Within the last decades, research has been able to identify several general predictors and precursors that contribute significantly to learners’ conceptual development. Working memory abilities are prominently discussed as predictors for mathematical development. Working memory enables us to retrieve and store information and control attention while working on a maths problem. Visual-spatial abilities are particularly predictive in preschool and early school age regarding mental arithmetic performance (Arndt et al. 2013; Barnes et al. 2014; De Smedt et al. 2009; Kroesbergen & Van Dijk 2015), whereas older children’s verbal working memory abilities outrun visual-spatial abilities when it comes to mathematical reasoning (De Smedt et al. 2009). Executive functions, such as inhibition and shifting, seem to predict mathematical development as well (Cowan & Powell 2014; Navarro et al. 2011). Inhibition is the ability to abort an ongoing action and to supress the urge of an action; shifting refers to changing between tasks fast and reliably (Miyake et al. 2000).

One of the oldest predictors discussed is inductive reasoning (Desoete 2015; Klauer & Phye 2008; Piaget 1965). Inductive reasoning contains general cognitive skills that allow finding patterns, regularities and rules that can occur in attributes or relations of items. Typical tasks are classification, seriation and pattern formation (Desoete 2015; Klauer & Phye 2008). In particular, patterns are considered as a central element of mathematical thinking (Devlin 2003). In empiric studies, classification and seriation skills could be identified as predictors for arithmetic achievement (Desoete 2015; Desoete et al. 2009). Furthermore, inductive reasoning predicted arithmetic performance in 6-year-old children, even if controlled for working memory (Nunes et al. 2007).

Language skills are on the edge between general and domain-specific predictors, depending on their relation to mathematics. However, even non-specific language skills predict mathematical development (Desoete 2015; Prediger et al. 2013). Particular attention was drawn to phonological awareness of young children, which showed predictive power in empiric studies (Barnes et al. 2014; Navarro et al. 2011; Passolunghi, Vercelloni & Schadee 2007). Empirical studies showed that the influence of phonological awareness on mathematical concepts is sustainable across the transition from kindergarten (same age as South African Grade-R learners) to Grade 1 and partially even increases (Langhorst, Ehlert & Fritz 2013; Navarro et al. 2011). Prediger et al. (2013) found high correlations between mathematical and reading skills that applied even for tasks that did not require reading competencies. Language skills get more important during the growth of mathematical skills (Dehaene 2011). As formal schooling involves verbal activities, it requires linguistic skills, which affect the acquisition and retrieval of mathematical knowledge in school (Prediger et al. 2013).

Research on domain-specific language skills is still rather rare. For instance, Göbel et al. (2014) revealed how knowledge of single and multi-digit number words at the beginning of Grade 1 is a good predictor of mathematical achievement 1 year later. However, there is still a lack of empiric research on mathematic-specific language skills and knowledge, for example grammatical competencies in prepositions or vocabulary. Indeed, research suggests that grammar skills predict later mathematical performance (Cowan & Powell 2014; Sarnecka 2014).

A general predictor of mathematical development that is often neglected is the emotional aspect of learning. Children’s attitudes to mathematics, whether they enjoy maths or are scared of it, affect their learning success enormously (Moore, Rudig & Ashcraft 2015). Children who feel anxious about mathematics show significant poorer performances in maths (Ashcraft & Moore 2009). Some researchers argue that negative attitudes towards mathematics and poor maths skills reinforce each other; maths anxiety leads to avoidance, which leads to poorer performance, which increases the fear to fail in maths and so on (Dowker, Bennett & Smith 2012; Krinzinger, Kaufmann & Willmes 2009; Moore et al. 2015). This relation can be found even in preschoolers. In contrast to this, Thomas and Dowker (2000) report that younger children get affected by positive attitudes towards maths rather than by anxiety. Most interestingly, the relation of maths anxiety and performance could be replicated with preschool teachers (Jenßen et al. 2015). Thus, teachers’ attitudes towards mathematics have direct influence on learners’ success, which sheds a broader light on the importance of teachers’ attitudes.

While general factors on learning promote all disciplines, domain-specific predictors and prerequisites specifically affect learning mathematics. All children are born equipped with a set of innate abilities to distinguish quantities, the so-called number sense (Dehaene 2011; Feigenson, Dehaene & Spelke 2004). This innate number sense could be identified as a predictor of mathematical development (Desoete 2015). Magnitude comparison is the ability to distinguish large quantities at one glance, as long as their difference is big enough. We distinguish non-symbolic (e.g. dots) from symbolic (numbers) magnitude comparison tasks, which have different predictive powers for mathematics during childhood (De Smedt et al. 2013). In general, symbolic magnitude comparison tasks show higher correlations with mathematics performance. This applies in particular to learners in primary school.

Another innate ability called subitising is to record small quantities up to four at one glance. Empirical research revealed a correlation between subitising and mathematical performance in Grades 1–3 (Desoete et al. 2009; Kroesbergen et al. 2009). Quantities bigger than four can be recorded at once only if they hold a structure that allows for subitising subsets, which can be added mentally afterwards (Starkey & McCandliss 2014). This process, which is known as conceptual subitising or groupitising, does not only require subitising skills, but also addition facts that can be retrieved easily. Thus, it is no surprise that groupitising is a good predictor of mathematical performance (Arndt et al. 2013; Starkey & McCandliss 2014).

Obviously, a central precursor skill for mathematical learning during preschool is counting. Children usually learn to recite the number word sequence (i.e. the number words in the correct order) before they know what the number words mean (Wynn 1990). This means that children at some age can recite the number words up to five, yet cannot give five counters or enumerate them. Children acquire the concepts of integers one at a time (Le Corre et al. 2006). Firstly, they learn to enumerate one while giving varying numbers when asked for two or more. Subsequently, they learn the meaning of two, three and four. According to Le Corre et al. (2006), this process takes more than 1 year, usually starting in the third year of life. In many studies, counting skills are a good predictor for mathematical achievement in school and outrun most other predictors (Desoete et al. 2009; Sarnecka, Goldman & Slusser 2015).

Counting routines facilitate solving small and simple addition tasks (e.g. 5 + 3 = 8) by counting. However, tasks that involve bigger numbers or are more complex (e.g. 8 + 7 = 15; 18 + 7 = 25; 8 - ? = 3) require more sophisticated effective solving strategies like breakdown (8 + 2 + 5 = 15; 18 + 2 + 5 = 25) or separating the place values. These strategies usually build up on a cardinal number concept (Fritz et al. 2013; Resnick 1983). With this concept, also called cardinality, learners perceive numbers not only as positions on a mental number line, but also as representatives of sets. In this way, a number like 5 gains a property of representing five items, or in other words a certain kind of ‘fiveness’. When numbers are closely related to sets, they can be decomposed on a conceptual level (Fritz et al. 2013). Cardinality is not acquired on its own; it depends on the instruction (Dehaene 2011).

Meerkat Maths contains several training units that not only cover general cognitive skills, but also introduce numbers and important first number concepts. The contents and structure of Meerkat Maths are derived from research results. Table 1 provides an overview of Meerkat Maths. The general and domain-specific knowledge mentioned above is covered in the first two sections of Meerkat Maths.

The aim of the first section (1.1–1.4) is to provide children with the cognitive concepts they need to develop basic numerical competencies. As described above, these concepts prepare for successful learning in Grade 1. Although the chapters deal with different issues, each chapter’s contents are hierarchically based on the preceding chapter’s concepts. As research has revealed, all parts of the first section are precursor skills for mathematical development (Desoete et al. 2009).

The first chapter (classification) is about characteristics of objects like shape or colour. This chapter is designed to enable children to learn how to recognise and compare characteristics. They learn how to combine objects that share characteristics in a group.

In the second chapter (differentiation), the children learn about differences in characteristics. These differences allow finding the odd one out. By detecting common characteristics and classifying them to a group, children are able to exclude objects that do not fit. This concept is important to distinguish relevant from irrelevant characteristics when it comes to counting.

Mathematics is referred to as the ‘science of patterns’ (Devlin 2003). The ability to detect, describe and transfer patterns is a crucial prerequisite for the abstraction that one encounters while learning maths. Thus, in the third chapter (patterns), children learn how to find, describe and continue a given pattern.

Chapter 4 (seriation) enhances the knowledge of patterns and introduces the principle of order within patterns. While patterns are arbitrary and have individual rules, the number word sequence is ordered in a certain way: by number size. In this chapter, the children learn how to seriate – that is, to sort a set of objects by size. This will help them with learning the number word sequence.

Before children learn numbers and counting, they should learn certain basic numerical concepts. These include the approximate comparison of quantities. These concepts are closely related to special verbal expressions that should be taught with the concepts, yet deserve their own section. In particular regarding younger children, magnitude comparison tasks predict later performance (De Smedt et al. 2013). One-to-one correspondence is a crucial prerequisite for the acquisition of counting skills (Sarnecka et al. 2015). Number-specific language is an important learning factor for the development of arithmetic concepts (Dehaene 2011; Prediger et al. 2013).

In the first chapter (2.1), children train their approximate non-symbolic skills (estimating quantities). All humans have innate neural structures that allow comparing quantities that differ in certain ratios. Usually, ratios of 1:2, 2:3 and 1:3 are possible for children to distinguish. As the ability to compare small sets of objects is known as a predictor of later success in numeracy, children train their core systems in this chapter.

Mathematics requires the precise distinction between amounts. Counting requires the precise assignment of number words to counted items (one-to-one correspondence). Before children learn to use number words for counting, they should learn strategies to develop the ability to compare amounts in two or more sets without counting. The one-to-one correspondence is such a strategy. One object in a set is paired with one object in another set. Remaining objects in one of the two groups indicate the bigger set. This strategy is a precursor skill for counting, when each number word is assigned to an individual object.

The third chapter (numerical vocabulary) focuses on verbal expressions that allow describing and handling quantities and their relations verbally. While children learn to distinguish sets by size, they should learn to express differences in sets verbally. As numerical vocabulary such as ‘more’, ‘many’, ‘nothing’, ‘less’, ‘few’ or ‘equal’ is very important, this chapter serves as a checkpoint before learning the number word sequence.

Usually, children between the ages of 3 and 4 learn how to count and thus acquire the first level. On level I (rational counting concept), children know the number word sequence and are able to count and enumerate a small amount of items. This skill is acquired successively for each number (Le Corre et al. 2006). Typical tasks such as ‘give-a-number’ and ‘how-many’ are solvable for them (Wynn 1990). Children on this level know what counting can be used for and how to perform counting. However, their counting routines are not yet flexible and therefore learners on level I can only count forward by one. In addition, they do not yet understand the cardinal aspect of numbers: A number like ‘5’ is seen as a representative for a result of a counting process, not as a representative for a set of five elements (Fritz et al. 2013). As they understand the principle of the one-to-one correspondence underlying the counting scheme, these children can share an even quantity of counters between two persons equally; while doing so, they rely on concrete manipulatives.

After having learned to count, children understand that numbers possess an orientation property: they get bigger. Each number has a predecessor and a successor forming a linear sequence of numbers. Thus, level II (number sequence concepts – line and lists) is characterised by a mental number line representation. To learners on this level, numbers line up on a directional number line (Nuerk, Moeller & Willmes 2015). This mental representation allows comparing numbers: the number that comes later in the number word sequence (and thus is farther right on the number line) is bigger. Children on level II can count backwards and forwards. This enables them to solve simple addition and subtraction tasks by counting (Siegler & Booth 2004). Note that computing by counting is considered a main characteristic and source for maths difficulties (Dowker 2005).

With level III (concept of cardinality), children construct a set-based representation of a number. While on the previous level, numbers were represented by a position on an ordinal number line, on the third level, numbers represent cardinalities (sizes) of certain sets. ‘Five’ now means the cardinality of a set consisting of five items; ‘five’ obtains a certain property – a kind of ‘fiveness’ (Fritz et al. 2013). This concept is seen as an important milestone during the development of arithmetic concepts as it forms the basis for following concepts (Dehaene 2011). The set representation of numbers also allows their decomposition. Five can be decomposed into three and two (or four and one) by splitting up the whole set of five into subsets which add up to five. For example, an addition task like 3 + 5 = 8 can be seen as the merger of the decompositions of 8. A sound and flexible knowledge of number decompositions supports addition task performances (Dowker 2005; Fritz et al. 2013). Besides, ordinal representation is no conceptual basis for number decompositions. The cardinal number concept facilitates the use of efficient computing strategies. Equipped with a cardinal number understanding, learners do not rely on counting as only computing routine, but are able to develop and use sophisticated strategies.

Children are supposed to gain the cardinal number concept during the first half of Grade 1 to have a resilient basis for the arithmetic contents of primary school. Keeping in mind the hierarchical structure of the concept sequence, all learners should acquire level II during Grade-R.- Including level III in Grade-R would clearly overload the Grade-R curriculum. To avoid this, Meerkat Maths Grade-R sticks to the levels I and II. Note that the last chapter (addition & subtraction) is not supposed to train computing by counting. This chapter rather aims at supplying a conceptual basis for a set-based representation of addition and subtraction tasks.

Learning the number word sequence and how to count is the first step into numeracy. Therefore, counting completes Meerkat Maths for Grade-R. It includes learning the first 10 number words; applying the one-to-one correspondence principle, the stable order principle (number words are always used in the same order) and the cardinal principle (the last number name indicates the amount in the set); and using counting to do very simple addition and subtraction calculations in a cardinal sense. Counting skills are one of the most powerful predictors for mathematical performance in preschool age (Desoete et al. 2009; Sarnecka et al. 2015).

Children start learning to count with the number words (Sarnecka et al. 2015). They learn the number word sequence like a poem by heart but do not necessarily have a conception about what the words mean. In the first chapter (number words), children learn to recite the number words correctly.

Knowing the number words does not imply being able to count. For this reason, this chapter (learning to count) specifically focuses on learning to count small amounts up to 10. Counting in this chapter means being able to enumerate a set and give the correct number of items when asked for.

In chapter 3 (flexible and structured counting), the learners learn to count more flexibly. The more counting experience the children get, the more flexible they are at reciting the number word sequence including counting forwards and backwards. While doing so, children understand the orientation of the number words: they get bigger. Hereby they build the representation of the mental number line that allows them to increase or decrease amounts by using the orientation of the numbers on the mental number line. Counting is easier and more reliable, when the set is structured. The children are given structured representations of numbers and learn to count in a structured manner and to structure sets before counting.

Applying the knowledge to addition and subtraction tasks is the heart of the last chapter of Meerkat Maths Grade-R (addition and subtraction). The mental number line coupled with the cardinal principle of counting enables children to solve simple addition and subtraction tasks. However, Meerkat Maths uses this skill to link addition and subtraction to cardinal representation (e.g. with concrete manipulatives). Addition and subtraction shall be seen as set-based operations embedded in number decompositions.

Mathematical training should pay respect to learners’ development to avoid too high as well as too low demands on them (Langhorst et al. 2013). The mathematical content should be structured in a way that suits the learners’ development. As Meerkat Maths is aligned to the hierarchical model by Fritz et al. (2013), its structure is according to children’s development. As seen above, the sections and chapters are clearly structured and build up on each other. Each chapter includes several exercises that are matched to the learners’ prior knowledge.

Why do we teach the way we teach? This question expresses how every training programme needs to legitimate itself. A theoretical basis explains and justifies the learning steps. Moreover, it organises the content in a way that the following contents can be derived from current contents. The theoretical basis of Meerkat Maths was validated empirically in South Africa (Fritz et al. 2014).

The best training programme is likely to fail if the conductors, in this case the Grade-R teachers, are not well educated (Hellmich 2007). As mentioned initially, South African Grade-R teachers often lack important professional knowledge (Van Rensburg 2015). To make a difference where it matters most – in the Grade-R classroom – it is necessary to ensure that certain requirements are met during the implementation of intervention programmes. It is imperative that training programmes consider teachers’ specific needs (Guskey 2002) and also their motivation to learn (Selter et al. 2015). In our implementation study (see below), we combine Meerkat Maths with teacher training. In regular sessions, the principles of Meerkat Maths and how it is used are explained to a group of Grade-R teachers in an active manner. By testing the programme, the teachers take the learners’ role and reflect their experiences subsequently. This approach is supported by a comprehensive manual.

Learning mathematics involves several different skills as described above. Therefore, maths training programmes are supposed to include these skills, too. General cognitive aspects of mathematical learning are taken up in the first section. An important part of Meerkat Maths is the linguistic aspect of mathematical learning. Each chapter starts with a story in which the central mathematical problem is introduced. The ongoing story line leads the learners through the programme and moderates the learning process.- Section 2 includes a complete section regarding mathematical language. In several creativity exercises, children can apply their acquired mathematical knowledge to musical, kinaesthetic or artistic activities.

Children ought to develop a positive attitude towards mathematics as it has deep impact on mathematical learning success (DBE 2011; Moore et al. 2015). Playful activities help to develop such a positive attitude (Barnard & Braund 2016). Therefore, Meerkat Maths offers such playful learning opportunities such as songs, games or a storyboard play in which the stories’ content is re-enacted. Research has shown that it is of great importance to perceive a domain as useful and valuable for learning outcomes (Harackiewicz et al. 2014). Although the problems that the protagonists solve by using mathematics are not part of the learners’ everyday life, they experience the utility of mathematics.

Mathematical instruction should not only convey processes (e.g. solving routines for mathematical problems), but also conceptual foundations and strategy knowledge such as breakdown or decimal structure-based strategies (Langhorst et al. 2013; Long & Dunne 2014). This prevents the development of the so-called inert knowledge (Whitehead 1929). For this reason, Meerkat Maths stresses metacognition and strategy reflection at the end of every exercise.

As children learn at different paces, mathematical training programmes should adapt to their learning speed (Langhorst et al. 2013). Meerkat Maths does not include a fixed time schedule for each chapter. The training takes as long as necessary and thus adapts to the learners’ individual learning speed. Although it is intended that all children of a class learn together, splitting classes and teaching them at different rates is possible. Figure 2 demonstrates how this adaptiveness is realised in Meerkat Maths.

As described, Meerkat Maths is designed in a way that can improve mathematical learning in Grade-R. However, the training programme has to be implemented and evaluated to investigate its efficiency. The focus of an evaluation study usually is to measure how efficient a certain training programme is by comparing the learning success depending on which programme was used. The only criterion is usually the learning outcome of the learners; how the programme can be used in school is mostly neglected (Balzer & Beywl 2015; Petermann 2014). Implementation means to reveal under which conditions Meerkat Maths can be used in schools by Grade-R teachers (Michie et al. 2005). Important questions include the following: does the training programme meet the teachers’ competencies (feasibility), does it meet the learners’ demands (appropriateness) and can schools afford the programme (costs)? Both aspects – learning outcomes and implementation conditions – are crucial to improve mathematical education successfully.

Therefore, we recently started an implementation study in five primary schools in a rural part of Mpumalanga. The implementation contains teacher training sessions that are run by an academic expert and convey the contents of the section the teachers conduct currently in school. The monthly training programme for these Grade-R teachers considers the matter of feasibility. The training is conducted to enable the teachers who work in Grade-R classes to conduct proper mathematical training. In between the training sessions, schools are visited by the academic expert to support teachers. The appropriateness of the course for the learners is gauged by the reports received from the teachers at the training sessions and during the workshops. Workshops are started with a reflection session and teachers are encouraged to share failures and successes. An evaluation study is planned subsequently.