Abstract thought in STEM education: an integrative literature review

Abstract

Purpose

Abstract thought builds the basis for problem-solving and knowledge consolidation across the disciplines of science, technology, engineering, and mathematics (STEM). Scientists across these fields acknowledge its significance and have approached the topic from their distinct perspectives, and yet, in STEM, there is a lack of a unified understanding and definition of abstraction.

Methods

To bridge this gap, in this study we employed the integrative literature review methodology to identify the most relevant literature in STEM education and unify the literature into a comprehensive model of abstraction for STEM.

Results

Abstraction is a process of consolidation common to all STEM fields. Starting from a specific point of view, empirical and reflective abstraction strategies, such as reduction, pattern recognition, generalization, interiorization, encapsulation, and coordination, work in parallel to manipulate objects across the concrete-abstract continuum. Beyond, we identified six discrete levels of abstraction in STEM and aligned the new model to the prevalent discourses on modeling, representation, problem-solving, and computational thinking.

Conclusion

Our model of abstraction for STEM contributes to a unified understanding of STEM education by highlighting the fundamental role that abstraction plays across all disciplines. In doing so, this study provides a foundational framework that can inform curriculum development and interdisciplinary teaching approaches across STEM.

Introduction

Let us start with a thought experiment. What comes to your mind if we say: “4 legs, orange and white stripes?” It is clearly an abstract description; however, probably enough for us to be thinking about the same thing. What if we change the prompt to, “whiskers and paws?” We are still not excluding the first object, but now we have expanded the set of possibilities.

Humans are good at categorizing the world based on empirical observations. With little doubt, most can recognize a tiger, or any member of the cat family based on their distinct features. This is how most of human knowledge has come to exist: empirical observations built into increasingly abstract concepts (Chambers, 1991).

Harder to explain is how we arrived at more complex concepts like evolution. Category creation based on commonalities seems insufficient as a sole strategy. Although Darwin’s conclusions are clearly based on observations, increasingly complex human ideas are rather the result of reflective thought on existing abstract concepts (Chambers, 1991), e.g., trait inheritance + natural selection = evolution.

Transitioning from biology, some readers may find the dynamics of the tiger’s hunt more compelling. A physicist, for instance, might focus on the kinematics involved: if we reduce the tiger into a point mass, we can model the force at collision with $F=m·a$. Mathematicians may delve into the second derivative of distance with respect to time, while engineers might prioritize the mechanics behind the tiger’s self-sharpening claws, essential for capturing prey. Meanwhile, computer scientists would have defined the class Tiger$\{\}$, a subclass of Cat$\{\}$, containing the function hunt().

Abstract thought builds the basis for all problem-solving and knowledge consolidation across the fields of science, technology, engineering, and mathematics (STEM; Gray & Tall 2007; Halmos 2007; Karch & Sevian 2006; Monaghan & Ozmantar 2021). Scientists have referred to abstraction as the key competency for student success in mathematics (Ferrari, 2003), science (Darwish, 2014) and computer science (Kramer & Hazzan, 2006; Bennedssen & Caspersen, 2008). Consequently, abstraction has been studied from different perspectives: STEM educators want to learn how to teach it, philosophers ponder its nature, and psychologists study its cognitive implications.

And yet, even experts will struggle to explain what abstraction means and how it manifests in the STEM classroom (Karch & Sevian, 2021). The whole concept is a haunting tautology: abstraction is too abstract to explain. It is used as a noun to refer to a summary or synopsis, as a verb to refer to the process of extraction or removal, and as an adjective to associate with the theoretical and even the complex (Wilensky, 1991). The different uses of the concept are not exclusive or contradictory, and yet a clear definition eludes us.

If we dive deeper into STEM literature, it gets more convoluted. Not only do we refer to abstraction with different terminologies, e.g., “general” or “essence,” but we also investigate adjacent concepts without clearly relating them to one another. The concepts of modeling (Capps & Shemwell, 2020), representation (Ferrari, 2003), and problem-solving (Ginat, 2021) stick out for their relevance in STEM education and close relation to abstraction. Furthermore, the education community has seen a steady growth in the literature concerning computational thinking (CT), and abstraction was repeatedly appointed as the cornerstone of CT (Gautam et al., 2020; Wing, 2006).

Abstract thought is a universal aspect of daily life, recognized by scientists across STEM fields for its importance. While each discipline has contributed to understanding this complex concept, to the best of our knowledge, no comprehensive, cross-disciplinary framework for abstraction currently exists. This literature review aims to tackle this challenge by reviewing the current body of literature in STEM education and integrating the results into a comprehensive model of abstraction in STEM. This work is guided by the following research question:

RQ:What are the key components and structure of a synthesized model of abstraction in STEM?

The contributions of this review are threefold. First, we gather and relate relevant literature across different STEM domains and offer an ontology of different concepts and naming conventions used across them. Second, we integrate the multiple perspectives into a joint view of abstraction in STEM. Our framework distinguishes six abstraction strategies that work in parallel to enable abstract thought. Third, we distinguish six discrete levels of abstraction within the concrete-abstract continuum.

By tackling these questions, we lay the theoretical foundation for a more targeted and interdisciplinary development of abstract thought in STEM curricula.

Conceptual definitions

Scientists have studied abstraction from all kinds of perspectives. The first contemplations into its nature originated in the philosophy of John Locke in the nineteenth century (Locke, 1847). However, the most influential contribution can be traced to the work of psychologist Jean Piaget on genetic epistemology (Piaget, 1950). Since then, science educators, philosophers, and psychologists have developed many more ideas on abstraction. Most definitions reduce abstraction to the concepts of generality and similarity.

Striking is the amount of vocabulary that surrounds the field. Not only across STEM fields, but within a field itself, multiple terms are used to coin the same ideas. This review seeks to untangle the diverse use of terminology. To avoid further linguistic confusion, in the current work the terms will be used in the following way:

• Abstraction (the noun) refers to both the result of a process of abstraction as well as the concept.

• The verb, to abstract, describes the cognitive process that culminates in an abstraction.

• As an adjective, abstract is the opposite of concrete, and it is a relative attribute of an object (see the discussion for exceptions).

• An object is any physical or mental entity that an individual can think or speak about (Cable, 2014).

• Abstract thinking is the competency by which an individual engages in abstraction.

Research design

We follow the Integrative Literature Review (IRL) methodology documented by (Torraco, 2016) and adhere to the latest Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines (Page et al., 2021) for the selection and documentation of relevant literature. As documented by Snyder (2019), the IRL allows us to combine empirical and theoretical research, combine perspectives and insights from different fields of research traditions, analyze the results qualitatively, and synthesize them into a new model or framework. This methodology has been well-documented and revised (Torraco, 2005, 2016) and has found extensive applications in education research (Jørgensen et al., 2023; Cherewka & Prins, 2023; Tsortanidou et al., 2022).

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Fig. 1

PRISMA diagram for study selection (Page et al., 2021). *Capped search as documented by (Haddaway et al., 2015)

Database and search strategy

The literature search was conducted using the PRISMA paradigm to ensure transparency [Page et al. (2021); Fig. 1]. All search queries were made on the 20th of September 2024. ERIC was selected as the main electronic database for its wide coverage of educational research. We looked for peer-reviewed entries combining the keywords (“Abstract Thinking” OR “abstraction”) AND (“STEM” OR “Science” OR “Physics” OR “Chemistry” OR “Biology” OR “Computer Science” OR “Engineering” OR “Mathematics”). ERIC returned 1250 peer-reviewed entries.

Complementarily, we searched the Scopus and PsycINFO electronic databases. We filtered the Scopus search for peer-reviewed STEM entries and limited the search for publications containing (“Abstract Thinking" OR “abstraction”) AND (“education” OR “competence” OR “cognitive”). At PsycINFO, we looked for peer-reviewed entries combining “abstraction” AND “education”. The searches returned 4542 and 1724 entries, respectively. Given the vast number of results, we only considered the 300 most relevant entries for each of the two supporting databases (Haddaway et al., 2015). Relevance is calculate based on the search parameters by the databases proprietary algorithms (Rovira et al., 2019).

Selection and screening process

After duplicate removal, 1346 records were selected for screening. The first author screened all entries according to a set of predefined criteria (Table 1). The second author screened a 20% random subsample of the total records (Cohen’s Kappa $\kappa =0.8$, Moher et al. 2007, O’Connor & Joffe 2020). Afterward, the first and second author met to adjust the screening criteria and align on discrepancies. 329 studies matched the selection criteria, out of which nine records could not be retrieved.

Next, we conducted a full-text review following the same selection criteria. The first author reviewed all 320 records and the second author reviewed a 10% random subsample (Cohen’s Kappa $\kappa =0.7$, Moher et al. 2007; O’Connor & Joffe 2020). Discrepancies were discussed until agreement. 247 articles were discarded, leaving 71 for extraction. During extraction, 11 additional articles were identified through citation searching and added to complement the original search. A total of 82 studies qualified for the review (Fig. 1).

Table 1. Inclusion and exclusion criteria

Inclusion criteria	Exclusion criteria
1. The studies were published in peer-reviewed journals. 2. The studies are reported in English. 3. The full-text is available. 4. The studies offered a theoretical or empirical contribution to the cognitive nature of abstraction.	1. The studies had no explicit reference to abstraction in the title or abstract. 2. The studies explored abstraction from a perspective other than STEM education. 3. The studies reviewed and/or applied theories of abstraction; however, abstraction is not the central topic of the study.

Characteristics of the qualifying articles

Table 2. Qualifying articles by field (N = 82)

Field	Qualifying articles	N
Science	Capps and Shemwell (2020), Darwish (2014), Fyfe et al. (2014), Lowell (1977)	4
Biology	Coleman et al. (2021)	1
Chemistry	Blackie (2014), Fackler and Capps (2024), Karch and Sevian (2020), Karch and Sevian (2021), Sevian et al. (2015)	5
Physics	Chen and Lee (1992), Widada et al. (2019)	2
Computer Science	Alexandron et al. (2014), Armoni (2009), Bennedssen and Caspersen (2008), Böttcher and Thurner (2023), Colburn and Shute (2007), Dordochi et al. (2021), Gautam et al. (2020), Ginat (2021), Hill et al. (2008), Kramer and Hazzan (2006), McMaster et al. (2010), Mirolo et al. (2021), Muller and Haberman (2008), Perrenet et al. (2005), Srinivasan and Te’eni (1990), Wang (2005), Wang (2008), Wilmont et al. (2012), Zehetmeier et al. (2019), Bilbao et al. (2021), Çakiroğlu and Çevik (2022), Ezeamuzie et al. (2022), Nicholson et al. (2009), Qian and Choi (2023)	24
Engineering	Gero et al. (2021), Hadish et al. (2023), Neuper (2017), Yagisawa and Iijima (2023)	4
Mathematics	Ahmadpour et al. (2019), Antonides and Battista (2022), Bakker and Hoffmann (2005), Breive (2022), Burton (1982), Cable (2014), Carrejo and Marshall (2007), Dixon and Bangert (2004), Dubinsky and Lewin (1986), Ellis et al. (2017), English and Sharry (1996), Ferrari (2003), Goodson-Espy (1998), Gray and Tall (2007), Guerrero-Ortiz et al. (2018), Hakim et al. (2019), Harel and Tall (1991), Hershkowitz et al. (2001), Hong and Kim (2016), Jao (2013), Kamii (2002), Mitchelmore and White (1995), Monaghan and Ozmantar (2006), Nemirovsky et al. (2020), Ozmantar and Monaghan (2007), Raychaudhuri (2014), Rich and Yadav (2020), Scheiner (2016), Simon (2019), Tall (2004), van Oers and Poland (2007), White and Mitchelmore (2010)	32
Interdisciplinary STEM	Cetin and Dubinsky (2017), Hazzan (2003), Krupczak and Bassett (2013)	3
Philosophy	Burgoon et al. (2013), Chambers (1991), Hampton (2003), Ohlsson and Lehtinen (1997), van Oers (2001)	5
Psychology	Bennett and Müller (2010), Chan and Chan (2023)	2

Table 2 presents an overview of the 82 selected articles. Over a third of the literature is from the field of mathematics (32 articles), a second third from computer science (24 articles), and the rest in science (12 articles), engineering (4 articles), and interdisciplinary STEM research (3 articles). In addition, we included seven articles in the fields of philosophy of science and psychology.

Approximately a third of the entries were purely of theoretical nature (29 articles) and two-thirds offered empirical contributions (53 articles). The articles reported a wide range of participant age groups, mainly 33 articles at university level, 8 high school, 14 middle school, and 8 at pre-school level.

Abstraction has mostly been explicitly studies in the fields of computer science and mathematics. Our selection of studies mirrors this distribution, with 40% of the articles coming from mathematics and 30% from computer science. Most the mathematics articles deal with school an pre-school students (77%), wile most of the computer science (80%) and engineering (60%) articles stem from university.

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Fig. 2

Analysis methodology

Analysis

We reviewed each paper and looked for the following types of insights:

• definitions of abstraction,

• characteristics of the abstraction process,

• comparisons between abstraction, modeling, representations, and problem-solving,

• descriptions specific to a STEM field or CT.

We extracted 953 insights, most of which are definitions and characteristics of abstraction. Insights varied from a single sentence to a full paragraph in length. In a first round of clustering, a tag was attached to each insight at the moment of extraction, categorizing it thematically (Fig. 2). The tags were developed “on the go” and refined in the further process. Each insight was assigned at least 1 and up to 4 tags. 318 different thematic tags were developed, e.g., “essence” or “context.”

In a second round of clustering, the tags supported the generation of themes. Each tag group (in alphabetical order) was revised, and each insight in the group was assigned to one and up to two emerging themes. The themes cluster important characteristics of abstraction. The generation of themes was done inductively and iteratively based on the conceptual similarity and relative frequency of the tags. For example, the insights under the tags "similarity", "commonality" and "difference" informed the creation of the theme "Identify Commonalities". After all insights were assigned at least one theme, a general description for each theme was formulated, taking care to include the nuances of all assigned insights.

Finally, the themes were recombined and structured, leading to a total of 36 themes (Table 3). In the process of definition, we inductively looked for a hierarchical categorization. This was especially challenging for the "empirical abstraction" group of themes, as many different authors have presented topics, such as "similarity", "generalization", "simplification", "essential", etc. in different relations to each other. Based on the integrated definitions for the themes, we recognized a new structure that efficiently related these elements to each other.

After all themes were finalized, we deductively reviewed them for consistency and re-assigned insights to more suitable themes when necessary. Hence we made sure that each insight was properly represented by a theme, and that all themes converged to a clear definition. By allowing more then one theme per insight, we represented the relationships and hierarchies between the themes.

Table 3. Hierarchy of themes developed during the second round of clustering

#	Theme name
1	Consolidation
2	Dialectic Hierarchy
3	Empirical Abstraction
4	Horizontal
5	Generalization
6	Identify Commonalities
7	Focus Attention
8	Reduction
9	Concretization
10	Decomposition
11	Pattern Recognition
12	Reflective Abstraction
13	Vertical
14	Interiorization
15	Action
16	Process
17	Encapsulation
18	de-encapsulation
19	Object
20	Coordination
21	Schema
22	Formalization
23	Recontextulaization
24	Embodiment
25	Socialization
26	Relatedness
27	Levels of Abstraction
28	Modeling
29	Representation
30	Problem Solving
31	STEM
32	Science
33	Computer Science
34	Computational Thinking
35	Engineering
36	Mathematics

The next section reports on the findings of these themes and integrates them into a comprehensive overview of abstraction in STEM.

Results

The EA and RA duality

In the introduction, we distinguished between two types of abstraction. Empirical abstraction (EA) accounts for the identification of the underlying structures in our sensory input, e.g., “4 legs, orange and white stripes” equals “tiger.” Its counterpart, reflective abstraction (RA), explains the assimilation of ideas (often gained through empirical abstraction) and the construction of higher-level knowledge, e.g., evolution (Chambers, 1991).

Both types of abstraction are fundamental across STEM; the relationship between the two, however, is far from obvious (Cable, 2014). Historically, research has tackled empirical and reflective abstraction individually, mathematicians focusing on the inference-focused abstraction in mathematics and computer scientists on the empirically grounded abstraction in CS (Colburn & Shute, 2007). Nevertheless, most modern approaches have recognized the importance of studying the strong interplay between the two (Hakim et al., 2019; Ozmantar & Monaghan, 2007; Scheiner, 2016).

Davydov (1990) was the first to suggest a dialectic relationship between EA and RA (Ozmantar & Monaghan, 2007). Ohlsson and Lehtinen (1997) followed by questioning the presumed order of abstraction: What happens first, does the thinker recognize the underlying structure in the information and then create the idea? Or do they recognize the need for an idea and then assign the underlying structure? In other words, does the knower need to already possess the abstraction to recognize an object as an instance of it?

Constantly switching between EA and RA is what makes abstraction a dynamic and powerful tool for knowledge consolidation and problem solving (Cable, 2014; Ozmantar & Monaghan, 2007). The two strategies are complementary and not opposing (Scheiner, 2016).

Empirical abstraction

Beth and Piaget (1966) first used the term empirical abstraction (EA) to describe the process of using perception (as opposed to reflection) to create a category based on the underlying characteristics of a group of objects [as cited by Cetin and Dubinsky (2017)]. Thereafter, Davydov (1990) kept the term for the process of observing differences and similarities to analyze features of reality using empirical thought [as cited by Ozmantar and Monaghan (2007) and Hershkowitz et al. (2001)]. Skemp (1987) defined instrumental understanding as a "knowing how" (rather then "knowing why") cognitive strategy [as cited by Mitchelmore and White (1995)]. Piaget and Inhelder (1969) also used the term operative mode of thought for experience-based reasoning, and along the same line of thought, abstract-general objects were defined as the result of identifying key common features in real-world situations [as cited by Mitchelmore and White (1995)].

Building on the perceptual component of EA, Tall (2004) introduced a new term, the conceptual embodied world, the first of three “worlds” that describe mathematical knowledge building. The conceptual embodied world uses sensory perceptions of the external (physical world) but also of the thinker’s internal world of meaning to construct knowledge. The other two “worlds” fall into RA. Similarly, Hong and Kim (2016) established three levels of mathematical abstraction. The first level recognizes the underlying mathematical structure using perceptual abstraction; the other two levels adhere to RA. Clements (2000) also used the term sensory-concrete knowledge to describe the use of sensory input in sense-making (as cited in Nemirovsky et al. (2020), and Freudenthal (1991) first coined horizontal mathematization to describe a move from the real world to the world of symbols [as cited in White and Mitchelmore (2010)]. Karch and Sevian (2021) used horizontal abstraction with equivalent meanings.

On the other hand, Peirce (1976) (as cited in Bakker & Hoffmann 2005) focused on the correspondence aspect. They used the term prescissive abstraction for the process of dispensing with unnecessary aspects to keep the “essence” of an object. In similar fashion, Simon (2019) talked about an Abstraction of Commonalities (AoC) for the process of identifying what is “the same.” Interestingly, Simon (2019) categorized AoC as a type of reflective abstraction, opposite to empirical. Their argumentation is based upon the inherent duality of abstraction (one needs to possess the category to assign the elements; Ohlsson & Lehtinen 1997). Acknowledging this duality as a key characteristic of EA allows us to categorize AoC as empirical abstraction based on the similarity criterion.

Scheiner (2016) building on the work of Tall (2013) expanded both the perceptive and correspondence notions. Their notion of structural abstraction shifts the focus from perceptual, physical-based understanding to one grounded in internal conceptual structures. In addition, building on the work of Piaget (2001), Scheiner (2016) used the term abstraction of objects. Both theories highlight the contextual aspect of EA: knowledge starts domain-bound and shifts across a variety of contexts, allowing the thinker to identify the underlying structure. Mitchelmore and White (1995) spoke of abstract-general ideas that are valid across multiple contexts, and Maton (2011, p.65) used semantic gravity to measure “the degree to which meaning relates to context” (as cited in Blackie 2014).

In addition, the RBC (and RBC+C) frameworks described abstraction not as a linear process but rather as a set of nested epistemic actions for abstraction (Hershkowitz et al., 2001). The nesting of the actions is cleverly designed to address the circular dependency between empirical and reflective thought. The second epistemic action, recognition, corresponds to the use of EA to recognize underlying structures.

All these perspectives on empirical abstraction build toward a better understanding of the concept. Altogether, empirical abstraction is an essential component in the cognitive process of sense-making and knowledge construction (Ozmantar & Monaghan, 2007; Gray & Tall, 2007). EA enables navigating between contexts and a bidirectional movement between the concrete and the abstract (Scheiner, 2016). At its core, it involves dissecting the underlying structures of sensory information. Rather than seeking similarities, the emphasis lies on recognizing complementarity, where diverse perceptions enrich understanding (Scheiner, 2016). This enables individuals to perceive dimensions through their experiences, often leading to unexpected discoveries (Dixon & Bangert, 2004). Ideas are distilled and condensed, ultimately alleviating cognitive complexity and thereby aiding comprehension and retention (Gray & Tall, 2007; Chen & Lee, 1992).

Reflective abstraction

In 1966, Beth and Piaget (1966) first used the term reflective abstraction (RA) to describe a different type of abstraction, one capable of explaining higher knowledge, such as numbers and mathematics, which do not casually follow from empirical observations [as cited by Cetin and Dubinsky (2017)]. A long tradition in mathematics didactics followed to further define the underlying process. Like empirical abstraction, multiple terminologies have been suggested to describe the same process.

The main idea behind RA is to draw meaning from the coordination of actions (as opposed to characteristics). An action is a step-by-step set of instructions to explicitly transform an object. Actions can be interiorized into processes, by which they become more dynamic and available across contexts. Actions and processes can also be encapsulated, transforming them into objects that can then be manipulated independently. Finally, RA allows for the coordination of objects and processes into complex knowledge schemas. Dubinsky (1991) first published this framework in 1991 as the action, process, object, schema (APOS) theory [as cited in Cetin and Dubinsky (2017)]. Scheiner (2016) refers to the same idea as abstraction from actions (Piaget, 2001).

Simon (2019) maintained this focus in his definition of coordination of actions. According to them, during RA, the learner progresses from applying a sequence of actions to anticipating the outcome without mentally carrying the actions out, thereby moving from lower-level actions to a higher-level object that automatically gives the result. An equivalent meaning is mirrored in the hypostatic abstraction described by Peirce (1976, p.124), where “a noun takes over a predicate"(as cited in Bakker & Hoffmann 2005).

The second and third worlds of mathematics described by Tall (2004) detail a cognitive process based on actions. In their approach, the proceptual-symbolic world grows out of embodiment through actions and symbolization, forming new concepts to think with. They further differentiate a third step, the axiomatic-formal world, specifically for knowledge structures defined solely by set-theoretic definitions. Similarly, the second and third levels (internalization and interiorization) in Hong and Kim (2016)’s structure mirror a stepwise move from the embodied toward the symbolic-mathematical. Furthermore, Maton (2011, p.66) used the term semantic density to describe “the degree to which meaning is condensed in symbols” (as cited in Blackie 2014).

The concept of theoretical abstraction is also relevant (Davydov 1990, as cited in Ozmantar & Monaghan 2007). Theoretical abstraction builds upon the empirical to establish relationships not available to the senses. Davydov (1990) emphasizes the analysis of the underlying structures of knowledge, relating it directly to vertical mathematization and the development of mathematical ideas and their symbolic representations (Freudenthal, 1991; White & Mitchelmore, 2010). Karch and Sevian (2021) use the term vertical abstraction to describe the process of reorganizing conceptual (or mathematical) knowledge within the problem space to create new meaning. Furthermore, Mitchelmore and White (1995) discussed abstract-apart concepts, which exist only within the mathematical system, apart from external references. They directly equated this concept with Skemp (1987)’s relational understanding. Similarly, Clements (2000) described integrated-concrete knowledge as resulting from the combination of ideas into new structures [as cited in Nemirovsky et al. (2020)]. Furthermore, RA can be mapped to the third epistemic action (Hershkowitz et al., 2001). In the process of construction, novel structures are pieced together from existing components, aligning well with Ellis et al. (2017)’s intra-contextual extending.

Overall, reflective abstraction describes the cognitive process responsible for synchronizing mental activity and prior knowledge (Goodson-Espy, 1998; Cetin & Dubinsky, 2017). RA is responsible for the compression and construction of higher-level knowledge (Gray & Tall, 2007).

An synthesis of EA and RA

Multiple authors have discussed the dialectic relationship between EA and RA using varying terminologies.(Hershkowitz et al., 2001; van Oers, 2001). As part of our synthesis of the existing literature, we offer Table 4 as an overview of the terms used to refer to equivalent ideas. The terms empirical and reflective stand out for their descriptive nature and are thus used in this review to signify all others. It should be noted that this table is designed to aid comparability across frameworks and ideas while at the same time acknowledging the nuances different authors have given to their terminologies (see the previous sections for a detailed review). In some cases the author intended these terms beyond the EA and RA duality, we have clearly marked them and engage with this re-interpretation in the discussion.

Beyond the duality, authors in the reviewed literature have recognized that abstraction is strongly dependent on the individual making the abstraction and their specific context (Nicholson et al., 2009). The next section reviews this dependency in detail.

Point-of-view

Abstract thinking is a highly context-dependent process. Through empirical abstraction, objects are constantly shifted across contexts (Ferrari, 2003; Scheiner, 2016). Reflective abstraction elevates processes and objects to become available across contexts. Nonetheless, abstraction is never a process of decontextualization. Removing context would impoverish a concept rather than enrich it (van Oers, 1998; Scheiner, 2016). Hence, we recognize abstraction as a process of re-contextualization.

Abstraction is triggered by an individual recognizing the need for a new structure (Hong & Kim, 2016). Beyond simply reproducing reality, the thinker establishes a point-of-view to guide the thinking process (van Oers, 2001; Cassirer, 1923). This is also referred to as the motive (Hershkowitz et al., 2001) of the abstraction. In the process, objects are stripped away from their original context and re-contextualized according to the individual’s personal history and the assigned purpose of the abstraction (Carrejo & Marshall, 2007). This process goes beyond merely knowing but is deeply entangled with individual embodiment (Breive, 2022). The thinker takes a position within the abstraction and in the process co-produces themselves. External manipulations, both haptic and virtual, can aid the abstraction process (Böttcher & Thurner, 2023; Hakim et al., 2019; Nemirovsky et al., 2020; Chan & Chan, 2023; Fyfe et al., 2014).

In addition, previously available knowledge and cultural conventions heavily influence the resulting abstraction (Wilmont et al., 2012). Transferability of the resulting abstraction requires knowledge of the constraints introduced by the individual’s context and the intended purpose of the abstraction (Capps & Shemwell, 2020; Nicholson et al., 2009).

Independently of the context or purpose of an abstraction, this complex cognitive process is aided by multiple cognitive strategies. The next section expands on the most common elements of abstraction.

Table 4. Terminologies used to refer to empirical and reflective abstraction in the reviewed literature

Empirical	Reflective	Citation
Empirical abstraction	Reflective abstraction	Beth and Piaget (1966) as cited in Cetin and Dubinsky (2017)
Empirical abstraction	Theoretical abstraction	Davydov (1990) as cited in Ozmantar and Monaghan (2007)
Instrumental understanding	Relational understanding	Skemp (1987) as cited in Mitchelmore and White (1995)
Conceptual-embodied	Proceptual-symbolic and axiomatic-formal	Tall (2004)*
Perceptual abstraction	Internalization and interiorization	Hong and Kim (2016)*
Sensory-concrete knowledge	Integrated-concrete knowledge	Clements (2000) as cited in Nemirovsky et al. (2020)
Horizontal mathematization	Vertical mathematization	Freudenthal (1991) as cited in White and Mitchelmore (2010)
Horizontal abstraction	Vertical abstraction	Karch and Sevian (2021)
Preccisive abstraction	Hypostatic abstraction	Peirce (1976) as cited in Bakker and Hoffmann (2005)
Abstraction of commonality	Coordination of actions	Simon (2019)*
Structural abstraction	Reflective abstraction	Tall (2013); Scheiner and Pinto (2014); Scheiner (2016)
Abstraction from objects	Abstraction from actions	Piaget (2001); Scheiner (2016)
Abstract general	Abstract-apart	Mitchelmore and White (1995)
Semantic density	Semantic gravity	Maton (2011), Blackie (2014)
Recognition and built-within	Construction	Hershkowitz et al. (2001)*

*Reinterpretation of the terminology which was originality intended beyond the EA/RA duality

The strategies of abstraction

Empirical and reflective abstraction are tightly intertwined, working in parallel to make sense of sensory information and consolidate knowledge (Scheiner, 2016). In the reviewed literature we found six strategies of abstraction. Generalization, reduction, and pattern recognition manipulate empirical information across different contexts, while interiorization, encapsulation, and coordination compress and organize information into complex knowledge structures.

Abstraction strategy 1: generalization

Generalization is probably the most associated component of abstraction. It has received considerable attention; for example, Harel and Tall (1991) refer to it as generic abstraction. The fundamental process of generalization is to extract the essence of empirically obtained information and compress it into novel mental objects (Muller & Haberman, 2008; Cetin & Dubinsky, 2017). It involves a transition from the concrete to the abstract, purposefully relieving the object from its original context and unnecessary details. It is the purpose of the abstraction that dictates what the essential entails (White & Mitchelmore, 2010). Following the definition of Burgoon et al. (2013, p.503), the essence of an object or group of objects are “the invariant central characteristics that will increase the likelihood of accurately identifying the original(s) when encountered across various contexts.” Smith et al. (1974) also referred to it as core features (as cited in Burgoon et al. 2013), and Fackler and Capps (2024) uses the term abstracting ideas to describe the same concept. Alexandron et al. (2014) suggested a level of meaning to measure how much information (number of bits) is “hidden” within a concept.

Hill et al. (2008, p.16) introduced the term descriptive abstraction in their taxonomy to refer to the same process of “discerning the characteristics of chief importance to construct generalized accounts.” Also, the path of procedure, described by Ahmadpour et al. (2019), equates to a similar process. Their work in mathematical proofs described a series of steps, where “procedural proofs” are formalized based on naive experience, without necessarily creating a high-level abstract concept. In addition, in their synthesis of CT, Qian and Choi (2023) highlighted filtering information and locating similarities as two complementary key elements for abstraction.

Two complementary (and overlapping) sub-strategies are distinguishable as part of generalization: Identifying Commonalities among different objects and Focusing Attention on the core features of an object or objects (Armoni, 2009; Chen & Lee, 1992).

Identify commonalities

Abstraction based on commonalities was part of almost all reviewed frameworks. This type of generalization works by extracting the common attributes across a group of objects to generate a distinguished set (Ferrari, 2003). These corresponding features can be of descriptive, relational, perceptual, or conceptual nature and can be part of the current perceptual input or existing prior knowledge (Karch & Sevian, 2020; Ferrari, 2003). Powered by flexible thinking and analogical reasoning (Bennett & Müller, 2010; English & Sharry, 1996), attention shifts across the elements to recognize similarities and ignore or normalize differences, resulting in a distinct classification (Sevian et al., 2015; Zehetmeier et al., 2019). Ohlsson and Lehtinen (1997) argued for an articulation process that goes beyond simple pattern matching and requires a recursive top-down expansion of the category until the perception matches the new set. A simple example can be found in the geometry classroom. Students are traditionally confronted with various examples of triangles and carefully prompted by their teachers to abstract the idea of, for example, an “equilateral triangle” (Ferrari, 2003).

Other terminology used to describe this process include similarity (Hakim et al., 2019), complementarity (Scheiner, 2016), and intra-contextual forming (Ellis et al., 2017). In addition, terms such as aggregation (Doyle 1986 as cited in Chen & Lee 1992), comparison-based abstraction (Reeves & Weisberg 1994 as cited in Dixon and Bangert 2004), type development (Hampton, 2003), and summarization (Nicholson et al., 2009) were used to describe the same cognitive process.

Focus attention

Generalization can also happen without comparison. Focus attention refers to the cognitive process of eliminating non-essential features to center on details most relevant to a specific purpose (Hill et al., 2008; Harel & Tall, 1991). Physics educators, for example, prompt their students to ignore “friction” to approximate the idea of “linear motion” (Chen & Lee, 1992).

A parallel idea argues not to discard but “hide” the unnecessary features, making the process reversible (Colburn & Shute, 2007; Zehetmeier et al., 2019). This is often referred to as black boxing (Armoni, 2009; Muller & Haberman, 2008). Software engineering students, for example, learn to use functions to hide the implementation and increase code readability (Colburn & Shute, 2007).

Other authors have also referred to this process as approximation (Doyle 1986 as cited in Chen & Lee 1992), selective storage (Hampton, 2003), condensation (Perrenet et al., 2005), formal abstraction (Hill et al., 2008), and eliminating specificity (Lewis 1986 as cited in Colburn & Shute 2007. Çakiroğlu and Çevik (2022) used focusing and elimination as two sides of the same coin. In the work of Ezeamuzie et al. (2022), the term decomposition is used to describe the same process of identifying irrelevant details, obscuring or removing them, and focusing on the relevant. In this review, we assign a different meaning to the word decomposition (see next section).

Abstraction strategy 2: reduction

The shift toward the more empirically concrete comes with an increase in external contextuality. Reduction is often tied to simplification, as the change in context can often ease the problem-solving process, e.g., by making an already existing solution more accessible (Hazzan, 2003; Raychaudhuri, 2014). Armoni (2009) found that students often struggle to acknowledge reduction as a valid strategy. Two sub-strategies of reduction exist, namely, Decomposition and Concretization.

Decomposition

Decomposition is often considered (together with generalization) as one of the main components of EA (Ezeamuzie et al., 2022). Decomposition involves breaking down a concept into sub-components to process them independently and reduce the complexity of the original concept (Curzon et al.2019, Liskov & Guttag 1986 as cited by Kramer & Hazzan 2006). It is applicable to different contexts and levels and is often employed as a problem-solving heuristic that facilitates the reuse of solutions. In the engineering classroom, for example, students learn to decompose complex mechatronic systems into their mechanical and electrical components.

Muller and Haberman (2008) used the term structure identification to refer to decomposition in the context of problem-solving. In addition, Srinivasan and Te’eni (1990) highlighted the importance of abstraction and decomposition in problem structuring.

Concretization

Complementary to decomposition, concretization allows for reducing the level of abstraction by adding detail and introducing constraints (in the form of context dependencies) to the original object (Hampton, 2003). It is a useful problem-solving strategy if the necessary mental representations to make sense of the situation do not exist, e.g., when communicating an abstract idea to others. Educators, for example, will be familiar with the benefits of giving examples after introducing an abstract idea in class (Hazzan, 2003).

Karch and Sevian (2020) described concretization as one of the core manipulations of the problem space. Furthermore, Hampton (2003) referred to the same process as instantiation.

Abstraction strategy 3: pattern recognition

As a cognitive process, pattern recognition is closely related to identifying commonalities and decomposition, as all three rely on analogical reasoning to find relational correspondences across structures (English & Sharry, 1996). Qian and Choi (2023) refers to it as a mapping of structures and Fackler and Capps (2024) as abstracting structures. Nevertheless, pattern recognition does not necessarily lead to the creation of new knowledge structures, making it more of a problem-solving activity (Muller & Haberman, 2008).

Dixon and Bangert (2004) made a similar distinction between automatic schema abstraction and comparison-based abstraction. The first uses memories to relate objects, while the second requires an explicit object-to-object comparison. Likewise, the building with nested epistemic action (Hershkowitz et al., 2001) allows the thinker to use available structural knowledge to build a solution without creating any new structure. Furthermore, in the structure proposed by Harel and Tall (1991), the first step, expansive generalization, leads to generalization (or generic abstraction) and implies expanding the application range without the need for reconstruction. Simon (2019) reports an example, where some children are able to recognize the same arithmetic operation in two different word problems and successfully transfer the solution across the two contexts.

Other terminologies have also been used. In the work by Sevian et al. (2015), pattern recognition is addressed as rule-based reasoning, while Duval (2006) used conversion for the process of connecting representations through superficial recognition (as cited in Mirolo et al. 2021). In the work of Ahmadpour et al. (2019), we found an analogous process. The path of form allows students to create “formulaic proofs” by “mimicry,” i.e., recognizing other instances that have a similar surface form. According to the authors, the path of form does not require understanding of the arguments, implying a lack of knowledge building. Ellis et al. (2017) recognized inter-contextual forms, in which students established relations of similarity across problems or contexts. In addition, Cifarelli (1988)’s idea of recognition fits this definition, despite them originally describing it as a form of RA (see discussion, as cited in Goodson-Espy 1998).

Abstraction strategy 4: interiorization

The first strategy in the reflective abstraction repertoire is interiorization, i.e., the gradual acquaintance that comes from applying and coordinating actions (Sfard, 1991; Goodson-Espy, 1998; Dubinsky, 1991). Through interiorization, the thinker becomes conscious of the action, reflects on it, and is able to combine it with other actions, ultimately transforming it into a process (Cetin & Dubinsky, 2017). Processes are cognitively more flexible, meaning the actions can be seen for their potential rather than as a sequence, and deviations from the step-by-step order become possible. In the process of interiorization, actions are detached from their original context. Processes can be recalled without the need of the perceptual material that originally created them (Steffe et al. 1988, as cited in Antonides & Battista 2022).

For example, children are traditionally introduced to arithmetic by counting and manipulating objects (e.g., dots, fingers). Repetition of the “combining and counting” set of actions soon leads to an unembodied and symbolic interiorization of the process of addition (Gray & Tall, 2001, 2007).

Sfard (1991) differentiated condensation as an abstraction step between interiorization and encapsulation (reification); here, we collapse both steps into one. Thus, we can assign Cifarelli (1988)’s re-representation and structural abstraction levels of abstraction (Goodson-Espy, 1998), and the internalized and interiorized one levels in the structure proposed by Battista (2007) and Antonides and Battista (2022) to hold equivalent meaning.

Abstraction strategy 5: encapsulation

A higher-order strategy is the process of encapsulation. Through encapsulation, actions and processes get transformed into objects (Ahmadpour et al., 2019; Scheiner, 2016). It is a gradual and delicate shift from seeing a procedure and its details to perceiving it as a concept of its own, independent from content and detail. The new object can then be manipulated, e.g., by other actions, and can be combined with other objects to form higher and more complex ideas (Ferrari, 2003). Alexandron et al. (2014, p.311)’s increase in level of meaning refers to a change in “seeing the how for seeing the what.” Similarly, Ginat (2021) documented how students see algorithms at a higher level of abstraction as an object rather than a set of instructions. Returning to the previous example, encapsulation of the process of “addition” allows students to think of “the sum” as an object independent of the underlying process (Tall, 2004). The new object can then be further manipulated, for example, to arrive at “multiplication” as the sum of sums.

The term encapsulation originated in the work of Dubinsky (1991) but has also been referred to as reification (Sfard, 1991), entification principle (Greeno 1983, as cited in Harel & Tall 1991), formal abstraction (Harel & Tall, 1991), and structural awareness (Cifarelli, 1988; Goodson-Espy, 1998). Similarly, Gray and Tall (2007) used the term compression to describe the process by which the essential in a phenomenon is conceived as an independent entity to think with. The resulting object is also referred to as a procept (Gray et al., 1999; Scheiner, 2016). Of similar nature is the idea of cognitive abstraction, whereby agents are able to “think through” possible scenarios and eliminate implausible ones without having to carry them out (Chen & Lee, 1992).

Abstraction strategy 6: coordination

Knowledge structures of the highest level stem from the process of coordination. Here, existing abstractions and prior knowledge objects and their relations are combined and coordinated to form more complex and powerful knowledge structures (Ohlsson & Lehtinen, 1997; Dubinsky, 1991; Dubinsky & Lewin, 1986). It is a dynamic and recursive process, where connections are made within and across different layers of abstraction. The emphasis lies in connecting different knowledge elements, rather than replacing them, leading to new knowledge insights (Hampton, 2003; Monaghan & Ozmantar, 2006). Kolb (2015) also used the term abstract conceptualization to refer to this process (as cited in Coleman et al. 2021). Other equivalent terms used are restructure (Karch & Sevian, 2020), assembly (Ohlsson & Lehtinen, 1997), and expand (Zehetmeier et al., 2019).

Knowledge gets structured into hierarchically organized schemas. Schemas compress all the actions and processes associated with an object, making them available across different contexts (Ferrari, 2003; Dubinsky, 1991). Ohlsson and Lehtinen (1997) provides an example from biology. To understand the theory of evolution through natural selection, the concepts of “repetition” and “selection” need to be deeply interconnected into an idea of “selection-carried-out-repeatedly.”

Some mathematicians differentiate an additional type of reflective abstraction, axiomatic-formal (Tall, 2004). Through formalization, knowledge structures are mentally represented using formal definitions and become accessible as axioms to specify mathematical structures. In our model, we consider formalization as a special form of coordination (see discussion).

A synthesized model of the EA strategies

Empirical Abstraction is a gradual process of organizing empirically won information across multiple contexts. Many different elements of empirical abstraction have been suggested. Building on the reviewed literature, we propose a new synthesized model of EA (Fig. 3). Depending on the use of context, we recognize three different strategies of abstraction. Generalization moves from the concrete to the abstract, away from the original context, while reduction moves toward the concrete and more contextual. Both of them have sub-strategies. In addition, a third strategy, pattern recognition, describes structure matching across contexts, without necessarily a shift in abstraction. All three strategies are tightly intertwined, working in parallel to achieve empirical abstraction (Haberman et al., 2002; Qian & Choi, 2023). We use a Venn diagram to represent overlaps between the different strategies. Pattern recognition functions both as a stand-alone strategy and as an enabling strategy for other sub-strategies, (commonality and decomposition).

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Fig. 3

Synthesized model of the main strategies involved in empirical abstraction

An aggregated model for the RA strategies

Reflective abstraction is the cognitive process responsible for the compression and construction of higher-level knowledge (Gray & Tall, 2007). It builds a hierarchical structures with increasing levels of abstraction, where at the higher (more abstract) levels, knowledge is less bound to physical specificity and is increasingly independent of the underlying instances and representations that originally created it. Three main processes moderate the ascent in reflective abstraction: interiorization, encapsulation, and coordination (Dubinsky, 1991). Figure 4 shows an adaptation of the APOS model for reflective abstraction, characterizing the ascent from the concrete to the abstract as an embodiment of ideas.

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Fig. 4

Strategies responsible for reflective abstraction, adapted from Dubinsky (1991)

So far, we have presented abstraction as a continuum between the concrete and the abstract. Nevertheless, many scientists have attempted to discretize the process and define levels in the abstraction hierarchy. The next section reviews and synthesizes their work.

The levels of abstraction

Table 5 provides an overview of different hierarchies and levels found in the revised literature. The table is constructed around a newly synthesized structure that unifies the multiple terminologies and characteristics we reviewed. We assigned each level a compound name, where the first term describes the nature of the knowledge at that level and the second one hints at the type of internal representation used.

Table 5. Inventories for the levels of abstraction found in the reviewed literature

Article	I: Perceptual-concrete	II: Logical-diagrammatic	III: Functional-modeled	IV: Conceptual-symbolic	V: Conceptual-formal	VI: Objective
Ye and Salvendy (1996) as cited in Gero et al. (2021)	Physical	Logical	Functional	Conceptual		Objective
Rasmussen (1986)	Physical form	Physical function	Generalized function	Abstract function		System purpose
dos Santos and Mortimer (2019) as cited in Karch and Sevian (2021)	Description	Explanation	Generalization	Abstraction
Wang (2008)	Analogue objects	Diagrams	Natural languages	Professional notations	Mathematics
Widada et al. (2019)	Concrete objects	Semi-concrete objects	Theoretical models	Language in example domain	Math languages	Inference model*
Dienes (1960) as cited in Jao (2013)	Free play	Generalization	Representation	Symbolization	Formalization
Johnstone (1991) as cited in Karch and Sevian (2021)	Macroscopic phenomenol	Conceptual macroscopic	Conceptual submicroscopic	Symbolic
Battista (2007) as cited in Antonides and Battista (2022)	Perceptual	Internalized I	Interiorized II (symbolic)	Interiorized III (encapsulated)
Perrenet et al. (2005)		Execution	Program	Object		Problem
Lowell (1977)	Attribute identification	Attribute recognition	Object recognition	Class recognition	Class of classes 1 and 2
Srinivasan and Te’eni (1990)	Properties	Basic entities	Cluster	Composite object	Composite object chain

*Thematically incongruent with level description, hierarchically correct

Level I in the hierarchy, perception-concrete, is based on an understanding of the physical form, appearance, and attributes of objects (Srinivasan & Te’eni, 1990). Representations of knowledge at this level are fully embedded in their original context and are thus concrete. Only Perrenet et al. (2005)’s hierarchy of algorithms excluded a physical property layer, which is not unexpected given the functional nature of algorithms. Their classification starts at the second level.

Level II, the logical-diagrammatical, captures the meaning of objects and their physical parts in a logical sense [Ye and Salvendy (1996), as cited in Gero et al. (2021)]. It uses shapes, abstract entities, and their relations to build the first mental representations of the sensed phenomena (Wang, 2008). Between levels I and II, there is a clear switch from description to explanation [dos Santos and Mortimer (2019), as cited in Karch and Sevian (2021)]. This switch mirrors the transition from the empirical toward the reflective.

Level III, the functional-modeled, builds upon level II to capture functionality (Rasmussen, 1986). The goal lies in understanding the function in structures, states, and processes of an object and its functional components. Leaving the concrete behind, the first theoretical models arise at this level (Widada et al., 2019).

Level IV is reached when mental representations transition from natural language toward professional notations and symbols (Wang, 2008; Johnstone, 1991). The conceptual-symbolic level displays abstract understanding at the conceptual level [Ye and Salvendy (1996), as cited in Gero et al. (2021)], moving from specific objects toward schemas (Lowell, 1977; Srinivasan & Te’eni, 1990).

Levels II, III, and IV are common to all hierarchies. As reviewed in the chapter on coordination, some mathematicians have differentiated between the (simple) symbolic and the formal mathematical. Here, we find Level V, the conceptual-formal as an elevation of the conceptual-symbolic (Wang, 2008; Widada et al., 2019; Dienes, 1960). Furthermore, Wang (2005, 2008) proposed a representation of software engineering relations at this level, as the currently used models (Level III) are rendered insufficient. Despite this level being characteristic of only a handful of definitions, Level V also represents the recursive nature of the knowledge hierarchy. Objects are constantly coordinated into infinitely higher compounds, all represented by Level V (Battista, 2007; Lowell, 1977; Srinivasan & Te’eni, 1990). Following this logic, we assign the fifth and sixth Class of Classes levels in the hierarchy of Lowell (1977) to Level V.

A few authors have recognized a sixth level, the objective. Level VI sees objects for their ultimate goal or potential instead of their detailed composition, culminating the process of reflective abstraction (Ye & Salvendy, 1996; Rasmussen, 1986; Perrenet et al., 2005). The last level in the hierarchy of Widada et al. (2019), the Inference Model, does not quite fit this categorization but rather describes a level based on metalanguages and statements about statements. Since it is still a level higher than the conceptual-formal, we assign it to Level VI, but clearly mark the discrepancy.

Table 6 summarizes the newly synthesized levels of abstraction. Each level includes certain details and features specific to that level. The purpose of the abstraction (the point-of-view) dictates which level is the most appropriate for an abstraction; i.e., the highest level is not always the most appropriate (Kamii, 2002). Expert abstract thinkers are better at navigating among and switching between different levels in the abstraction hierarchy (Hartmann et al. 2006, as cited in Zehetmeier et al. 2020)

Table 6. Synthesized levels of abstraction

I. Perceptual-concrete	II. Logical-diagrammatic	III. Functional-modeled	IV. Conceptual-symbolic	V. Conceptual-formal	VI. Objective
Understanding of the physical form and appearance of an object	The meaning of an object and its physical parts in a logical sense. Use of icons	Understanding of functional structures, states, and processes of an object and its components	Abstract understanding at the conceptual level. Use of symbols to encapsulate meaning.	High-level abstraction of objects, attributes, and their relations and rules, which are generically true in a domain	Understanding of objectives and intentions of an object. Black box on implementation

In the next section, we relate the levels of abstraction back to the abstraction strategies and synthesize the results into a full model for abstraction in STEM education.

The model of abstract thinking in STEM

Figure 5 consolidates the reviewed literature on abstraction in STEM education. At its core, abstract thought is a long-term process of consolidation, by which constructs become more easily available (familiar) to the thinker. It is a cognitive process by which information is compressed and enriched to form and store new ideas. Through abstract thought, individuals are able to organize and represent information across multiple contexts and levels, significantly aiding the communication and problem solving process. The repetitive use of abstraction is responsible for the acquisition of knowledge.

The spiral arrow in Fig. 5 represents this one-directional consolidation as an interplay between EA and RA. While abstract thought allows free navigation between the abstract and the concrete on an empirical level, reflective abstraction flows only one-directionally, constantly aggregating new information into the existing knowledge structures.

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Fig. 5

Synthesized model of abstraction in STEM. The model combines the six strategies and levels of abstraction and the dialectic relation between EA and RA that leads to a consolidation of knowledge

Every abstraction is made from a specific point of view. This includes the abstraction’s guiding purpose and the personal context of the individual building the abstraction. Empirical abstraction employs three main mechanisms to shift across contexts: reduction increases the external contextuality of objects, pattern recognition identifies underlying structures and matches them across contexts, and generalization identifies the essence of empirical information. In parallel, reflective abstraction interiorizes, encapsulates, and coordinates abstract ideas into a hierarchical knowledge structure. In the process, ideas become increasingly detached from the physical world and more intertwined within the existing knowledge hierarchy. EA and RA are strongly dependent on each other; knowledge advances in a spiral manner, touching many contexts at various levels. In the continuum between the concrete and the abstract, six levels of abstraction, each characterized by a specific level of detail and purpose, outline the ascent across the knowledge hierarchy. The transition from Level I to II marks a change from pure empirical to dialectic abstraction.

Abstraction has been studied explicitly mostly in the contexts of mathematics and computer science education. Other disciplines, however, have explored similar questions from different perspectives.

Abstraction across STEM

Abstraction plays an important role in many fields, including art and music (Kramer, 2007; Bilbao et al., 2021). In STEM, however, abstract thought is a foundational competency for the acquisition and advancement of knowledge and for problem-solving (Krupczak & Bassett, 2013; McMaster et al., 2010). Different STEM fields have different traditions surrounding abstraction (Colburn & Shute, 2007). Furthermore, a long tradition in STEM has explored abstraction through the lens of modeling, representation, problem solving and computational thinking. Here, we briefly review abstraction across these lenses and each of the STEM fields.

Science

Abstraction plays a prominent role in the generation of scientific concepts (Chambers, 1991). Scientists use abstraction to understand the empirical world and build theories that describe it. Hence, there is a general move from the concrete toward the abstract (Krupczak & Bassett, 2013). Modeling is key for the validation and communication of science, and representational competence is a key element of scientific education (Colburn & Shute, 2007). Darwish (2014) found a positive correlation between high abstract thinking levels and good scientific achievement. Looking more into detail, the physical sciences stand out for the deep hierarchy of their knowledge structures (micro to macro) and the highly abstract nature of their theories (Blackie, 2014; Karch & Sevian, 2021). There is often a clear jump from the concrete to the abstract, as most concepts have a direct link to the physical world (Verhoeff, 2011). The life sciences, in comparison, are much more concrete. A long tradition in Science education has explored abstract thought through the lens of modeling, representation (Fyfe et al., 2014; Karch & Sevian, 2021, 2020; Fackler & Capps, 2024; Capps & Shemwell, 2020; Blackie, 2014) and problem solving (Sevian et al., 2015).

Modeling and representation

All models and representations are (by definition) purposeful abstractions of reality and are thus built using abstract thought (Capps & Shemwell, 2020; Nicholson et al., 2009). Models consist of internal and external representations, their interaction limited by the model’s constraints. The agent uses the epistemic aim to define what is preserved and how it is modeled (van Oers & Poland, 2007; Knuuttila, 2011; Capps & Shemwell, 2020). Yagisawa and Iijima (2023) refereed to the identification of the scope of a model as a bird’s eye view.

A model will always have an internal representation but does not necessarily need an external one (Kamii, 2002). A model may also consist of multiple representations, and constructing multiple representations may aid the abstraction process (Jao, 2013). Fluidity to connect external and internal representations at different levels and correctly interpret the purpose of a model are signs of abstraction proficiency (Carrejo & Marshall, 2007; Hadish et al., 2023; Kozma, 2003; Gautam et al., 2020) and fundamental for learning (Bakker & Hoffmann, 2005; Carrejo & Marshall, 2007). For example, Guerrero-Ortiz et al. (2018) showed that auxiliary material can aid the externalization of ideas and the formation of new abstractions in the context fo mathematics.

Problem solving

Problem-Solving is deeply convoluted in the narrative surrounding abstraction and modeling (Ginat, 2021; Hong & Kim, 2016; Karch & Sevian, 2020). It is understood as "the practice of overcoming obstacles through information and reasoning" (Cardellini 2010, p.43, as cited in Sevian et al. 2015). In the context of abstract thought, problem-solving can be seen as a context-bound modeling practice (Karch & Sevian, 2021). Problem solvers use abstraction to switch between levels of abstraction, each appropriate for a specific stage of the process, ultimately reducing the complexity of the problem (Lyon & Magana, 2021). In most problem-solving processes, there is a general move from the concrete toward the abstract (analysis phase) and then back to the concrete [implementation phase; Sevian et al. (2015)].

Computer science

Computer science is inherently abstract due to its lack of physicality (Bennedssen & Caspersen, 2008; Colburn & Shute, 2007). Verhoeff (2011) argues that computer science is the most abstract of the STEM disciplines. Computer scientists constantly construct, manipulate, and reason with abstraction at multiple levels (Hazzan & Kramer 2007, Kramer & Hazzan 2006; for a review see Mirolo et al. 2021). Abstraction plays a key role in both implementation and theory building. Modeling and representations at multiple levels of abstraction lay the foundation of software engineering (Dordochi et al., 2021; Alexandron et al., 2014). Böttcher and Thurner (2023) found a positive correlation between abstract thinking skills and success in computer science-related study programs. Bennedssen and Caspersen (2008), on the other hand, did not find a correlation; however, they and many other educators extensively argue for the importance of abstract thinking in a successful CS education (Kramer & Hazzan, 2006).

Computational thinking

Since first formulated by Wing (2006), the literature on computational thinking (CT) has grown exponentially (Tekdal, 2021). At its core, CT refers to the thought process involved in formulating problems so that their solutions can be represented by computers (Wing, 2006). It includes a cluster of tools for analytical problem-solving, building upon a long tradition of analysis across all STEM fields. Abstraction stands out as the most important of the competencies included in CT (Wing, 2008; Cetin & Dubinsky, 2017; Qian & Choi, 2023; Bilbao et al., 2021). Hence, the literature between the two is tightly interwoven.

Engineering

Abstraction in engineering is primarily represented by its modeling and representational components (Kramer, 2007; Gero et al., 2021). Krupczak and Bassett (2013) made a distinction between science and engineering, noting that engineering often starts with abstract ideas, with the goal of progressing toward concrete physical products. Hence, both fields employ abstraction but in opposite directions. In addition, Hadish et al. (2023) observed how learning at various levels of abstraction improves engineering education, and Neuper (2017) suggests engineering education that guides students through the process of abstraction.

Mathematics

Mathematics is arguably the most abstract of all STEM disciplines, as it constantly develops higher and higher levels of abstraction from relationships among abstract objects (van Oers & Poland, 2007). Mathematicians model highly abstract internal representations, mediated by deeply condensed symbolic representations (Bakker & Hoffmann, 2005). Darwish (2014) observed a positive correlation between abstract thinking skills and mathematical performance, and Ferrari (2003) described a lack of abstraction skills as the main reason for failure in mathematics learning. Furthermore, Rich and Yadav (2020) propose a framework to support mathematics by contextualizing the abstraction process of mathematics word problems. In comparison with the other fields, abstraction in mathematics is increasingly studied at a school and pre-school level. It is here that students are first confronted with a formal abstraction instruction.

Abstract thought is a foundational competency for the acquisition and advancement of knowledge and problem solving in STEM (Krupczak & Bassett, 2013; McMaster et al., 2010). Scientists across all fields have explored the inner workings of this complex cognitive ability. This review integrates the most influential ideas across different fields and unifies them into a framework to understand abstraction across STEM education (Fig. 5). In the next section, we summarize the proposed framework and discuss how existing work supports and deviates from it.

Discussion

Abstraction is a complex and rich cognitive process, responsible for the consolidation of knowledge and the creation of new ideas (Gray & Tall, 2007; Tabach et al., 2017). It is one of the fundamental reasoning processes of natural intelligence (Wang, 2005). Knowledge arises through concatenations of empirical and reflective abstractions, both of which are important and inseparable parts of the learning process. Through abstraction, information is compressed and constructs become progressively more available to the learner. This progression can be discretized in six levels of abstraction. Mastery of abstraction goes well beyond building the most abstract idea; it involves understanding the correct motive, context, and level required for each task (Nicholson et al., 2009).

Despite the heterogeneity in the abstraction literature across STEM, we managed to relate existing research in different fields to one another and synthesized the results into a model of abstraction for STEM education (RQ). Before putting everything into a common framework of abstraction, we first defined and synthesized the different elements. Here we discuss the integration decisions that went into each step.

A new categorization of EA

Empirical abstraction enables individuals to identify the underlying structures in their sensory experiences (Fig. 3). Focus attention and identifying commonalities are generalizing, knowledge-building cognitive processes that remove objects from their original context, resulting in more empirically abstract objects. Focus attention does this by removing details while identifying commonality constructs new sets. Correspondingly, concretization aids the thinker in adding details, and decomposition is responsible for deconstructing concepts. Both are reductive activities, often employed in problem-solving, that lead to more empirically concrete structures. In addition, pattern recognition aids empirical abstraction by matching structures across contexts without building new knowledge structures. It functions both as a stand-alone strategy and an enabler for the commonality and decomposition strategies.

In the reviewed literature, we found high variance in definitions offered for empirical abstraction. Most authors agree on a general notion of generalization and simplification, but different relations and hierarchies between these and other adjacent concepts (e.g., essence, decomposition, and reduction) are offered. While each of the reviewed frameworks has its validity, we believe our categorization has two main strengths.

First, our model makes a fundamental distinction between generalization and reduction as opposing yet complementary strategies that enable bidirectional navigation. The model goes beyond a simple abstract/concrete classification and highlights a context-dependent, complex spectrum of empirical abstraction. Second, our model proposes pattern recognition as a stand-alone practice able to produce abstractions by itself while at the same time representing the cognitive similarity between commonality, decomposition, and pattern recognition, all of which rely on recognizing the underlying structure of information for success (Burton, 1982).

An enriched APOS model for RA

Reflective abstraction is responsible for the compression and construction of higher-level knowledge (Fig. 4). Actions are interiorized into processes, which in turn can be encapsulated into objects. Through coordination, a dynamic and complex knowledge hierarchy is constructed and maintained. As reflective abstraction moves from the concrete toward the abstract, objects become less bound to the physical context, where they originated and increasingly embodied.

While our representation of reflective abstraction is mostly based on the ’APOS model (Dubinsky, 1991), other important pieces of work went into the synthesis of our model. First, an important group of mathematicians described a reflective abstraction strategy specifically tailored for gaining mathematical knowledge (Gray & Tall, 2007; Tall, 2004). Tall (2004) spoke of a “reverse” meaning construction, from known definitions to formal concepts based on those definitions. We acknowledge this as a recursive form of coordination and assign a dedicated level (V. Conceptual-Formal) on the abstraction hierarchy to represent formal mathematical knowledge.

Second, our model presents the ascent from the concrete to abstract in reflective abstraction as unidirectional. In the surveyed work, only Gray et al. (1999) explore a reverse conversion, where an object is turned back into a process (de-encapsulation). If we understand RA as exclusively aggregating to the knowledge structure (not eliminating), the process remains available in the knowledge hierarchy even after encapsulation, making a reverse conversion obsolete. That is not to say that using the process instead of the object cannot be a useful problem-solving strategy (Hazzan, 1999, 2003; Raychaudhuri, 2014).

Finally, we use embodiment as the measure of reflective abstraction; i.e., the more reflectively abstract a concept is, the more embodied it becomes (Breive, 2022). This integrates the idea of establishing a point of view, where abstraction goes beyond knowing and becomes a process of becoming (van Oers, 2001). Therefore, while EA allows bidirectional navigation in context, RA coordinates those concepts into existing knowledge structures, making them more dependent and thus more embodied. In the process, knowledge becomes more familiar (Wilensky, 1991).

Abstraction is an interplay of EA and RA

An important tradition in the study of abstraction recognizes a third, previously unmentioned type of abstraction: pseudo-empirical abstraction (PEA; Piaget, 2001). PEA is commonly characterized as a combination of EA and RA, something “in the middle” (Scheiner, 2016). We believe this intermediate category to be trivial. While certain cognitive tasks might be dominated by one or the other, most abstractions are dependent on the dynamic interplay between the two (Ozmantar & Monaghan, 2007), making PEA synonymous with abstraction; i.e., all abstractions are pseudo-empirical.

This is constantly reflected in the reviewed literature. We offer Table 4 as an overview of different terminologies and frameworks we encountered. The table is designed to aid the comparability across literature, without invalidating the individual nuances intended by the respective authors. Beyond the clear dichotomies, authors have approached the interplay between EA and RA from multiple perspectives. For example, Ohlsson and Lehtinen (1997) pointed out the circular dependency between set creation and assignment, Hershkowitz et al. (2001) cleverly designed the nested epistemic actions to circumvent the duality, and Cable (2014) challenged the object-concept distinction, highlighting the dynamic advantages of a dialectic approach. Furthermore, we found a couple of instances, where authors originally categorized their concepts as EA and RA; however, the definitions they provided point to the opposite categorization in our model (Karch & Sevian, 2021; Simon, 2019). We see these examples not as contradictions but as evidence of the complex and dynamic interplay between EA and RA.

Our model aligns with the embodied cognition framework of Lakoff and Núñez (2000), who argue that even advanced mathematical concepts are grounded in sensorimotor experience and structured through metaphor. This perspective supports our finding that empirical and reflective abstraction are not separate stages but dynamically intertwined. Rather than representing a detachment from experience, abstraction involves the reorganization and compression of embodied knowledge. By emphasizing context, point-of-view, and purpose, our model reinforces the idea that even the most abstract thinking in STEM remains rooted in human experience—a view with important implications for how we teach and assess abstraction.

Consolidation is an increase in familiarity

All definitions in the review are embedded in a specific frame of reference: Empirical abstraction allows for a bidirectional shift between the concrete and the abstract, where (by definition) the abstract is less context-bound. Likewise, reflective abstraction is defined by an ascent from the concrete toward the abstract, where abstract refers to more detached concepts. Hence, by definition, complex high-order ideas are more abstract and less concrete.

Wilensky (1991) proposed an alternative perspective, switching this reference frame around (Hazzan, 2003). Abstraction is not an inherent property of an object but a description of an individual’s relation to that object. As an individual builds an abstraction, they gradually become more familiar with the concept. The object that, upon first encounter, appeared unfamiliar and abstract becomes more accessible and thus, concrete. Hence, from this perspective, abstraction is a process of familiarization that moves from the abstract toward the concrete. This opposing perspective helps explain the intricate relation between abstraction and complexity (Blackie, 2014). When an individual shares an abstract idea with others, e.g., a teacher explains the concept of evolution in class, the students will rightfully perceive the concept as abstract. First, the concept is objectively abstract (the teacher built it using a long process of abstraction). Second, it is bound to the personal context of the teacher, i.e., to the preconceptions and available knowledge it was built around. The student recognizes the objective abstractness (lack of physicality) of the concept and the complexity of integrating the new concept into their own knowledge structures. Hence, the idea appears unfamiliar, complex, and abstract.

Ideally, the teacher will ease the consolidation process and provide the students with the necessary instruction to build the abstract idea from their individual points of view (Monaghan & Ozmantar, 2006; Dubinsky & Lewin, 1986). Students use abstraction to make new connections to their previously available knowledge, resulting in the concept becoming more familiar and thus more concrete. White and Mitchelmore (2010) proposed the ABC (abstract before concrete) method to teach abstraction, following a similar logic. Starting with the abstract, familiarity is followed by similarity and reification, culminating in a concrete application.

Despite representing opposing angles, the two frames of reference are complementary. The first describes the contextuality of an idea (more detached ideas are more abstract). The second details how familiar an individual is with an object (more familiar ideas are more concrete). Clements (2000) makes the distinction between the sensory-concrete (first frame) and the integrated-concrete (second frame). Higher-order abstract concepts are thus both abstract (more detached) and concrete (more familiar) at the same time.

Our model treats concrete and abstract as traits of objects (first frame), while acknowledging that knowledge consolidation is equivalent to an increase in familiarity with the object and a reduction in perceived complexity.

Limitations and future directions

While this integrative review offers a comprehensive framework for understanding abstraction in STEM education, several limitations constrain the scope and generalizability of our findings. First, the distribution of reviewed literature is uneven across disciplines, with mathematics and computer science dominating the data set. This reflects broader research trends, where abstraction has been more explicitly theorized and studied in these fields. As a result, our synthesized model may over-represent abstraction processes and terminology specific to these disciplines, potentially limiting its direct applicability in more experimentally grounded fields, such as biology, chemistry, or engineering. Future work should aim to validate and extend this model using literature and empirical data from under-represented STEM domains and directly relate it to the more prevalent discussion on modeling, representation and problem solving in science and engineering.

Second, although the integrative literature review methodology allowed for the inclusion of both empirical and theoretical sources, the thematic coding and model development process is inherently interpretive. While steps were taken to ensure reliability, including inter-rater agreement, and iterative refinement, the clustering of insights into abstraction strategies and levels involves subjective judgment. Moreover, the assignment of overlapping or competing terms to unified constructs (e.g., generalization vs. abstraction of commonalities) inevitably simplifies nuances found in the original works.

Third, integrative work across a field as broad as STEM inevitably involves certain limitations. To keep the scope manageable, we followed Haddaway et al. (2015) and limited search results in Scopus and PsycINFO to the top 300 entries ranked by relevance. While Scopus’s relevance algorithm is well-documented (Rovira et al., 2019; Scopus, 2024), PsycINFO’s ranking criteria are less transparent. This reliance on proprietary relevance algorithms limits transparency in our protocol and may introduce bias into the selection of literature.

Finally, while our model incorporates conceptual alignment with reflective frameworks such as APOS (Dubinsky, 1991) and theoretical perspectives such as those of Lakoff and Núñez (2000), it has not yet been empirically tested. Future studies could empirically assess the efficacy of this model in supporting instruction, curriculum design, or learning assessment, particularly by tracing how students move between abstraction levels or apply specific strategies in solving STEM problems.

Conclusion

Despite the many different perspectives and terminologies that STEM fields have used to describe the process of abstraction, it is possible to arrive at a common understanding. Abstraction is a key component across all fields. It is a process of consolidation by which ideas become more easily available to the thinker. Abstraction starts with an intention and a point of view from which it is built. Empirical and reflective abstraction are two sides of the same coin; both cognitive processes work in parallel to manipulate the level of abstraction to achieve the desired outcome. Six abstraction strategies are distinguishable: reduction, pattern recognition, generalization, interiorization, encapsulation, and coordination. Within the concrete-abstract continuum, six discrete levels of abstraction can be identified. Abstraction is deeply influenced by the personal context of the thinker. It enables the use of models (epistemic tools used to externalize, organize, and make sense of knowledge), and all models are abstractions by definition. Problem-solving is a context-bound modeling practice, and computational thinking is a computer-specific problem-solving skill set. At the core of both lies abstraction.

Implications for research and educational practice

Our model offers a new perspective to measure and teach abstract thought as an interdisciplinary competency. We bridge the gap between different STEM disciplines, showing that abstract thought is fundamental to all of them and universally defined. We believe our synthesis of the EA strategies (Fig. 3) to be of great value for the ongoing abstraction debate in computer science. Furthermore, our synthesis of the levels of abstraction (Table 6) has the potential to support the teach modeling and representation. We hope this review will spark increasing dialogue across fields, as scientists and educators combine experiences and efforts to foster abstract thinking abilities in their students. The work of Cetin and Dubinsky (2017) is a clear example of how combining theories of abstraction across mathematics and CT benefits both.

As we showed in the final part of our analysis, abstraction has deep implications for modeling, problem-solving, and CT. By taking the abstraction lens, we might be able to adjust how we teach and practice these activities. Beyond this, we hope to spark a conversation in the CT community on what this “twenty-first century” competency entails and how it differs from the “traditional” abstraction fostered in traditional STEM education.

With this review, we have set up the theoretical foundation to measure abstraction. Our model clearly defines the underlying strategies behind abstraction. In follow-up work, we intend to use this dissection of abstraction to develop a psychometric instrument to measure the abstract thinking abilities of students across different STEM fields.

Acknowledgements

We are grateful for the constructive feedback provided by the editor and reviewers.

Author contributions

Jose Vega (JV; first author) and Daniel Pittich (DP; last author) designed the review. JV reviewed the full literature and extracted the results. Anna Trikoili (AT; second author) supported the literature selection by reviewing a subset of the records. JV synthesized the results and was a major contributor in writing the manuscript. All figures are attributed to JV. All authors read and approved the final manuscript.

Funding

Open Access funding enabled and organized by Projekt DEAL. Not applicable.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare no competing interests.

Abbreviations

CT

Computational thinking

EA

Empirical abstraction

ILR

Integrative literature review

MRS

Mental Representation of the Situation

PEA

Pseudo-empirical abstraction

PRISMA

Preferred Reporting Items for Systematic Reviews and Meta-Analyses

RA

Reflective abstraction

RQ

Research question

STEM

Science, Technology, Engineering, and Mathematics

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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