1 Introduction

The understanding of place value systems, especially the base-ten place value system, is one of the most important prerequisites to develop numeracy (Benz 2005; Verschaffel et al. 2007). There is evidence that place value understanding is predictive for later performance in arithmetic (Moeller et al. 2011). Moreover, various difficulties in arithmetic can be ascribed to a lack of understanding of the base-ten place value system, such as errors in written arithmetic (e.g. Cox 1974; Jensen and Gasteiger 2019) as well as difficulties with decimal fractions (Heckmann 2006).

Several authors consider the ability to connect a (structured) quantity with the written numbers using the place value system and with number words as crucial aspects of place value understanding (e.g. Fromme 2017; Fuson et al. 1997; Van de Walle et al. 2020) analogous to the triple-code model (Dehaene 1992) (see Fig. 1).

Fig. 1
figure 1

Three components of “relational understanding of place value” as named by Van de Walle et al. (2020, p. 250) (creation S.J.)

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In Fig. 1, the numbers written on the base of the place value system are called “written names” and the number words are called “oral names”. The base-ten concept means grouping by tens and understanding the relation between tens and ones, so that ‘53 ones’, ‘5 tens and 3 ones’ and also ‘4 tens and 13 ones’ can be regarded as equivalent (Van de Walle et al. 2020, pp. 248f.). As the base-ten concept is seen as a component of place value understanding, grouping activities and the part-whole-concept are named as prerequisites of place value understanding (Fromme 2017, pp. 57ff., see also Van de Walle et al. 2020).

Taking a closer look at the written names, this symbolic representation is defined mathematically as a sum of powers of tens. Every \(n\in \mathbb{N}\) can be uniquely represented as:

$$n=a_{m}g^{m}+a_{m-1}g^{m-1}+\ldots +a_{2}g^{2}+a_{1}g+a_{0}$$

with g, ai \(,m\in \mathbb{N}\),\(g>\) ai \(\geq 0\). g is called the base of the place value system. Short this sum is notated as: \(n=(a_{m}a_{m-1}\ldots a_{2}a_{1}a_{0})\)g (In the base-ten place value system the base \(g=10\) is not noted).

Mathematics education deals with the question what one needs to know and understand in order to be able to interpret and work with the symbolic representation of numbers based on our base-ten place value system (the written names in Fig. 1). In the tradition of German subject-matter didactics, mathematical content is analysed from a mathematics education perspective to be able to teach the content appropriately to the subject and the learner (Hefendehl-Hebeker 2016). According to this tradition, place value systems commonly are ascribed to the regrouping principle and the place value principle (e.g., Büchter and Padberg 2020; Krauthausen 2018; Scherer and Moser Opitz 2010)Footnote 1. The regrouping principle describes that the same number of units is bundled continuously (as also described in the base-ten concept by Van de Walle et al. 2020, see also Fig. 1); the place value principle states that each digit gives two pieces of information by its position, which indicates the bundling unit it stands for, and its face value, which tells the number of the represented bundling units.

Studies show that many children have difficulties in understanding the base-ten place value system (e.g., Herzog et al. 2019; Scherer 2014) and because of that there is a need for support. Several studies deal with the development of an understanding of the base-ten place value system and of multi-digit numbers (Cobb and Wheatley 1988; Fuson et al. 1997; Herzog et al. 13,14,a, b, 2019; Ross 1986, 1989; Sinclair et al. 1992), focussing on different aspects of the components of place value understanding as shown in Fig. 1. In these studies, the place value principle and the regrouping principle are not addressed explicitly, but several conclusions about some developmental steps regarding both principles can be drawn. The question arises whether the two principles can be used systematically for an effective identification of gaps in student’s understanding.

Following this line of argumentation, we first clarify the meaning of the place value principle and the regrouping principle and their contribution in understanding the base-ten place value system. We then give a brief overview of findings regarding both principles that can be derived from the studies resp. the development models mentioned before (Cobb and Wheatley 1988; Fuson et al. 1997; Herzog et al. 13,14,a, b, 2019; Ross 1986, 1989; Sinclair et al. 1992). In our study, we analysed children’s errors for disregarding the two principles. We thus aim at clarifying, whether and how the two principles (regrouping, place value) can contribute to a differentiated diagnosis of difficulties as basis for appropriate individual support in the context of place value understanding.

2 Theoretical Considerations

2.1 The Regrouping and Place Value Principle

To understand place value systems, it is—following the tradition of German subject-matter didactics—crucial to understand the regrouping principle and the place value principle (Büchter and Padberg 2020; Krauthausen 2018; Scherer and Moser Opitz 2010, see also Van de Walle et al. 2020).

The regrouping principle describes that the same number of units is bundled continuously; for example, in the base-ten place value system ten ones are combined to form the new bundling unit ‘tens’, ten tens build a ‘hundred’, and so forth. The regrouping principle refers primarily to (material) actions—concrete or mentally (in Fig. 1 named as base-ten concept). In the written numbers using the place value system, the regrouping principle is not visible at first sight. But each digit is related to the respective power of ten (tens, hundreds, …) and therefore an understanding of regrouping is needed to deal with written numbers. For example, 245 stands short for 2 ∙ 102 + 4 ∙ 101 + 5. Because in the definition is stated that \(g>\) ai \(\geq 0\) (see Sect. 1, g =10 in this case) a new bundling unit with the next greater power of ten has to be formed when there are more than nine units. Children must therefore understand that each bundling unit contains ten of the next smaller bundling units and that, in turn, if there are 10 bundling units, it is convenient to regroup them into the next larger bundling unit. The regrouping principle is also mirrored in the oral names in several languages (Fig. 1), because the bundling units are named. For example, 444 is fourhundredfourtyfour in English. Fuson (1990) deals with the question how the irregularities in the English number words cause trouble in understanding multi-digit numbers. But on the other hand, the number words seem to give the children that are learning the words hints about the decade structure of numbers (Fuson et al. 1997, pp. 140f., see also Sect. 2.2).

The regrouping principle is not only characteristic for symbolic number representations in place value systems. There are also symbolic number representations that are based solely on the regrouping principle. The ancient Egyptian numerals, for example, used bundling units of ten, but each bundling unit was identified by a different character (e.g., a finger representing 10,000 or a tadpole representing 100,000) and as many of these characters were noted as there were bundling units (for example 500,000 were noted with five tadpoles, Ifrah 1993, p. 231). Because different characters stand for different bundling units, it is not necessary to note these characters in a particular order (see Ifrah 1993, pp. 233ff. for examples). This also applies to some manipulatives like base-ten blocks, that are often used in the context of learning the base-ten place value system. Hence base-ten blocks do reflect the regrouping principle of place value but not all the characteristics of written numbers using the place value system (see also Varelas and Becker 1997, p. 269). Therefore, being able to apply the regrouping principle is fundamental, but alone insufficient for understanding place value systems.

As described, in today’s base-ten place value system, the number and type of bundling units is only indicated by digits and their positionsFootnote 2. Hence, in place value systems it is crucial to use a specific order to write down the digits. This is described by the place value principle. It states that each digit gives two pieces of information: Firstly, the position indicates which bundling unit the digit stands for, and secondly, the value of the digit indicates the number of represented bundling units. Therefore, the digit ‘0’ is required to indicate if there is no bundling unit of a particular size—otherwise the order of the digits would not correctly represent all bundling units sequentially. Furthermore, in case of more than nine bundling units regrouping is not only convenient but necessary: 2 hundreds, 13 tens and 1 one cannot be transferred into place value notation without regrouping the 10 tens to 1 hundred. Hence, the place value principle relates primarily to numbers written in the place value system and the meaning of the digits—Varelas and Becker (1997) call this the semiotic aspects of understanding of place value. To be able to read the symbols, one must be able to interpret the value of the digit and to deduce from its position, which bundling unit (power of ten) it refers to (and multiply the face value with the power of ten). When writing numerals, the order of the digits must be respected.

In summary, the regrouping principle is reflected in all three different representations that are regarded in understanding multi-digit numbers (the number words, the written numbers using the place value system, and a structured magnitude representation resp. the base-ten concept), whereas the place value principle only has to be considered in the written numbers using a place value system (see also Fromme 2017). Therefore, one does not need place value understanding for dealing with number words or the base-ten concept, but only in using the written numbers.

The place value principle is linked to the regrouping principle, because the digits represent the kind of bundling unit and the number of the respective bundles. Only by connecting the two principles a sound understanding of the (base-ten) place value system can be developed. This understanding becomes visible in the fact that the two principles can be applied in dealing with the written numbers: In being able to connect the positions in a multi-digit number to a magnitude by multiplying the face value of the digit with the represented bundling unit, the place value principle is applied. In cases of arithmetic operations and representation change of numbers into the written numbers using the place value system, the regrouping principle comes into play by being able to use the relations between the bundling units to regroup.

2.2 The Regrouping and Place Value Principle in Developmental Models of the Understanding of Multi-digit Numbers

There are various studies on the development of the understanding of the base-ten place value system and of multi-digit numbers (Cobb and Wheatley 1988; Fuson et al. 1997; Herzog et al. 13,14,a, b, 2019; Ross 1986, 1989; Sinclair et al. 1992). Through the lens of the two principles, various insights into the gaps in students’ understanding of the base-ten place value system and multi-digit numbers can be derived from these models.

In some of the developmental models, an initial stage is described in which children understand two-digit numbers as a single sign for a given quantity. Therefore, no understanding of the meaning of the digits and their positions (place value principle) or for the representation of continued bundling (regrouping principle) becomes apparent (Fuson et al. 1997; Herzog et al. 13,14,a, b, 2019; Ross 1986, 1989).

The understanding of the regrouping principle contains the understanding of the connection between tens and ones. To examine this understanding counting processes that allow using the connection between tens and ones can be informative (the written names do not have to be used). Some studies describe a growing understanding of the connections between the bundling units (Cobb and Wheatley 1988; Fuson et al. 1997). At first, the children cannot see ten ones as a ten and the other way round but see them as different objects (Cobb and Wheatley 1988; Ross 1986). In a counting task using material similar to base-ten blocks, the children cannot count a ten, but count the visible ten ones instead (Cobb and Wheatley 1988). Therefore, the children do not show an understanding of the regrouping principle. Later in their development children can use the equity of ten ones and one ten and count in tens instead of counting the ones (Cobb and Wheatley 1988). In addition, studies indicate that children initially need visual support like base-ten blocks to identify the relationship between the bundling units (Cobb and Wheatley 1988, Herzog et al. 13,14,a, b, 2019). The number range in which children can use the relationship between the bundling units seems, however, to be limited to numbers up to 100. The understanding of the relationship between tens and ones therefore seems to be the basis for transferring this understanding of the regrouping principle to larger number ranges (Herzog et al. 13,14,a, b, 2019).

Understanding the place value principle seems to begin by an understanding that the two digits in a two-digit number must be considered separately, but with no conceptual understanding of place value (Herzog et al. 13,14,a, b, 2019; Ross 1986, 1989). Similarly, the ‘decade and ones conception’ (Fuson et al. 1997) describes a beginning awareness for the different parts of a two-digit-number, possibly because of hearing a tens-part and a ones-part in number words (e.g. ‘fifty-three’) and linking “the decade and the ones part of a number word to written marks” (p. 141). This understanding is not yet definitively related to the respective places of the digits. Consequently, the individual word fragments are sometimes translated isolated one after the other when writing down the number (for example, ‘fifty-three’ becomes 503) (Fuson et al. 1997)Footnote 3.

Fuson et al. (1997), Ross (1986, 1989) and Sinclair et al. (1992) describe children who interpret the digits of a two-digit number in isolation only with regard to its face value, but the reference to the bundling units is missing. These children for example take two sticks when asked of the meaning of the ‘2’ in ‘25’ instead of 20 sticks. They did not understand the place value principle, because the reference of position and face value and therefore the link between the two principles is a crucial characteristic of the place value principle. This linkage becomes especially clear by a task Ross (1989) used: The children were asked about the meaning of the digits of a two-digit number and in a special task Ross (1989) used a bundling in groups of four as a distractor. The understanding of the link between the two principles becomes apparent, when children assign two bundles of ten to the tens digit of the two-digit number representation and six ones to the six, even if the material offers two ones and six bundles of four (Ross 1989, see Fig. 2).

Fig. 2
figure 2

26 objects partitioned in two ones and six bundles of four (creation S.J.)

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In summary, indications of an understanding of both of the two principles can be found in the different studies. But until now it is not clear, if the two principles can be used systematically—as a kind of diagnostic tool—to describe difficulties or a lack in understanding the base-ten place value system precisely. Because the understanding of the base-ten place value system has high importance and at the same time is difficult to achieve, it could be very helpful to understand students’ difficulties better. The question arises whether the two principles can be used systematically to effectively identify specific gaps in students’ understanding. If we can identify students’ lack of understanding accurately, it would be possible to foster them adequately to their individual difficulties and their individual stage of understanding.

3 The Present Study

Following our line of argument, that indications of specific difficulties of the children could become visible through the lens of the two principles, we conducted an empirical study using tasks in which both the regrouping principle and the place value principle have to be used. Our research question was:

Which diagnostic information can the regrouping principle and the place value principle provide when analysing primary school children’s errors in using symbolically represented numbers in the base-ten place value system?

Because understanding the base-ten place value system is one of the most important prerequisites to develop numeracy, it is important to foster children in acquiring this understanding. If the two principles help to analyse the children’s difficulties in more detail, this may be a good basis for a diagnostic tool for teachers. Teachers would then be able to better narrow down difficulties and therefore provide more appropriate support maybe focussing of one of the principles.

3.1 Methods

We conducted a cross-sectional study with 100 third graders (8–10 years) from 7 classes in 3 German elementary schools. We chose children in the second half of third grade, because they already worked with numbers up to 1000. In this number range, the place value understanding can be examined in more depths, because applying the regrouping principle is more demanding (see Sect. 2.2). The children also already learnt written addition, in which the function of the base-ten place value system is applied. We therefore expected children with a sound understanding of the base-ten place value system, but as well children showing difficulties.

In march 2020, the students were asked to work on 7 tasks. The tasks regard the translation of named units like 5T 3O, which are commonly used in German school books (see for example Balins et al. 2016), into written numbers using the base-ten place value system (in this case ‘53’). We chose this task format because it allows to operationalise the place value principle, the regrouping principle, or both principles simultaneously, so that an error analysis regarding the two principles individually seems possible:

For the regrouping principle, the literature emphasises the ability to translate bundling units into each other while also being able to deal with non-canonical representations (e.g., Scherer 2014) and also the ability to regroup mentally (e.g., Cobb and Wheatley 1988). We therefore use tasks with more than nine bundling units like 3T 14O = 44. To test the understanding of the place value principle, we chose tasks, in which the correct order of the bundling units and in some cases the use of a zero had to be taken into account. The following 7 tasks were used:

  • place value principle:

$$7\mathrm{O}1\,\mathrm{H}4\,\mathrm{T}=\_ \_ \_ \_ \_ \_ ,4\,\mathrm{H}6\mathrm{O}=\_ \_ \_ \_ \_ \_ ,6\,\mathrm{T}9\,\mathrm{H}=\_ \_ \_ \_ \_ \_ \_ \_ \_$$

  • regrouping principle:

$$4\mathrm{\,H}15\,\mathrm{T}6\mathrm{O}=\_ \_ \_ \_ \_ \_ ,4\,\mathrm{H}3\,\mathrm{T}12\mathrm{O}=\_ \_ \_ \_ \_ \_ \_ \_ \_$$

  • both principles:

$$7\,\mathrm{H}3\mathrm{O}19\,\mathrm{T}=\_ \_ \_ \_ \_ \_ ,3\,\mathrm{H}15\,\mathrm{T}=\_ \_ \_ \_ \_ \_ \_ \_ .$$

Though all children were familiar with the abbreviations T, H, T, O, at the beginning of the test, the abbreviations were explained again to help the children to remember, e.g., O = OnesFootnote 4.

For the data analysis, we firstly coded students’ answers dichotomously (true/false). To differentiate the difficulties of the children we did a qualitative analysis of the students’ errors made in each task: For each error, we analysed whether the regrouping principle, the place value principle or both were disregarded. For example, in the task regarding both principles \(7\mathrm{\,H}3\mathrm{O}19\,\mathrm{T}=\_ \_ \_ \_\) the solution 7193 indicates that the place value principle was considered in ordering the numbers regarding the bundling units. Because the child did not consider the continuous bundling, the regrouping principle was not applied, so we coded a regrouping error. Note that as a result, the child has also not applied the meaning of the places and accordingly there is also a disregard of the place value principle, but this seems not to be the decisive problem of the child. Solution 749 shows a disregard of the right order of the bundling units and therefore of the place value principle (place value error), but the child regrouped (because of not regarding the right order it regrouped falsely ten tens into one one). In 7319 both principles are violated (two errors were noted here: a regrouping error and a place value error). Mostly, the place value errors occurred in tasks regarding the place value principle resp. both principles according to the classification made above and regrouping errors occurred in tasks regarding the regrouping principle resp. both principles. Three errors were in that sense ‘atypical’ (two regrouping errors in tasks regarding the place value principle, according to the classification made above, one place value error in a task regarding the regrouping principle). For example, in the task \(7\mathrm{O}1\mathrm{H}4\mathrm{T}=\_ \_ \_\) a child gave the answer 111 probably in adding up 7O and 4T without the necessity to regroup and therefore made a regrouping error.

Based on these error classifications we added a further qualitative analysis in building groups of children that made mainly the same kind of errors. Following Kuckartz’ (2010) four phases for typification, we decided to use as basis characteristics (phase 1) firstly the kinds of errors made by the children regarding the two principles and secondly, we analysed the errors assigned to each principle more detailed.

For the typification (phase 2), we started with the task \(7\mathrm{\,H}3\mathrm{O}19\mathrm{T}=\_ \_ \_ \_\) where the children need to use both principles. We grouped the children according to the results they noted. The errors 7193 and 722 both show a disregard of the regrouping principle, but in different ways: The first error shows no regrouping at all, the second shows a wrong kind of adding up bundling units. The error 749 shows a disregard of the place value principle disregarding the right order. Because the children could make errors disregarding each of the two principles in this task, a first grouping of children according to the concrete results was a good basis for further grouping. We described the types that resulted based on the first grouping. When we were able to recognise further patterns, the types were further differentiated and we were able to describe the types in more detail (phase 3). At last (phase 4), the assignments of all children were checked.

3.2 Results

The mean value of the solution rate of the whole test instrument is 57.6% (see Table 1 for solution rates of all items), 19 of the 100 children made no error at all.

Table 1 Mean values of solution rates
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The solution rates for the regrouping-principle-tasks are a little bit lower as for the place-value-principle-tasks (62.5% vs. 68.3%, see Table 1), the solution rate for the tasks in which both principles have to be considered is lowest (36.5%).

Four of the 297 wrong solutions were missing solutions. The results of the 293 remaining incorrectly solved tasks we analysed qualitatively according to the type of error. There were a total of 158 regrouping errors and 153 place value errors (18 wrong solutions were assigned to the two kinds of errors at the same time). In the tasks addressing only one of the principles, the majority of the errors were attributable to the addressed principle (there were only three atypical errors: one unexpected place value error and two unexpected regrouping errors).

A comparison of the numbers of errors each child made on each principle (Fig. 3) shows that there are great differences in the difficulties the students have in regarding one or the other principle. 22 children showed at least 2 errors of each kind. Some children disregarded one of the principles in each task where it had to be applied, but could apply the other principle: 7 children made 4 regrouping errors, but no place value error. 6 children made 5 place value errors, but no regrouping error. To check whether the two kinds auf errors are statistically independent of each other, we used Spearman’s rho. The number of place value errors does not correlate significantly with the number of regrouping errors (rs = 0.161, p = 0.114, N = 100).

Fig. 3
figure 3

Number of place value errors compared to number of regrouping errors for each child

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The further qualitative analysis revealed different error patterns of children across all tasks. Based on the classification of all children according to their solution behaviour, we formulated types that describe wrong approaches and therefore different types of difficulties the children have in considering the two principles (see Fig. 4). The children assigned to one group did not make the same number of errors but showed the same wrong solution behaviour. They showed this behaviour in all or only in some of the tasks. Some children only made one error, but if this one error was typical for an error pattern, these children were assigned accordingly (for example, two children only had the wrong solution “749” for the task \(7\mathrm{H}3\mathrm{O}19\mathrm{T}=\_ \_\); they were assigned to group c). Four children only made few atypical errors and therefore were classified as “others” like also one child with errors because of missings (type g, see Fig. 4).

Fig. 4
figure 4

Typification after error patterns. Note: A representative example of the error pattern is given, but not all of the children assigned to the type followed this exact pattern esp. in regarding the zero, number of errors variates in each group (regrouping error: pale grey; place value error: dark grey; both principles disregarded: medium grey)

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The type a did not show any understanding of both of the principles. The nine children mostly just copied the digits and therefore did not consider any principle at all, so that they solved at least 6 out of 7 tasks incorrectly. Two of them (type a.2) seemed to know that there should only be one digit per bundling unit and solved this problem in dropping digits instead of regrouping and therefore solved each task wrongly.

Type b (16 children) was aware of the necessity of only one digit per bundling unit and solved this problem in adding up different numbers of bundling units. Interestingly, the number of false solutions varies between 2 and 7 tasks: 4 children had no problem with the tasks only regarding the place value principle but showed place value errors in the tasks regarding both principles. Therefore, on the whole these children showed problems with both principles either.

The error analysis also revealed that some children had more difficulties with one or the other principle. The analysis and typification of the solution behaviour provides further information about the kind of difficulties.

The types c and d (28 children) were able to use the idea of the regrouping principle and only made place value errors. Type c (17 children) applied it in regrouping ten bundling units on the right position into one of the bundling units on the left position, but (sometimes) did not apply the right sequence (and therefore the regrouping was not completely right either). The amount of errors varies between one (two times, wrong solution: \(7\mathrm{H}3\mathrm{O}19\mathrm{T}=749\)) to five. In seven cases the three tasks needing a zero made no difficulties at all.

Type d (11 children) regarded the regrouping principle and used the right order (place value principle) but showed some difficulties in applying the place value principle in case of using the zero, so that overall, the children showed some difficulties in applying the place value principle.

Type e and f (23 children) mostly regarded the right order and therefore showed an initial understanding of the place value principle, except in some cases the children could not handle the zero (6 children). Type e and f could not apply the regrouping principle: The type e (9 children) used no regrouping at all. Type f (14 children) was aware of the need of regrouping but added up the numbers of tens and ones instead. Eleven children used this solution behaviour only in cases where regrouping of tens was necessary, the other four also in the task \(4\mathrm{H}3\mathrm{T}12\mathrm{O}(=415)\), where the ones needed to be regrouped.

Type g was formed of four children who mostly regarded both principles and did not show a typical wrong solution of one of the other types and one child only had errors because of not answering (three missings).

4 Discussion

The aim of this study was to analyse, how the regrouping and place value principle can contribute to a differentiated diagnosis of difficulties in understanding the base-ten place value system. In studies examining and describing the development of understanding multi-digit numbers some indications of the growing understanding of the two principles can be found. But so far, the two principles were not used systematically to describe difficulties in understanding the base-ten place value system. Our study indicates that the analysis of errors in different tasks regarding the translation between named units into written numbers using the base-ten place value system regarding the two principles can give some indications of specific difficulties in understanding the base-ten place value system.

The mean values of the solution rates show that children had more difficulties in solving the tasks regarding both principles than in solving tasks regarding only one principle (36.5% (both) vs. 62.5% (regrouping principle) resp. 68.3% (place value principle), see Table 1; Sect. 3.2). It seems that in the tasks regarding both principles the difficulties regarding the two principles sum up. The comparison of the numbers of each kind of error per child also shows that there are great differences in the difficulties individual students have in regarding one or the other principle. This is further emphasised by the fact that the two kinds of errors do not correlate significantly.

In addition, we were able to form types of error patterns that show different types of misconceptions concerning one or both of the principles in different ways. Type a mirrors the least ability of applying both principles. Type d on the other hand only shows problems with using the zero. That leads to some implications regarding an adequate support.

4.1 Implications

The diagnosis of various misconceptions based on error patterns offers opportunities for tailored support. Certainly, the information provided by the tasks is not exhaustive and further observations need to be made. In the following, we try to make some suggestions for an initial support, but these ideas to foster children also offer possibilities to deepen or expand the diagnosis.

The types of error patterns differ in having problems with both or only one of the principles. Types a and b showed difficulties in both principles and therefore the most problems in the tasks. Only writing down the given digits without regrouping like the children of type a.1 is a solution behaviour that Herzog et al. (2019) also describe in a lower level of their model, where the children cannot deal with non-canonical representations and “apply the routine of canonical representations” (p. 575). Children of the types a.2 and b did not just copy the digits, but could not regroup. Therefore, children of type a and type b should be engaged in bundling activities in the number range up to 100 using different kind of materials to support the base-ten concept. They should describe the result after bundling (for example, “I now have nine tens and seven ones”) resp. counting the sum using the bundling units as described in Fuson et al. (1997): The children should count on the one hand the bundling units as wholes (counting for example one—two—three—four tens) and on the other hand count in tens (ten—twenty—thirty—fourty) to be able to connect the two ways of counting. It is important that the teacher observes the children and checks whether they understand the idea of forming new bundling units and are able to name them. Because the children disregarded the place value principle either, they should use a place value table to note the result of the bundling process before noting the written number.

The typification of error patterns also mirrors that some children have specific problems with one of the principles while being able to (mostly) apply the other one. Types c and d only disregard the place value principle. They could use the idea of the regrouping principle, but because they did not sort the bundling units they regrouped wrongly, for example ten tens into one one. This could be interpreted as only regarding the face value (Ross 1986, 1989; Fuson et al. 1997; Sinclair et al. 1992), because the reference to the bundling units seems to be missing. The children might only use the mechanism of regrouping in a procedural way. Tasks to translate written numbers in ten blocks—and answer the question “What do you notice?”—could direct to the discovery, that the amount of the largest bundling unit always is presented by the digit on the left. To emphasise the importance of the positions for the represented magnitude, it could be helpful for the children to learn to translate written numbers that use the same digits in different combinations, e.g. 749, 974, 497 in ten blocks, and also numbers with zero like 305, 35. Because type d only showed problems with using the zero, they should get mainly the second kind of tasks.

Type e and f mainly made regrouping errors. It may be helpful to present iconic non-canonical representations of the ten blocks and to ask children to use the concrete material of base ten blocks to regroup. Afterwards the children note the results of their regrouping in a place value table. If these children note numbers larger than 9 in one column, they can be asked to find another way to represent the number in the place value table. Children with these error patterns (type e and f) also could work on tasks as “Explain: 20 O equals 2 T”. A diagnostic conclusion here might be, that some type-e-children should engage in regrouping activities as type a. Type f mainly made regrouping errors when it was necessary to regroup tens into hundreds. The children therefore seem to be able to regroup ten ones into one ten but cannot transfer this idea to regrouping ten tens into one hundred, like also Herzog et al. (2019) describe. Children of this type should work mainly on tasks where tens have to be regrouped in hundreds using ten blocks. A more demanding task could use named units as used in our study. Translating these named units in a place value table could point out the problem that one has to regroup tens into hundreds in the same way as ones into tens. In this process, intermediate steps as described by Wartha and Schulz (2011) can support the children in building mental images (after handling the ten blocks for themselves, children describe the actions with a view of the material when translating into the place value table, then they do the translation without using the ten blocks).

Overall, few similar task formats can be used for further support and diagnosis, and the teacher can save resources by focussing on specific aspects based on the diagnosis.

4.2 Perspectives

The children who took part in the study were introduced to the number range up to 1000 some months ago (at the beginning of third grade). In spite of that, they only solved 57.6% of the tasks correctly (see Table 1; Sect. 3.2), only 19 children made no errors at all, 7 children only had difficulties applying the zero. This comparably low solution rate confirms results of other studies that show insufficient understanding of the base-ten place value system (e.g., Fuson et al. 1997; Herzog et al. 2019; Scherer 2014). Our study emphasises the importance of addressing the understanding of the base-ten place value system.

In our study we used relatively few tasks and tasks of a special kind. In order to examine the diagnostic potential of the two principles in greater depth, replication with other item formats or with supplementary interviews would be useful. But all in all, our analysis indicates that these few tasks are an easy and fast possibility to provide information about individual difficulties and therefore an opportunity for teachers to gain diagnostic information.

We found that students may have specific problems with one of the two principles (regrouping principle or place value principle) and make more errors of one type. In our study this seems to be related to individual gaps in the conceptual understanding of the base-ten place value system rather than to the tasks. Since Moeller et al. (2011) identified place value understanding as a predictor of later computing performance, our results therefore give some important indications for teaching the base-ten place value system. Firstly, it can be useful to keep these two principles in mind when selecting specific tasks and materials for a lesson with the aim of generating understanding of the base ten place value system. Secondly, it also seems reasonable to consider both principles separately in the diagnosis of students’ individual problems in understanding the base-ten place value system. As we showed, the tasks used in our study could be used for this purpose. Finally, students might benefit more from support in developing an understanding of the base-ten place value system, which is adapted to their specific problem with one of the principles.

For further research it could be beneficial to look on the development of the two principles separately to differentiate development models. In our analysis of the studies presenting developmental models of understanding multi-digit numbers resp. place value understanding (see Sect. 2.2), we could identify one or both principles in these models. For example, Cobb and Wheatley (1988) describe above all the relation between ones and tens and we were thus able to find indications for the development of the regrouping principle. Herzog et al. (13,14,a, b, 2019) regard this relation between bundling units as important for conceptual place value understanding, but they locate the place value principle only in a lower level of their model as procedural place value understanding. Our study points out that students show significantly greater difficulty with one or the other of the two principles by making more errors of one type. This could be seen as an indication that understanding of the two principles might develop differently for individual students. Therefore, for future research it could be worth to consider both principles separately in development models of the understanding of the base-ten place value system and to analyse children’s development separately for both principles.