## Abstract

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way.

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## Notes

- 1.
Both authors have made substantial and direct intellectual contributions to this work in equal terms.

- 2.
For an insightful overview of the relation of Marr’s levels of analysis and relevant research questions in cognitive science, see Varma (2014).

- 3.
As an anonymous reviewer pointed out, this analytic strategy is particularly salient in Bayesian models of cognitive phenomena that have been influenced by Anderson’s (1990) work. On these accounts, the computational level is of particular importance for analysing cognitive processes in terms of Bayesian conditionalisation (Eberhardt and Danks 2011; Griffiths et al. 2010). A notable exception is recent work on predictive processing, which depicts perception, action, and cognition as cases of Bayesian conditionalisation in a hierarchical generative model that is implemented in the brain (Clark 2016; Hohwy 2013). Predictive processing is committed to the view that explanations need to make specific assumptions about the algorithmic and implementational details of probabilistic prediction error minimisation (for an example, see Friston 2005).

- 4.
The most common way of doing this is to frame the problems as

*decision problems*, so that the output for each input is either yes or no (Kozen 2012). - 5.
For particular cases of general problems (such as a particular integer multiplication instead of the general problem of integer multiplication), complexity is often characterized in terms of the number of computational steps it takes to reach the solution.

- 6.
It needs to be noted that there is not one unique optimal algorithm for solving a particular problem. In theoretical computer science, an algorithm is called asymptotically optimal if it never performs more than a constant factor worse than the best possible algorithm. Hence there can be several, even an infinite number of optimal algorithms for solving a problem. Furthermore, Blum’s (1967) speedup theorem shows that it is not always possible to define the complexity of an arbitrary problem in terms of an optimal algorithm for solving it. Namely, the theorem says that there are computable functions for which any algorithm computing that function can be sped up so that it demands less computational resources. Although this possibility is important to remember, it does not imply that optimal algorithms cannot be discussed coherently in most cases.

- 7.
It is interesting to note that Anderson (1990) is in disagreement with Marr (1982) about the relationship of competence and performance on the one hand and the computational, algorithmic, and implementational levels of analysis on the other hand. In particular, he argues that ”[u]nlike Marr’s case, performance is not just a matter of implementing the goals of competence. Indeed, unlike Marr’s computational-level, Chomsky’s competence is not concerned with the goals of the system” (Anderson, 1990, p. 9). Nevertheless, Anderson continues to concede that ”Chomsky used the competence level to serve the same role in theory building as Marr used computational theory. Under both analyses, the scientist should first work out the higher level. Both felt that this was a key to making progress. Also, the lower levels are constrained somehow to reflect the higher levels” (op. cit.). In what follows, we are committed to this correspondence view of Marr’s and Chomsky’s contributions to the idea that theory formation—with the help of computational models—should proceed in a top-down fashion.

- 8.
- 9.
Hayes’s (2006) detailed survey indicates that three texts report that Gauss employed a different strategy. However, the reported details of the problem differ from the ”folding” and ”two rows” solutions, because they mention a different interval of integers (i.e., 81,297, 81,495,… 100,899). All three accounts state that Gauss solved the problem by using the following formula: n\( \frac{{\left( {n + 1} \right)}}{2} = x \). For the ease of exposition, we focus on the “folding” and “two rows” solutions here.

- 10.
- 11.
- 12.
Peirce (1960–1966) is perhaps the most influential proponent of a theoretical account of the representational properties of diagrams and of diagrammatic reasoning more generally. For assessments of the Peircean account of diagrams, see Carter (2017), Giardino (2017), Stjernfelt and Østergaard (2016), Tylén et al. (2014).

- 13.
In any case, computational steps for operations like multiplication can be defined explicitly in terms of state transitions of Turing machines.

- 14.
“Schoolbook algorithm” refers to the pen and paper method of multiplication. For mental arithmetic, different algorithms are often used. To give just one example, a multiplication like 25 * 16, can be calculated quickly and reliably by breaking it up as 25 * (4 * 4) = 100 * 4 = 400. While the schoolbook algorithm is used in the same way for all numbers, in mental arithmetic such “shortcuts” can be used when the numbers are appropriate (like in this case the easy multiplication of 25 * 4 making the quicker solution possible). See, e.g., Baroody (1984) for details.

- 15.
Another potential problem with the P-cognition thesis is that the asymptotic complexity measures may be misleading for the kind of finite inputs we are interested in when modelling human cognitive tasks. Thus, a better solution could be to consider complexity with suitable parameters concerning the size of the input (Van Rooij 2008).

- 16.
The 100 addition steps solution is the case when each addition of multi-digit numbers is considered as one addition. In reality, for human subjects the addition of multi-digit numbers is processed as several additions of single-digit numbers, with possible carrying increasing the amount of steps (Nuerk et al. 2015). The exact number of addition steps depends (among other factors) on the number base.

- 17.
Menary (2015) builds his assumptions about learning-driven plasticity on Dehaene’s (2010) neuronal recycling hypothesis. According to this hypothesis, the ontogenetic acquisition of cognitive practices such as arithmetic and reading is rendered possible by the recycling of already existing cerebral regions. The implication is that recycled regions lose their original function and cease to contribute to formerly established neural circuitry. We will see later on in this paper that the neuronal recycling hypothesis is at odds with the empirical evidence of the realisation of arithmetical cognition throughout ontogenetic cognitive development. In the domain of reading, the neuronal recycling hypothesis has led to the claim that the left ventral occipito-temporal region is recycled and contributes to the neural circuit realising visual word recognition (Dehaene, 2010; Dehaene et al. 2005; McCandliss et al. 2003). Importantly, Dehaene and his collaborators have argued that this region ceases to contribute to the realization of processes other than reading, for example to the recognition of faces, objects, and visuo-spatial patterns. However, there is ample evidence suggesting that this is not the case and that the ventral occipito-temporal area continues to be involved in the neuronal realisation of these other processes (Price and Devlin 2003, 2004; Vogel et al. 2012, 2014). In other words, the ventral occipito-temporal area appears to be

*reused*,*not recycled*. For this reason, we prefer to focus on the neural reuse account. For details, see Fabry (2019). - 18.
It should be noted that many languages are more confusing than English in this sense. In German, for example, the order of tens and ones is always reversed (sixty-four, for example, is

*vierundsechzig*, literally “four-and-sixty”. In French, ninety-eight is*quatre-vingt-dix-huit*, literally “four-twenty-ten-eight”. - 19.
This suggests the functional changes to the bilateral IPS is a matter of neural reuse, and not neuronal recycling, since these brain regions continue to contribute to neuronal processes associated with number approximation, rather than being exclusively associated with arithmetical processes. This lends additional support for our view that neural reuse is preferable to neuronal recycling as a mechanism that underlies LDP (see footnote 17).

- 20.
As expected, given the large cultural differences in educational systems and mathematical learning strategies, there are culture-specific differences in the neuronal profiles of numerical processing, even when the same number symbols are used. Tang et al. (2006), for example, report an fMRI study according to which native Chinese speakers show more tendency to engage in visuo-premotor association of numbers than native English speakers as evidenced by activation levels in the premotor association area and the supplemental motor area. By contrast, mental calculation processes in native English speakers are associated with increased activation levels in the left perisylvian cortex, which plays an important role in the neuronal realization of language processing. Tang and colleagues suggest the wider use of the abacus in China as one reason for this, alongside factors such as shorter number words in Chinese.

- 21.
Amalric and Dehaene’s (2016, 2018) studies are in part motivated by the question whether or not mathematical cognitive practices are derived from phylogenetically older linguistic capacities that capitalize on the recursivity of linguistic structures. They attribute the assumption that mathematics is a derivative of language to Chomsky (2006): “According to Noam Chomsky, ‘the origin of the mathematical capacity [lies in] an abstraction from linguistic operations’” (Amalric and Dehaene 2018, p. 1; see also Amalric amd Dehaene 2016, p. 4909). This clearly misconstrues Chomsky’s (2006) position. The entire sentence, which is only partly quoted, reads as follows: “Speculations about the origin of the mathematical capacity as an abstraction from linguistic operations are not unfamiliar” (pp. 184-185). Irrespective of this misconstrual of Chomsky’s (2006) position, the question about the dependency of mathematical competence on linguistic competence has sparked interest in cognitive neuroscience recently. Contrasting the evaluation of mathematically meaningful and of mathematically meaningless linguistic statements (Monti et al. 2012) and of mathematically meaningful and random symbol strings (Maruyama et al. 2012), the main finding of two independent fMRI studies is that the neuronal activation patterns associated with the evaluation of mathematically meaningful symbolic representations is markedly different from the neuronal activation patterns associated with the evaluation of mathematically meaningless symbolic representations. In both studies, the evaluation of mathematically meaningful symbolic representations is associated with activations in brain regions that have been found to play an important role in mathematical cognition (e.g., the right IPS and other parietal regions). By contrast, the evaluation of mathematically meaningless linguistic statements in the study by Monti et al. (2012) is associated with activations in brain regions that are typically associated with language processing (e.g., left inferior frontal gyrus, left middle and superior temporal gyri). This indicates that the neuronal realization of competence in mathematical problem solving and in written language processing are clearly distinct.

- 22.
The most robust effect detected by these studies is the

*spatial-numerical association of response codes*(SNARC) effect (Dehaene 2011; Everett 2017; Hubbard et al. 2005; Shaki et al. 2009; Tschentscher et al. 2012). When asked to indicate which numeral of a pair is greater by pressing the assigned key on a keyboard, cognitive agents who have been enculturated in a socio-cultural environment in which left-to-right is the prevalent direction of writing is the norm, both for linguistic and mathematical symbols, they are significantly faster in identifying the larger numeral if the appropriate key is on their right hand-side. This effect is reversed in cognitive agents who are enculturated in an environment in which right-to-left writing is the norm (Shaki et al. 2009). Intriguingly, Shaki et al. (2009) also found that Israelis are not subject to the SNARC effect, because in this cultural community “words are read from right to left and number from left to right” (p. 330). Overall, the SNARC effect has been interpreted as evidence for the idea that the relationship between numerical and spatial cognition is often characterized by “cognitive intertwinement” (Everett 2017, p. 207). - 23.
Duncker’s (1945) radiation problem is represented by a diagram that shows a tumor which is enclosed by healthy tissue. The healthy tissue is surrounded by the skin, which in turn is surrounded by the outside local environment. Grant and Spivey’s (2003) instruction reads as follows: “Given a human being with an inoperable stomach tumor, and lasers which destroy organic tissue at sufficient intensity, how can one cure the person with these lasers and, at the same time, avoid harming the healthy tissue that surrounds the tumor?” (p. 462).

- 24.
Here "standard algorithm" and "alternative algorithm" can be understood either as being descriptive or prescriptive. In the descriptive case, "standard algorithm" refers to the algorithm most commonly used by the members of a certain socio-cultural community. In the prescriptive case, “standard algorithm” refers to the algorithm that is favoured by the cognitive norms governing mathematical problem solving in a certain socio-cultural community.

- 25.
In fact, there is evidence that enculturation plays a crucial role in the development of mathematical cognition already in the acquisition of number concepts. The Amazonian tribe of Pirahã do not have a stable numeral system in their language (Gordon 2004). Frank et al. (2008) have argued that this prevents them from representing exact quantities, which renders learning even basic arithmetic almost impossible for mono-lingual members of the Pirahã (see also Everett 2017).

- 26.
In this way, the present account can be seen as compatible with, e.g., Simon (1968/1996), according to whom behaviour is optimized to “reflect characteristics largely of the outer environment” (p. 53). In our account, mathematical problem solving strategies should be assessed in terms of limitations due to our cognitive architecture, but they are also determined by cultural factors. Therefore, it would be mistaken to state that the problem solving strategies are optimized to reflect the characteristics of the outer world in a strong realist sense. But if “outer environment” is understood in a culturally sensitive fashion, the optimization of problem solving strategies can be characterized in terms of reflecting the outer environment. We thank an anonymous reviewer for pointing out this connection.

- 27.
This is in line with Krämer’s (2016) following assumption: “Embedded in normatively shaped practices, which do not need to be explicit, but are often implicitly anchored in cultural habits, diagrams organize shared epistemic experiences” (p. 80; italics removed; our translation).

- 28.
It helps sharpen our theoretical understanding of diagrammatic problem solving to briefly compare it to Giardino’s (2016, 2017) account. Our and Giardino’s account differ in two important respects. First, Giardino (2017) highlights the

*heterogeneity*of reasoning and continues to argue that “humans happen to rely on many different sorts of instruments with the aim of*externalizing thought*, diagrams being among them” (p. 500; emphasis added; see also Giardino 2010, 2014, 2016). The important difference between Giardino’s account and ours is that we do not subscribe to the view that the bodily manipulation of diagrams*externalises thought*. Rather, our assumption is that the bodily manipulation of diagrams is deeply integrated into mathematical problem solving and directly contributes to the completion of cognitive tasks. Second, we part company with Giardino’s (2016) suggestion that “[d]iagrams as well as other kinds of cognitive tools can serve as*convenient inferential shortcuts*if the space they display is correctly interpreted and permissible actions are performed on it” (p. 97; emphasis added). We suggest that the bodily manipulation of diagrams and other cognitive tools accomplishes more than providing “convenient inferential shortcuts”. Rather, they augment and transform mathematical competence and have a direct impact on the normatively constrained reasoning capacities of proficient and expert mathematicians. This is in full agreement with Krämer (2014b), who argues that “[k]nowledge is not only represented, transmitted and disseminated through the diagrammatic; it is*produced and expanded*by it” (p. 3; emphasis added). - 29.
The integer addition problem is solved by exploiting what Krämer (2014b) calls the

*operative iconicity*of public representations in mathematics and other domains. - 30.
An important task for future research on humanly optimal algorithms will be a reassessment of existing computational models of mathematical problem solving.

- 31.
Although our focus is on enculturated competence, one potential advantage of our account over computational-level explanations is that it might be better at predicting errors that human problem solvers may systematically commit. By studying enculturated competence through enculturated performance, the research can be sensitive to the ways in which performance may be flawed as a negative side-effect of enculturation. The harmonic series case discussed in Sect. 3 would be a good candidate for systematic errors in this context. As with enculturated mathematical problem solving processes in general, there is nothing to suggest that the systematic errors could not be computationally modelled. In fact, research on errors that are predicted by computational models was already conducted decades ago for procedural skills such as arithmetical problem solving (Brown and VanLehn 1980). Errors in carrying and borrowing in arithmetical problem solving by the schoolbook methods, for example, can be predicted by computational models. It is likely that such errors can be traced back to failures in following enculturated problem solving strategies and therefore the enculturation account can be helpful in predicting, and repairing erroneous problem solving methods. This is a highly interesting topic that should be tackled in future research. We thank an anonymous reviewer for making this point.

- 32.
It should be noted that this approach of studying competence through performance is nothing new. Clearly, the Chomskyan notion of competence requires that an account of competence is grounded in the empirical and theoretical study of performance (Fitch et al. 2005).

- 33.
This framework bears important similarities with Clark's (1990) description of

*rogue models*. Clark is interested in the ways in which connectionist explanations based on neural networks can provide the resources to model human’s competence in the completion of a certain cognitive task under ecologically valid conditions. In contrast to*Newtonian models*, in which “the connectionist network is*itself*capable, under idealised conditions, of behaving in all the ways specified by the competence theory”, rogue models take the important functional role of the manipulation of symbolic (or diagrammatic) structures into account (Clark, 1990, p. 208; italics in original). More specifically, “[i]n a rogue model […] the basic connectionist network does not itself have the capacity (even under idealizations of processing time and well-posed problems) to produce the full range of results required by (i.e., derivable) the competence theory” (op. cit.). The implication of our conceptual framework for computational modelling is entirely consistent with this idea of rogue models. In rogue models, the cognitive competence postulated by a certain theory is modelled by specifying the complementarity of the model itself and the symbolic (or diagrammatic) structures that serve as input and output of the connectionist network. Unlike computational models in the Marrian paradigm, rogue models encompass computational and algorithmic levels of analysis and are informed by implementational-level considerations, for example on distinct eye movement and neuronal activation patterns in proficient and expert mathematicians reviewed in Sect. 6. In fact, Clark’s (1990) consideration of rogue models is an important theoretical precursor of our proposal to explore enculturated cognitive optimality, and to study enculturated competence in mathematical problem solving by accumulating evidence about enculturated performance. - 34.
Our view is largely consistent with Cobb’s (1994) integration of constructivist assumptions on the one hand and assumptions about the ontogenetic cognitive development of mathematical practices that emphasise the importance of socio-culturally situated processes on the other hand. In particular, we share Cobb’s (1994) claim that “mathematical learning should be viewed as both a process of active individual construction and a process of enculturation into the mathematical practices of wider society” (p. 13). We are grateful to an anonymous reviewer for bringing this to our attention.

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## Acknowledgements

We would like to thank an anonymous reviewer for very helpful comments. M.P. would like to thank the Academy of Finland and professor Gabriel Sandu for making this reseach possible as part of the project “Dependence and Independence in Logic”.

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Fabry, R.E., Pantsar, M. A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving.
*Synthese* **198, **3221–3263 (2021). https://doi.org/10.1007/s11229-019-02276-9

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### Keywords

- Mathematical problem solving
- Levels of analysis
- Computational modelling
- Competence
- Performance
- Computational complexity
- Enculturation
- Neural plasticity
- Embodied cognition