This article reflects critically on the guidance offered to South African teachers in two canonical texts: the Curriculum and Assessment Policy Statements (CAPS) and Mathematics teaching and learning framework for South Africa: Teaching mathematics for understanding (TMU). I make explicit my philosophical orientation, and how ‘teaching mathematics for (relational) understanding’ is evident in both documents. Distinctions are drawn between strategy, representation and procedure, and the progression towards efficient calculation strategies is emphasised (neither of which is clear in the CAPS). The suggestion made in the TMU framework is that teachers can shift from bundling concrete manipulatives for multi-digit numbers to the standard written algorithm in Grades R–3, which contradicts both the CAPS and insights gleaned from mathematics education literature and two learning programmes that have shown positive results at large scale. In particular, the justifications for delaying teacher introduction of ‘break up both numbers’ strategies are discussed. Further, when the ‘break up both numbers’ strategy is used, alternatives to the standard written algorithm which have been found to be more accessible to learners – expanded notation, write all totals, and new groups below – are offered. Ways of using empty number lines and ‘drawing numbers’ to show the 5-wise and 10-wise structure are suggested. These alternative representations for breaking up both numbers are expected to be accessible to teachers for whom the standard written algorithm is a familiar calculation strategy.

Any mathematics education community (researchers, teacher educators, government officials, subject advisors, materials developers, instructional leaders, teachers, trade union officials, etc.) ought to seek out, and provide, theoretically sound guidance on how to approach the teaching of particular topics in mathematics. Ernest (

the aims, goals, and overall philosophy of the curriculum;

the planned mathematical content and its sequencing, as in a syllabus;

the pedagogy employed by teachers, and

the assessment system. (p. 480)

The four aspects are clearly related to each other, and ought to cohere. Within such mathematics communities it is expected and appropriate that there are debates, discussion and contestation over what mathematics is, why it has value, and how it ought to be taught. Absence of disagreement – with various positions and their underlying rationales being openly and hotly debated – ought to be cause for concern. Despite the inevitable contestation, there is simultaneously a need for ‘sufficient consensus’ to steer the way mathematics is approached in schools.

In South Africa the legislated policy framework provides this ‘sufficient consensus’ and is articulated through a national curriculum policy. Such curriculum policy is expected to be subject to revisions over time while maintaining sufficient stability to avoid disruptions to a large, yet fragile, public schooling system. Currently, South Africans are guided by a national curriculum and assessment policy statement (CAPS) for mathematics which gives specific learning outcomes for Grade R – Grade 12 (Department of Basic Education [DBE],

This (TMU) Framework is

As such at the heart of the TMU framework is an intention to support a transformation in how mathematics is taught in South African schools. From the Minister of Education’s perspective, the TMU framework is a contribution to ‘the urgent need to pay particular attention to the development of a new curriculum for initial teacher education, induction and continuing professional development’ (DBE,

It is worthwhile reflecting on the nature of the transformation envisaged for ‘teaching mathematics with understanding’. In this regard I draw on Hiebert (

Most characteristic of traditional mathematics teaching is the emphasis on teaching procedures, especially computation procedures. Little attention is given to helping students develop conceptual ideas, or even to connecting the procedures they are learning with the concepts that show why they work. (p. 12)

In contrast, drawing across numerous successful studies in the teaching of primary arithmetic, Hiebert (

Notwithstanding its articulated focus on Ernest’s (

Given the potential influence of the TMU framework, it is therefore imperative that both it and the CAPS are reflected upon critically. Their similarities and differences should be noted and motivated for. This article focuses on one particular aspect of guidance offered to teacher educators and mathematics teachers in the TMU framework which is a clear departure from that which is offered in CAPS: the ‘standard written algorithm (SWA)’ for addition and subtraction, as illustrated in

An illustration of the standard written algorithm (SWA) for addition.

Following Fischer et al. (

In South Africa, this particular aspect of the mathematics education policy debate relates to how to teach ‘context free calculations for addition and subtraction’ in the Foundation Phase (DBE,

The research questions being reflected on for this article are theoretical. I therefore first make clear my philosophical orientation towards mathematics which underlies my response to two research questions: First, what does research offer in relation to whether, when and how to approach teaching vertical algorithms? And second, how do the teacher guidelines on addition calculation strategies in the CAPS accord with, and differ from, those offered in the TMU framework? These questions are answered in order to reflect critically on the CAPS and TMU framework in order to inform the expected process of strengthening CAPS.

In pondering these questions I drew on mathematics education literature pertaining to learning-teaching trajectories into number and multi-digit addition to frame the article and inform my document analysis.

I conducted a detailed content analysis of the two DBE documents: the CAPS and the TMU framework. I ‘identified visual-quantitative learning supports and written-numeric aspects’ (Fuson & Li,

On receiving feedback from the peer review process for this article, I revisited both of the canonical texts again and engaged further with the literature. In addition I met with the government officials responsible for mathematics curriculum. This meeting provided further details about the purpose of the TMU framework and its status in relation to existing policy. Following this further research and personal engagements I realised the need to, firstly, clarify the conceptual distinctions I made with regard to my use of the terms ‘procedure’, ‘strategy’ and ‘representation’, and, secondly, to include a learning programme (considering textbooks, learner books and teacher lessons plans) perspective. Although the theoretical findings from academic research were of value, I felt I had been remiss in not considering how a couple of large-scale mathematics improvement interventions approached multi-digit addition.

In order to include some engagement with guidance relating to multi-digit addition as offered to teachers (rather than researchers), I purposively selected two well-known early grade mathematics learning programmes:

There is some conceptual fuzziness with regard to distinguishing a range of terms: procedure, method, model, strategy, algorithm, representation and technique. The various interpretations of, and contestations relating to, each of these terms is beyond the scope of this article. As such, I simply make explicit the ways in which these terms are used in the South African CAPS and TMU framework, and then how I use ‘strategy’, ‘procedure’ and ‘representation’ to analyse the documents, in this article.

The CAPS explain that ‘in the early grades children should be exposed to mathematical experiences that give them many opportunities ‘to do, talk and record their mathematical thinking’ (DBE,

To me, the CAPS does not adequately distinguish a strategy (way of thinking), from a representation (how such thinking is recorded), from a procedure (a generalised step-by-step rule or process on how to create a particular representation to depict a particular strategy). Drawing on Kilpatrick, Swafford and Findell’s (

I view a procedure as a specific step-by-step process which can always be followed to implement a particular strategy for a calculation (and hence, over time, procedures can be performed fluently). Procedures may be followed mentally (as internal representations) or communicated using words (mathematics talk to self or others) or by drawings, number symbols, operations, gestures and so on (and so making use of external representations).

The notion of a ‘representation’ as used in mathematics education becomes important here as this draws attention to the particular way in which a procedure is recorded:

As most commonly interpreted in education, mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships… To call something a representation thus includes reference to some meaning or signification it is taken to have. (Goldin,

Thomas (

I view a strategy as a mental process – a particular way of thinking or approach to a calculation. As such strategies are internally represented to self, and then a strategy may be externally represented to others in numerous ways. Importantly, there is not a one-to-one mapping between a strategy and a representation. A particular strategy may be represented in multiple ways. For example to calculate 34 + 7 = … a child may use a ‘count on in ones strategy’, and represent this in many ways: orally, using mathematics drawings or on a number line, as shown in

Exemplar representations and related procedures for a ‘count on in ones’ strategy for 34+7.

The standard written algorithm for addition, as shown in

When critiquing canonical texts or policy documents, the critical commentary ought to make clear its philosophical positioning. There are underlying philosophies of what mathematics is, which mathematics is worth knowing, and how one might expect children to learn. These approaches are based on values and beliefs and necessarily differ by country, by national curriculum and by individual. It is therefore worthwhile to make explicit my philosophical orientation to mathematics learning.

I take mathematics learning to involve learning ways of thinking. This approach has been succinctly defined by Carpenter, Franke and Levi (

Learning mathematics involves learning ways of thinking. It involves learning powerful mathematical ideas rather than a collection of disconnected procedures for carrying out calculations. But it also entails learning to generate those ideas, how to express them using words and symbols, and how to justify to oneself and others that those ideas are true. (Carpenter et al.,

My view on mathematics learning is compatible with how mathematics is defined in the South African CAPS:

Mathematics is a language that makes use of symbols and notations for describing numerical, geometric and graphical relationships. It is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making. (DBE,

Notice the slight shift in emphasis in these definitions. While Carpenter et al. (

It is worth emphasising that I take mathematics learning to involve learning powerful ideas rather than a collection of disconnected procedures for carrying out calculations (Carpenter et al.,

In society today, there is general acceptance that ‘drill and practice’ of taught routines will not prepare children for life in technological society and that teaching approaches need to focus on the links that demonstrate the logical structure underlying numbers and number operations. Rather than being shown how to do written calculations, children are to be encouraged to work mentally to observe patterns to predict results and to talk about the connections that can be made. (Anghileri,

This view has driven much of the mathematics reform agenda, as evident in the NCTM Standards in the USA (see, for example, Kilpatrick, Martin, & Schifter,

The TMU framework emphasises teaching mathematics with ‘understanding’, which could refer to either relational or instrumental understanding. However, the TMU framework refers to a purposeful move to conceptual understanding rather than memorisation of procedures in this extract:

Although the four key dimensions (conceptual understanding, mathematics procedures, strategic competence, and reasoning) are interdependent and should be properly linked to optimise effective teaching and learning of mathematics, it could be argued that more emphasis should be placed on conceptual understanding since this is the metaphorical foundation on which all other dimensions build. The emphasis on conceptual understanding is a purposeful move to address the common teaching and learning practice which is characterised by memorisation of mathematical procedures with little understanding of how they were derived, why they work and when they are relevant. (DBE,

Further evidence of support for relational rather than instrumental understanding appears in other parts of the TMU framework too:

Modern societies and economies are in a constant state of flux. It is no longer sufficient for learners only to learn how to reproduce the steps in the calculations that they are shown by teachers. (p. 15)

If children learn procedures without understanding, their knowledge may be limited to meaningless routines. (p. 16)

Mathematics is not simply a collection of isolated procedures and facts; it consists of a web of interconnected concepts and relationships. If learners are taught mathematics as a series of disconnected procedures that need to be learnt off by heart, they are likely to experience mathematics as meaningless. It will also mean that they have more to memorise which deprives them of the opportunity to develop higher order thinking skills. (p. 18)

These quotes reveal that the TMU framework hopes to contribute to a shift away from reproduction of steps to calculations shown by teachers and meaningless routines, towards mathematics as a web of interconnected concepts and relationships.

The attempt to shift away from learning mathematics as instrumental understanding is not new in South Africa. Spending significant time on mastering taught written routines for quick and accurate calculation in school mathematics has been questioned for over three decades. Olivier (

I do not share Kamii and Dominick’s (

Firstly, I concur with Olivier (

Secondly, having a reliable ‘go-to’ algorithm for enacting a particular calculation strategy (strategic competence) frees up time and attention for other important mathematical processes such as adaptive reasoning, proof and explanation. I emphasise that I do not consider the ‘go-to’ algorithm to imply that the same strategy and representation must be used by all children in a class. I expect that children have their own go-to algorithm which is appropriately efficient for their grade level.

Thirdly, if we consider teachers to be products of their own mathematical learning experiences (see Roberts,

I identified what, in my view, are fundamental insights about the developmental progression of young children’s sense of numbers from counting to efficient calculation strategies. Each key insight is briefly discussed in turn, drawing on the literature informing it. I first consider the various strong justifications to delay the introduction of the SWA as a taught way of recording the ‘break-up both numbers’ strategy. I then discuss the important considerations on how to use columns methods (of which SWA is a very condensed form) when a ‘break up both numbers’ strategy is used.

Researching young children’s thinking in a project referred to as Cognitively Guided Instruction, Carpenter and Fennema (

Count all.

Count on (from the first number, and then from the larger number).

Count up to reach a target.

In this case counting strategies refer to unit counting, rather than counting in groups (such as twos, fives or tens). Calculating refers to more sophisticated strategies which do not use unit counting. Calculating strategies may use:

Counting in groups or counting on in groups (using the medium and large number sequences of counting in tens and hundreds, from any number).

Building on known facts (often knowledge of bonds of five and ten to ‘fill up or make tens’ and doubling or halving).

The relationship between the numbers in the calculation for solving.

All of these single-digit calculation strategies are also used for multi-digit calculations. At first learners need to ‘fill up the ten’ for calculations resulting in a solution that is more than ten. Later the same strategies are used to fill up any multiple of ten. A ‘make a ten’ (or multiple of ten) strategy is included in ‘building on known facts’. Knowledge of the bonds of ten (the whole number pairs which sum to 10: 1 and 9, 2 and 8, 3 and 7, etc.) is therefore central, as is knowledge of breaking down single digit numbers. Using ‘the relationship between numbers’ includes solving a subtraction calculation using an unknown addend. So 9 – 7 = …, is solved as 7 + … = 9. Here any addition or subtraction fact is seen as belonging to a family of equivalent number sentences. In line with this trajectory, Fuson and Li (

It is important to notice that these trajectories do not expand on the shift into formal written calculations. Their focus is primarily on the early grades (R–3).

Efficient calculation strategies – which include use of formal written algorithms – take more than 3 or 4 years of formal schooling. From the Netherlands, Van den Heuvel-Panhuizen (

Working within this Dutch tradition, Treffers and Buys (

Treffers and Buys’s trajectory from counting to calculating.

Level | Descriptor | Approximate age |
---|---|---|

1 | Learning to count | Approximately age 2 |

2 | Context-bound counting-and-calculating | - |

3 | Object-bound counting-and-calculating | - |

4 | Towards pure counting-and-calculating via symbolisation | - |

5 | Calculation by counting where necessary by counting materials | - |

6 | From counting to structuring | - |

7 | Calculation by structuring with the help of suitable models | - |

8 | Formal calculation up to 20 using numbers as mental objects for smart and flexible calculation without the need for structured materials | - |

9 | Counting up to 100 | - |

10 | Calculating up to 100 | Approximately age 9 (Grades 3–4) |

The Treffers and Buys (

This is not the first, nor the only, learning-teaching trajectory for number concept development. For example, Murray and Olivier (

The South African context is important here: most Foundation Phase children learn mathematics in African languages and are introduced to English as an additional language. Starting in Grade 2 children are expected to count orally with meaning up to 100, and to write the number symbols up to 100. Counting orally with meaning up to 100 means that children have to create an association between five aspects of each number: the visual stimulus, the chain of sounds for the spoken number word, the ten-wise place value structure of the number words, the written number words, and the number symbols (see Fuson & Li,

Five aspects of oral counting multi-digit numbers with understanding.

Using multi-digit number names with meaning requires children to notice word parts and associate these with decades and ones (Fuson & Li,

Becoming secure with these five associations between the visual stimuli (of the real or imagined aggregate, the written number word, the written number symbol) and oral stimuli (of the chain of sounds for the spoken number name) takes time. When reading and writing is not yet fluent this is more difficult. When two languages are at play, additional time is required. All of these associations – together with fluency in reading and writing number symbols and operators – are prerequisites for formal written calculation methods. As a result, the introduction of formal written methods ought to be delayed until use of written number symbols and spoken number names is secure.

There is much research that coheres on the finding that using a column method for addition and subtraction encourages a digit-wise conception rather than a multi-digit conception of large numbers. Fuson et al. (

Various studies have shown that children may have an adequate multi-digit conception which they use for addition and subtraction calculations that are presented horizontally or in word problems. However, when presented with the same calculation vertically, they use a digit-wise conception and make errors (Fuson et al.,

Common errors in using the column algorithm are also documented by Kamii and Dominik (

In their small-scale study of 21 Grade 4 students, Flanders, Torbeyns and Verschaffel (

I therefore note that the SWA in the early grades (Grades R–3) has been found to encourage the incorrect digit-wise conception of the place value of multi-digit numbers and has been found to be error prone. Retaining numbers as whole numbers (such as 37 being 30 and 7) and not as digits (3 tens and 7 ones) in any partition is encouraged to avoid the digit-wise conception of multi-digit numbers.

Given the above insights, it is not surprising that if it is to be used, the introduction and development of SWA needs to be slow and incremental to ensure that a robust conceptual understanding of place value is first established. Several alternative representations to record a ‘break up both numbers strategy’ appear in the education literature. Some of these make reference to manipulatives or concrete materials. Shifts towards arranging discrete objects or using structured materials in 5-wise and 10-wise groups are encouraged (Roberts,

In my experience, particularly with an urban small class setting, disruptive learner behaviour was a significant feature of the mathematics classroom (See Roberts & Venkat,

Mathematics drawings as a representation of place value.

Futon and Li’s (2009) argument and suggested structured drawings draw on a review of Asian texts (Chinese, Korean and Japanese) as well as their empirical research with children using the

Mathematics drawing and variations on SWA.

Rather than have children overload their working memory, teachers should encourage children to:

record intermediate steps (rather than expecting these to be held mentally)

work flexibly in terms of either from ‘right to left’, or ‘left to right’

use structured drawings, together with expanded written methods.

Methods that record intermediate steps (rather than expecting these to be held mentally), and that allow for flexible working in terms of from right to left, or left to right, are advocated for.

The two learning programmes reviewed, which have shown positive effects at a large scale, both prioritise learner invented strategies when operations are first introduced. They then guide learners towards multi-digit addition using a ‘break up the second number’ strategy.

In the

The authors of

Example of ‘Write all totals’ or partial sums algorithm in

Notice that ‘write all totals’ is a four-step procedure (add the 100s, add the 10s, add the 1s, add partial sums) which is recorded using columns. A digit-wise conception is avoided, as the 2 hundreds + 4 hundreds is recorded as 600 (and not as 6 in the hundreds column). Similarly 60 + 80 is recorded as 140 and not as 1 hundred, 4 tens and 0 ones. This representation in columns also allows learners to work either from left to right (starting with the 100s), or from right to left (starting with the 1s). Bell et al. (

Algorithms are harmful to children’s development of numerical reasoning for two reasons: (a) They ‘unteach’ place value and discourage children from developing number sense, and (b) They force children to give up their own thinking. Children’s natural way is to think about numbers from left to right. However, algorithms require them to give up this thinking and to proceed from right to left and to treat each column as ones. (p. 58)

The procedure here also allows flexible working from left to right, or from right to left. Notice here that there are two important reductions of cognitive load (compared to the SWA). Firstly, all addition is done before any exchanges take place. So rather than alternating processes (add then exchange as needed) the child can focus first on adding, and only later on exchanging. Secondly, the way of recording exchanges is documented in the relevant column to try and avoid a digit-wise conception of place value (so 16 ones is recorded in the ones column, before it is exchanged to be 1 ten and 6 ones).

As a result of the above distinction between strategy and representation, when analysing the CAPS and TMU framework, strategies for addition were distinguished from the various representations that could be used as means of communication, as shown in

Strategies and representation for addition in CAPS.

Strategies for addition calculations | Representations of addition calculations (how the strategy is recorded) |
---|---|

Count all Count on Break down the second number Break down both numbers Use known or derived facts |
Concrete apparatus (counters, bead - strings, abaci) Number lines (structured or empty) Number sentences using number symbols and operations A whole-part-part diagram A triad or number triple Calculator Horizontal number sentences Column or vertical methods |

CAPS, Curriculum and Assessment Policy Statements

The addition strategies (and their related grade progression) offered in the CAPS are shown in

Grade progression for addition calculation strategies in CAPS.

The strategies, and the related progression, in CAPS are recognisable from the mathematics education literature. There are three important specifications relating to the expected progression for addition and subtraction techniques which are clearly stated in the CAPS and then exemplified in the clarification notes offered to teachers:

Adding and subtracting are considered together – with emphasis placed on using these operations as inverses. As such the structural relationship between a number triple (like 5-3-2) is made explicit. Teachers are expected to emphasise that subtraction can be checked using addition (5 – 3 = …, can be considered as 3 + … = 5). There is therefore an emphasis on families of related number sentences for each addition or subtraction fact.

In Foundation Phase ‘break down the second number’ is a strong focus as this can be utilised for both addition and subtraction without the need for regrouping.

Adding and subtracting in columns is delayed to Grade 5. In term 1 a ‘write all totals’ representation for the ‘break up both numbers strategy’ is presented. From term 2 onwards this is complemented by a SWA.

More detail about the CAPS suggested progression for addition and subtraction strategies is evident in the clarification notes for teachers.

Addition strategies presented in CAPS clarification notes and the TMU framework.

It should be noted, however, that progression in relation to the strategies for addition and subtraction (and the flexible use of various representations to record these strategies) is not made explicit in the CAPS document. It only becomes apparent when the entire document is analysed in relation to the expectations at each grade level.

When comparing CAPS to the TMU framework it should be noted that the latter only offers a limited number of exemplar tasks. The examples chosen for Foundation Phase relate to addition and subtraction of whole numbers, whereas the examples chosen for Intermediate Phase relate to rational numbers (common fractions in particular). It is therefore to be expected that the TMU framework is less comprehensive than the CAPS. It is nevertheless instructive to note which example types were omitted from the TMU framework exemplars. The five main strategies in the CAPS each feature in the framework. The particular strategies not referred to in the TMU framework are made explicit in

Appearance of Foundation Phase strategies in CAPS and TMU framework.

Main strategies | In both CAPS and the TMU framework | Only in CAPS | Only in TMU framework |
---|---|---|---|

Count all | build up and break down collections |
build up numbers in ones |
- |

Count on | in ones |
from the bigger in tens and ones |
- |

Break up the second number | fill up the tens use expanded notation |
into any parts |
- |

Break up both numbers | use place value horizontally |
- | use place value vertically (SWA) |

Build on known or given facts | pairs that make ten near doubles rounding off and compensating use given facts (inverse relationship and ‘if… then…’) |
use the relationship between addition and subtraction halves |
- |

CAPS, Curriculum and Assessment Policy Statements; TMU, Mathematics teaching and learning framework for South Africa: Teaching mathematics for understanding; SWA, standard written algorithm.

The TMU framework has much in common with the CAPS. By way of example, both the CAPS and the TMU framework recognise the need for learners to use any strategy, but that teachers should encourage them away from unit counting, towards more efficient strategies, and ways of recording calculations.

The TMU framework offers an example of 29 + 15 = … where learners are expected to ‘use any method’. It also includes examples of ‘break up both numbers’ strategy using manipulatives (p. 26 and p. 27), ‘break up the second number in expanded notation’ strategy recording this using number sentences and on a number line (p. 27), and ‘break up the second number’ to fill up the tens, recording this on a number line (p. 28).

A major distinction between the TMU framework and CAPS is the suggestion in the TMU framework for teachers to introduce the SWA in the Foundation Phase (Grades R–3). In the TMU framework the SWA is offered as a way to record 34 + 37 = … using a ‘break up both numbers’ strategy (shown in

An example of SWA to solve 34 + 27 = … in the Foundation Phase exemplars of the TMU framework.

This differs from the guidance offered to teachers in the CAPS, where any use of written methods in columns is delayed to Grade 5 (DBE,

First example of SWA in CAPS for Intermediate Phase.

In addition, in the TMU framework the SWA is the only representation for the ‘break up both numbers’ strategy, and no examples of expanded written methods are offered. As such the TMU makes a major departure from CAPS when it suggests that teachers should make use of concrete apparatus to work with ‘breaking down both numbers’ into expanded notation in order to introduce the SWA in Foundation Phase.

I have shown that both the CAPS and the TMU framework emphasise the need to teach mathematics for understanding. I have argued that the desired understanding is relational understanding (‘knowing what to do and why’) not instrumental understanding (‘rules without reasons’). I have argued that this is in line with the mathematics reform agenda where the desired transformation of mathematics teaching is away from teaching meaningless procedures towards teaching mathematics that builds directly on learners’ entry knowledge and skills, provides opportunities for inventions and practice, focuses on the analysis of (multiple) methods, and expects learners to provide explanations.

I point out that the CAPS does not clarify the distinction between strategies, representations and procedures. I suggest, drawing on the TMU framework, that distinctions between a strategy (way of thinking), a representation (how such thinking is recorded), and a procedure (a generalised step-by-step rule or process on how to create a particular representation to depict a particular strategy) ought to be made explicit. I also note that the expected progression in relation to the strategies for addition and subtraction (and the flexible use of various representations to record these strategies) is not made overt in the CAPS document and is only evident when a grade by grade level analysis of strategies is conducted.

I make explicit my belief in the value of inculcating both conceptual understanding and procedural fluency. I emphasise that this does not mean I think algorithms should be the starting points of mathematical learning. Nor does my belief imply that the elegant and formal SWA ought to be a desired destination for all children. In fact, we know that the highly condensed SWA was used in a time when accuracy and efficient were most prized. In our current context – with ubiquitous access to calculators (in computers and mobile phones) – efficiency and accuracy are not as highly valued. The process of inventing, refining and reflecting on algorithms coupled with the ability to communicate with others about multiple strategies, and various ways of representing these, is of far greater importance and value. It is for this reason that I attend carefully to how I think formal written algorithms ought to be introduced to young children, and why I attend to various options for recording these (likely to suit a wider range of children), which I present in

We know from both local and international literature that early introduction of the SWA for addition and subtraction supports the incorrect digit-wise conception of place value for multi-digit numbers. We also know that deep conceptual understanding of place value is predictive of future mathematics attainment, that there are five correct conceptions of place value, each of which takes time to develop, and that the development from rote counting to efficient calculation takes about 7 years. The South African curriculum (CAPS) seems to build on this literature as it encourages a slow progression in relation to the development of the place value concept and progression with regard to efficiency of calculation strategies. In contrast, the TMU deviates from both the CAPS and the international literature. It proposes that the SWA for addition can be modelled using manipulatives, and then used as a written method with digit-wise tens and ones columns in Foundation Phase.

In addition, the TMU does not include the expanded column methods (such as ‘write all totals’ – either horizontally or vertically, which appear in CAPS) as possible alternatives to the SWA. The expectation created in the TMU framework is that teachers can shift from bundling concrete manipulatives for multi-digit numbers to the SWA in Grades R–3. This contradicts both the CAPS and insights reported on above. The TMU framework is also discordant with the two learning programmes –

Given how much conceptual knowledge for place value is to be constructed – and its critical importance in mathematical progression – the move from concrete manipulatives to the SWA suggested in the TMU framework is too rapid. The introduction of the SWA in Foundation Phase, when it is only in Grade 2 that notions of place value are beginning to be constructed by learners, is too soon. This creates a lack of clarity in our policy landscape, which is only partially clarified by the repeated assertion that the TMU framework does not replace CAPS.

So, if asked whether, when and how to teach the SWA, I would respond as follows. While some mathematics educators reject the teaching of SWA in its totality, I think exposure to algorithms in their historic context is worthwhile. As the SWA is familiar to South African teachers, I think it is worthwhile for children to understand it, without emphasising speed and accuracy. However, I do not think it should be taught in Foundation Phase. I concur with the CAPS that the SWA should only be introduced in Grade 5 when place value in two languages is secure. In Foundation Phase I would teach multi-digit addition by first encouraging children’s invented strategies, then offering structured mathematics drawings and expanded methods. I would use ‘write all totals’ as a ‘go-to’ algorithm for the whole class. Only once ‘write all totals’ is secure would I introduce the SWA as an optional condensed method, which was used at a particular time in history, when speed and accuracy were valued.

This article results from a symposium at the National Science and Technology Forum in August 2018. I wish to thank all the mathematics education colleagues who took part in the robust debates at, and following, this symposium. In addition, I received invaluable critical feedback, which substantially changed this article, from colleagues within the curriculum unit of the DBE, and academic colleagues who reviewed the first version of this article. Discussion with international colleagues emanating from an oral communication on this issue at the Psychology of Mathematics Education (PME-43) conference was also very helpful. All of these role players share a common concern for improving mathematics in South Africa which was evident in their critical engagement and generosity in sharing ideas.

I declare that I have no financial or personal relationships that may have inappropriately influenced me in writing this article.

I declare that I am the sole author of this article.

No children were involved in collection of empirical data for this article; therefore no ethical clearance was applied for.

This research received no specific grant from any funding agency in the public, commercial of not-for-profit sectors.

Data sharing is not applicable in this article as no new data were created or analysed in this study.

The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliate agency of the authors.

The number symbols 0 to 9 which are used to represent all numbers. | |

Naming and thinking about a multi-digit number as comprising of its digits. 356 is ‘three five six’. |

Depicting a number as comprising units (or ones). | |

Depicting a number as comprising groups of 5. | |

Depicting a number as comprising groups of 10. This may include showing each ten as comprising 2 fives. |

7 | |

6 | 1 |

The SWA, which is familiar to many adults, is a very condensed version of a ‘break up both numbers’ strategy. But the mathematics education literature, and the example learning programmes, encourage invented strategies, and a ‘break up the second number’ strategy, before children are exposed to the ‘break up both numbers’ strategy. In this Annexure, I show how each strategy may be represented.

If being introduced by a teacher (and not brought or invented by a learner), this strategy should precede the ‘break up both numbers’ strategy. This strategy can be used for addition and subtraction and no regrouping (exchanging, borrowing or carrying is needed). This strategy can also be used flexibly for missing addend and missing subtrahend problems (such as 189 + … = 346; or 346 - … = 189).

Notice that the ‘break up the second number’ strategy means you don’t break up the first number as well. In these examples you always start with 189, which is the first number in 189 + 157 = …

If being introduced by a teacher (and not brought or invented by a learner), this strategy should be used after learners are secure with the ‘break up both numbers’ strategy for both addition and subtraction. Depending on the numbers, this strategy may require regrouping (carrying or borrowing). Also, this strategy can be difficult to use for missing addend and missing subtrahend calculations, such as 189 + … = 346; or 346 - … = 189).

Notice that both 189 and 157 are broken up for 189 + 157 = …

A ‘break up both numbers’ strategy, rearranging manipulatives or drawing quick 100s with squares, quick 10s tens with lines, and quick 1s with dots, then regrouping.

A ‘break up both numbers’ strategy, rearranging manipulatives or drawing quick 100s with squares, quick 10s tens with lines, and quick 1s with dots, then regrouping.

A ‘break up both numbers’ strategy, represented using rows and writing all totals (as whole numbers), then adding in columns.

A ‘break up both numbers’ strategy, represented using columns and writing all totals (as whole numbers) below.

A ‘break up both numbers’ strategy, represented using columns, writing all totals first (in their place value column), and then exchanging one place at a time.

A ‘break up both numbers’ strategy, represented using columns, exchanging using digits at the top of in place value columns and making use of digit-wise place-value (1 hundred 8 tens and 9 ones).