselves without human intervention.” —Student 29
Mechanistic Ultimate	An explanation in which several aspects of computing are combined together by cause-and-effect mechanisms to illustrate a goal or outcome of an AI system.	“A computer program that learns and grows depending on the data that it is given, which comes from saved answers. [AI] can be used for human-like tasks, such as social media.” —Student 13

Novices Students’ Explanations of FaceID	Summary of Students’ Hypotheses	Illustrative Quote
Anthropomorphic or Teleological (n = 11)	A: Attributed FaceID’s operations to human-like characteristics. T: Attributed FaceID’s operations to capabilities of visible hardware.	“AI looks at the user’s face from all different perspectives.” —Student 6 “AI reads multiple points from your face to see if it matches with the given information” —Student 15 “It uses a camera that recognizes stuff.” —Student 24
Mechanistic Proximate (n = 12)	Attributed FaceID’s operations to a coordinated system containing a program, data collection, and/or a task to be accomplished, without explaining how	“To get accurate reads of what makes your face unique, data is collected that is specific to your face.” —Student 35 “AI would be shown the user’s face and be quickly trained to recognize that person’s face specifically.” —Student 11
Mechanistic Anthropomorphic (n = 6)	Attributed FaceID’s operations to a coordinated system with human-like properties (e.g., ability to make a decision, reason, think) containing a program, data collection, and/or a task to be accomplished, without explaining how	“Face ID asks for many angles so it gets all the miniscule details, and when you unlock your phone from any angle, it works. Face ID processes all the images, internalizing them to recognize faces at any angle.” —Student 32
Mechanistic Ultimate (n = 8)	Attributed FaceID’s operations to a coordinated system with human-like properties and explains how each element of the system contributes to the completion of the task.	“When you move your head around, the AI is trying to combine different angles of you into what one person looks like. So you’re training the machine to produce several portraits of you. When you look at your camera, if your facial features match, your phone unlocks.” —Student 12