In this section, we present a number of counting situations. We want to show that several other factors beside the cardinality principle determine whether a child will find it easy or difficult to count in an accurate manner.
We analyze some ways in which a collection can be physically structured and some ways in which counting can be performed. We follow the same method as did Zhang and Norman (1994) in their analysis of different versions of the Tower of Hanoi problemFootnote 9. They write that
”[t]he basic principle of distributed representations is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task” (Zhang and Norman 1994, p. 87).
They define the concepts as follows:
“Internal representations are in the mind, as propositions, productions, schemas, mental images, connectionist networks, or other forms. External representations are in the world, as physical symbols (e.g., written symbols, beads of abacuses, etc.) or as external rules, constraints, or relations embedded in physical configurations (e.g., spatial relations of written digits, visual and spatial layouts of diagrams, physical constraints in abacuses, etc.). Generally, there are one or more internal and external representations involved in any distributed cognitive task” (Zhang and Norman 1994, p, 89).
The division between internal and external representations is important because as the amount of external representations increases in terms of what is represented in the physical structure of the task, the amount of internal cognitive work that needs to be done by the agent decreases. In relation to the task of counting, the main question concerns to what extent the steps (i) – (vi) involve a load on working memory and attention. The more memory and attention are required, the greater is the risk that the counter makes an error in some of the steps. Here we note that in the cases where verbal numerals are used in counting, these are internal representations, while visual numerals are external.
Some of the cases we present below have parallels in the work of Fuson (1988). She provides a detailed error analysis of the cases that support our arguments concerning the role of the situatedness relative to the counting steps. In contrast to her work, our focus is on the interplay between internal and external representations and how this affects the load on memory and attention. For all cases, we assume that the counted collection has already been identified, so that step (i) is fulfilled.
Mapping Time to Space
In this subsection, we analyze several cases of counting situations, where reciting verbal numerals is coordinated with pointing at the elements of a collection. These cases involve establishing a one-to-one mapping between the temporal domain of the recital and the spatial domain of the collection. As we shall show, the differences between the domains is a major source of the difficulties that the counter encounters while generating the mapping.
The collection is linearly ordered. Counting is done with the aid of a finger.
In this case, it is easy for the counter to move a finger stepwise from one end of the collection to the other. The finger functions as a spatially located memory extension. Selecting the uncounted object (step ii) then becomes automatic. The physical presence of the finger pointing at the objects is a visual scaffold for remembering where in the establishment of the one-to-one mapping the counter is. Without a physical pointing device, this task must be managed by the working memory (see Case 3). The counted item is also externally “marked” when the finger moves past it (iv). Similarly, once the finger passes the last object, the procedure stops and thus also (v) is fulfilled without any particular attention. In brief, only the incrementing in reciting numerals (step iii) when the finger passes an object must be attended to. The cognitive load in this situation is thus mainly determined by step (iii) (Fig. 1).
In her analysis of this case tested on children of age 3 to 6 years, Fuson (1988, p. 64) writes that because
“words are located temporally and objects are located spatially, some sort of intermediary is needed to connect the two. The counter establishes this correspondence by using an indicating act, often the act of pointing, which has both a temporal and a spatial location.”
In this quote, she brings out that the mapping is from the time domain to the space domain. She notes (Fuson 1988, p. 177) that a fruitful correspondence between an indicating act and the indicated object must satisfy four requirements:
• Each indicating act must be directed toward an object;
• Each indicating act must not be directed toward more than one object;
• Every object must be indicated (this is our step (v));
• No object is indicated more than once.
Violating these requirements leads to various types of errors. Fuson finds that errors in counting decrease with age, and that the most prevalent error subtype consists in pointing to one object while saying two count words, or skipping objects while pointing. The errors indicate that at least younger children have problems of attending simultaneously to both the process of reciting the list of numerals and the process of moving the finger step by step along the row of objects.
The collection is ordered, but not linearly ordered. Counting is done with the aid of a finger.
This case results in the same analysis as above except that the counter must make sure that the finger is following the order of the collection. The cognitive load is thus mainly determined by step (iii), but some attention must be given to (ii). In the case the collection forms a closed curve, for example a circle, the counter must also attend to (v) so that working memory keeps track of which is the last object to be counted. Thus, this case taxes the working memory more than Case 1 (Fig. 2).
Fuson (1988) made two studies of this case, one with children aged 3½ to 6 and one with children aged 2½ to 3½. In the first study, she finds that the only recount errors are those where the children continued around the circle and recounted object they had counted at the beginning. The children were aware of the rule saying that counting should stop just before they arrive again at the object where they started the counting. However, their working memory obviously had problems remembering the location of the first object. She notes (Fuson 1988, p. 198), that this increase in memory demand also affected the production of number words. In the second study, one of the dots to be counted had a different color than the rest, and the children were instructed to start counting that dot. Almost all children (17 out of 19) stopped counting at the correct point, giving evidence that they understand the stop rule (v). Marking the first dot thus helped off-loading working memory and the results thereby improved.
The collection is linearly ordered. Counting is done without the aid of a finger or any other physical indexing device.
This case is also similar to Case 1, except the last item counted must be kept in visual working memory while the next item is selected. Thus step (ii) is not externally fulfilled by the position of a finger as in Case 1, but it adds to the cognitive load (Fig. 3).
The collection is not ordered. Objects are not movable. Counting is done without the aid of a finger or any other indexical device.
This case is cognitively more demanding than the previous ones, since all of the steps (i) – (vi) in it are internal and must be kept in working memory. Even if a finger is used, it may be difficult to fulfill step (iv). One strategy to achieve the marking is to impose an external linear ordering on the collection, say by counting the objects from left to right. However, if the collection is extended in the vertical direction and comparatively dense, it may be difficult to visually determine which object is to the left of another (Fig. 4).
In a study, Fuson (1988, Sect. 4.2) compares this case with Case 1. As expected, she found that the error rate for non-ordered collections increase considerably. Children both miss some objects when counting and count some objects twice. Remembering which objects are counted thus puts a higher load on working memory.
The collection is not linearly ordered. Objects are movable.
In this case, step (iv) can be fulfilled externally by moving the objected from left to right, say, while counting them so that the counted and uncounted objects are visually divided. By situating the task in the physical location of the objects that is changed by a bodily action, the working memory is partially offloaded. Hence, this case is cognitively less demanding than Case 4 (Fig. 5).
Fuson (1988, Sect 4.4.1) investigated the case where children could (but were not asked to) move the blocks that were counted. About half of the younger children (age 3½ to 5½) and almost all of the oldest children (age 5½ to 6) moved the blocks while counting. Most of those who moved the block used the movement as an indicating act sorting the counted from the uncounted blocks. There were some differences in the error rates made when compared to counting objects in a row: errors with more than one pointing per object were frequent when pointing to objects in a row, but it never occurred that one block was moved and then moved again. This can be interpreted in the following way: when an object can be moved it becomes easier to determine – visually and motorically – that it has been counted than when an object is just pointed to.
The collection is not linearly ordered. Objects are mobile while being counted.
This task is cognitively more challenging than all the previous ones since fulfilling steps (ii), (iv) and (v) involves an extremely heavy load on working memory even if the collection only involves a few objects.Footnote 10 (Fig. 6).
Counting in a mental spatial representation.
If a person is given the task of counting the numbers of windows in the apartment where he or she lives while being at a distant location, the person must mentally imagine the rooms of the apartment and from memory reconstruct the windows in the rooms. This task thus involves objects in a mental representation of space rather than objects in the present spatial surrounding as in all the previous cases. In particular, this makes step (ii) more difficult than in Cases 1–5, but also steps (iv) and (v) depend on the accuracy of the mental representation.
More variations of counting can be considered, but we hope that the cases we have presented illustrate how an analysis in terms of attention and working memory can determine the cognitive difficulty of a particular counting situation.
Mapping Space to Space
We next turn to the case where external visual symbols are used instead of internal verbal numerals. If the numerals are presented in a linear spatial layout, then the working memory involved in reciting the numerals in the correct order (step (iii)) can be offloaded. New methods for counting can then be exploited. On the other hand, such cases presume that the counter is familiar with the visual symbolic representations of the numerals. This knowledge is different from the ability to recite numerals (Dehaene and Cohen 1995).
Numerals are linearly ordered from left to right on a piece of paper. The objects to be counted are linearly ordered below the numerals. The counter has a pencil.Footnote 11
In this case the one-to-one mapping can be created by drawing lines from the numerals to the objects. The counter first draws a line from “1” to the leftmost object, then a line from “2” to the second leftmost, etc. The procedure is stopped when there are no more objects without lines in the collection. Applying the cardinality principle just means identifying the last numeral. In this setup all of (i) – (vi) are more or less automatically fulfilled. The counter just has to make sure that there are not two lines going to the same object, that there is no object without any line from a numeral, and that no numeral is skipped in the procedure. All this can be visually determined. Hence the temporal domain is not involved and the working memory used for reciting numerals is not needed. This case thus involves less cognitive load than all the previous cases since the spatial locations of the numerals and the objects help constructing the one-to-one mapping and thereby fulfilling the steps of the counting procedure. The counter doesn’t even need to be able to recite the verbal numerals.Footnote 12 (Fig. 7).
Case 8 can be modified in several ways in parallel to cases 1–6 in the previous subsection. However, for all such extensions, the fact that the numerals are located in physical space (say on a sheet of paper) makes these cases cognitively less demanding than the corresponding cases where the numerals are produced as internal representations by counting verbally in a temporal sequence. In brief, mappings space-to-space are cognitively less demanding than corresponding mappings time-to-space. Most of the counting errors documented by Fuson (1988) concern the establishment of the one-to-one mapping, but her studies are about mappings from time to space. We predict that if they were repeated for space-to-space mappings, significantly fewer errors would be observed.
Mapping Time to Time
Finally, we turn to the third type of one-to-one mappings – those from the temporal domain to itself.
Case 9. Runners passing a finishing line
Counting objects that pass a fixed place in space, by reciting numbers, involves a mapping from time to time. In such cases the spatial locations of the objects are irrelevant, except for the moment they pass the fixed place. Steps (i) – (vi) are more or less externally fulfilled (given that there is only one type of objects that pass the place). However, objects should not pass the place faster than they can be counted, and it may be difficult to determine when the step (v) is fulfilled. If there are long temporal intervals between successive objects, the last numeral must be preserved in working memory.
The nine cases that have been analyzed here are all instances of applying the cardinality principle. As the analysis shows, however, the steps (i) – (v) must also be performed. We have seen that counting can be more or less difficult depending on the available external representations that can help offloading the working memory of the counter. We predict that the more working memory is involved in a particular case of counting, the more errors will the counter make. To some extent, Fuson’s (1988) experiments support this prediction, but more tests involving variations of the counting situation can be made to test it.
Davidson et al. (2012, p. 167) write that Sarnecka and Carey (2008) “may have overestimated the knowledge that results from becoming a CP-knower, since they did not explore whether differences between children were meaningfully related to [the experience of counting] … and did not test numbers beyond five”. To this we would like to add that our analysis shows that variations in children’s working memory and attention will lead to different performances in counting situations where the physical constraints more or less support the one-to-one mapping procedure.
The distinction between the temporal and spatial domains that generate the three types of one-to-one correspondence, and also the feasibility of different situated cognition contexts could serve as a base for an educational program, such as digital games enhancing children’s math development (see e.g. Haake et al. 2015). We predict, for example, that once a child has learned the symbols for the numerals, the space-to-space mapping in Case 8 is easier to master then a corresponding time-to-space mapping. We also predict that the bodily involvement in drawing lines between external representations of numbers and objects supports the learning of the counting process more than the reciting of numerals and thereby leads to fewer matching errors.
In our proposal for using situated counting in educational programs, we follow a recent study by Johnson et al. (2019) who suggest that children are way more competent when it comes to solving mathematical problems than it is usually believed. The authors highlight that even very young preschool children can engage and make sense of sophisticated mathematical ideas in a favorable contextFootnote 13. We suggest that contexts for counting situations can be systematized by the situatedness and the loads on attention and working memory they necessitate.