Several studies have shown the influence of mathematical knowledge on both individual opportunities and chances for a self-determined and prosperous life as well as the welfare of nations. Against this background, the contents of maths education in the foundation phase as well as the way in which it is conveyed gain importance. While competence-oriented approaches (e.g. the Curriculum Assessment Policy Statements [CAPS]) state learning goals that all learners should achieve, developmental approaches (e.g. developmental models) describe typical learning trajectories of learners. As both approaches are quite separated, there is a need for bridging the gap between them.

This article aims at revisiting the CAPS critically and comparing contents for early numeracy instruction. A possible alternative to the CAPS is intended.

In this article, we describe a maths learning programme for Grade-R (

Contents of the training programme and the CAPS are compared against the background of empirical research on numerical development and predictors for arithmetic performance.

The results reveal that research based math instruction can be conveyed in a formal training programme.

Keeping in mind the qualifications and training of Grade-R teachers, teacher training is necessarily embedded in the programme. Thus, the described programme is a comprehensive application of recent research for maths classes in the early grades.

International studies reveal that South African learners still show poorer performances in maths than most of their peers worldwide (Reddy et al.

Poor mathematical knowledge implies enormous individual disadvantages for learners. They earn less, are more often unemployed and have fewer chances to work in the field of their choice (Parsons & Brynner

However, research suggests that additional schooling time does not affect learning outcomes positively. It is not only the time learners spend in the school that determines their progress, but also the knowledge they obtain during this time (Hanushek & Woessmann

South African policy is aware that poor maths knowledge of learners leads to severe individual and economic problems. As a reaction to the maths performance misery amongst others, Grade-R was established in 1998 and efforts enhanced within the last years (Van Rensburg

This is of particular interest for maths learning as the acquisition of mathematical competencies is a complex learning process that sets in long before formal schooling (e.g. Carey

Regrettably, empirical findings underpin that South Africa’s Grade-R has only little effect on learners’ school readiness. In particular regarding maths, Grade-R does not substantially improve learners’ competencies or school readiness (Reddy et al.

The main reason for the failure of the current Grade-R is seen in the insufficient professional education of the majority of the Grade-R teachers regarding content and pedagogical content knowledge (Van Rensburg

As Grade-R in South Africa is not yet able to promote learners’ early numerical knowledge, the question how this can be done remains urgent (Long & Dunne

The first question to solve, when originating a maths training programme, is how to choose its contents. The aims determine the contents of a training programme. Theoretical considerations might justify the selection of contents. As research indicates several different predictors and precursor skills, mathematical training can and should be derived from research. In the case of mathematics, central precursor skills are necessary as they are required to understand the fundamental arithmetical operations.

Children entering school have a kind of ‘learning history’ that describes what they learned in their first years (Fritz, Ehlert & Balzer

Abilities and skills that are important for mathematical learning in school can be divided into predictors and precursors. Predictors are abilities that allow – for a group of learners and within a certain range of confidence – forecasting the development of mathematical concepts. The expected development as derived from the predictors is more likely to happen, yet is not determined. Precursor skills are directly linked to mathematical concepts. They precede important mathematical knowledge and are therefore necessary prerequisites for the learner’s development.

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.

One of the oldest predictors discussed is inductive reasoning (Desoete

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

Research on domain-specific language skills is still rather rare. For instance, Göbel et al. (

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

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

Another innate ability called

Obviously, a central precursor skill for mathematical learning during preschool is

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

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.

Contents and structure of Meerkat Maths.

Section 1 | Section 2 | Section 3 |
---|---|---|

1. Basic cognitive concepts | 2. Basic numerical concepts | 3. Pre-cardinal concepts |

1.1 Classification | 2.1 Estimating quantities | 3.1 Number words |

1.2 Differentiation | 2.2 One-to-one correspondence | 3.2 Learning to count |

1.3 Patterns | 2.3 Numerical vocabulary | 3.3 Flexible and structured counting |

1.4 Seriation | - | 3.4 Addition and subtraction |

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.

The first chapter (

In the second chapter (

Mathematics is referred to as the ‘science of patterns’ (Devlin

Chapter 4 (

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.

In the first chapter (2.1), children train their approximate non-symbolic skills (

Mathematics requires the precise distinction between amounts. Counting requires the precise assignment of number words to counted items (

The third chapter (

Predictors and precursor skills allow finding crucial abilities and knowledge that promote learning mathematics in school. However, these skills do not provide a structure, how they should be taught. Empirically validated learning trajectories allow structuring the contents in a way that meets learners’ typical development. This way, a training programme does not force mathematical knowledge onto a child that is not yet prepared, but rather is suited to the learner’s schooling demands.

Fritz et al. (

Usually, children between the ages of 3 and 4 learn how to count and thus acquire the first level. On level I (

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 (

With level III (

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 (

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.

Children start learning to count with the number words (Sarnecka et al.

Knowing the number words does not imply being able to count. For this reason, this chapter (

In chapter 3 (

Applying the knowledge to addition and subtraction tasks is the heart of the last chapter of Meerkat Maths Grade-R (

Mathematical training should follow certain principles (Hellmich

Typical structure of a chapter.

Mathematical training should pay respect to learners’ development to avoid too high as well as too low demands on them (Langhorst et al.

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.

The best training programme is likely to fail if the conductors, in this case the Grade-R teachers, are not well educated (Hellmich

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

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.

As children learn at different paces, mathematical training programmes should adapt to their learning speed (Langhorst et al.

Circular teaching strategy following Fritz and Ehlert (

The training programme aims at supplying a sound conceptual basis for mathematical learning in school. However, learners in Grade-R are supposed to meet certain learning goals as described in the Curriculum Assessment Policy Statements (CAPS) (DBE

Comparison of Curriculum Assessment Policy Statements and Meerkat Maths.

CAPS field | Topic in CAPS | Section of Meerkat Maths |
---|---|---|

Numbers, operations and relationships | Counting forwards (1.1) | 3.2 |

Counting backwards (1.2) | 3.3 | |

Number symbols (1.3) | 3.1 | |

Describe, compare and order numbers (1.4) | 3.2, 3.3 | |

Manipulatives (1.6) | 3.2, 3.3, 3.4 | |

Addition and subtraction with contexts (1.7) | 3.4 | |

Addition and subtraction without contexts (1.13) | 3.4 | |

Mental maths (1.16) | 3.4 | |

Patterns, functions and algebra | Geometric patterns (2.1) | 1.3, 1.4 |

Space and shape | Position (3.1) | 2.3 |

2-D shapes (3.3) | 1.1, 1.2, 1.3 | |

Measurement | Time (4.1) | 2.3 |

Length (4.2) | 2.3, 3.2 | |

Data handling | Collect and sort objects (5.1) | 1.1 |

Represent sorted objects (5.2) | 1.1, 1.3, 3.3 | |

Discus and report on sorted collection of objects (5.3) | 1.2, 1.4, 3.3 |

CAPS, Curriculum Assessment Policy Statements.

The most crucial field of the CAPS is

Most of the topics mentioned in the CAPS are included in Meerkat Maths (see

The aim of this article was to present a training programme that can meet the learners’ development of early numerical concepts. Research has shown the importance of successful mathematical education in school for individual life opportunities as well as social economic development (Hanushek & Woessmann

To develop a sound foundation for mathematical learning, children need basic mathematical knowledge that is appropriately structured. This means that the instruction in Grade-R has to address the development of basic numerical concepts. In particular, numerical, conceptual knowledge, for example cardinality and a resilient operation understanding, is a challenge for mathematical education in school.

Emotional aspects of mathematical learning have a great influence on the development of resilient arithmetic concepts. Arithmetic education in Grade-R has to convey a positive attitude towards mathematics to prevent math anxiety. Otherwise learners are likely to avoid mathematics and thus lack basic concepts because of missing learning opportunities. In conclusion, the contents of Meerkat Maths Grade-R demonstrate which aspects (i.e. cognitive and emotional) of early mathematics education deserve particular attention in Grade-R.

Teaching maths should not focus on the expected outcomes, but – in first line – consider the typical development, and how children acquire mathematical conceptual knowledge, including their precursor skills. Research consistently reveals hierarchies within the mathematical development of children during their first years. Precursor skills determine following developments and learners acquire arithmetic concepts successively following a certain hierarchy (Dehaene

Cumulative structure of learning mathematics.

To summarise, meaningful learning means to build up new learning content on previous knowledge and to link them together. In this sense, Ausubel (

However, instructions have not only to follow developmental trajectories, but also to pay respect to the learners’ individual learning state; otherwise, the new knowledge is likely to be isolated. If newly acquired knowledge is not related to previous knowledge, learners are unable to apply it to new situations (Fritz & Ehlert

All these learning processes need time. If we want learners in Grade-R to develop a real understanding of mathematics, we must not overload them. Giving learners the time they need has two dimensions. Generally, a suitable curriculum should not be too packed, but focus on the most important topics that research indicates. On an individual level, all learners have to be given the time

Finally, the schools and, even more so, the teachers have to be educated for implementing an appropriate curriculum. Teachers need theoretical background knowledge regarding the development of arithmetic concepts. To apply their knowledge to teaching, they need pedagogical content knowledge. Diagnostic competencies allow them to determine learning states and adapt teaching to the learners (Fritz & Ehlert

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

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.

This publication has been developed through the Teaching and Learning Development Capacity Improvement Programme which is being implemented through a partnership between the Department of Higher Education and Training and the European Union.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

E.J.v.V., M.H. and A.F. created the training presented, A.F. provided its theoretical basis in advance, A.F. and M.H. supplied the theoretical background, E.J.v.V. and M.H. described the training programme and the integration of research and curricular demands, while its implications were discussed by A.F. and M.H.