devika.naidoo @wits.ac.za

Abstract

Both South African and UK, US and Australian researchers cite inadequate pedagogy as a cause of poor learning and performance in numeracy in the early years. This paper examines a numeracy lesson to show one aspect of inadequate pedagogy. Following a qualitative research approach data collection included observation of lessons, writing fieldnotes and video-recording of lessons. The analysis has been framed by the concept of random and controlled variation derived from variation theory (Watson & Mason, 2006). Variation theory proposes structured variation as a mechanism for supporting learners to observe and discern central features or patterns or regularities in numbers. In relation to key levels of how variation can be structured to support numeracy learning, data is presented from a lesson to suggest that the dominance of random variations in consecutive steps of the pedagogic practice are key aspects of inadequate teaching in the lesson. The random variations, lead to the lack of provision of perceptual experiences that make conceptual learning, generalization and derivation of answers through sense realtions ?? possible. Together, these weaknesses negate firstly possibilities for learners immediate discernment/sense-making, conceptual learning, generalisation and secondly, the grounding necessary for cumulative learning to come. The implications of the analysis for teacher lesson planning are raised.

Random variations and unproductive range of change in a Grade 2 Numeracy lesson

Introduction

There has been an attempt to raise standards of numeracy in first world countries by placing emphasis on educational policy and practice (Wright et. al., 2006). While initial gains were made they have not been sustained as children continue to experience difficulty in numeracy. The causes of such difficulties include individual learner characteristics, inadequate or inappropriate teaching, absence from school and lack of pre-school or home experience with mathematical activities and language (Wright et al., 2006). Within the broader context of ongoing poor performance in numeracy in South Africa, this paper focuses on one aspect of what has been cited as one of the causes of such poor performance – inadequate and inappropriate teaching (Wright et. al., 2006). In this paper the focus is on one aspect of ‘inadequate and inappropriate teaching,’ the orientation towards basic numeracy skills as ends in themselves, rather than as means towards more sophisticated conception of patterns and regularities and generalizations that are transferable to other situations.

The poor performance of foundation phase learners in numeracy in historically disadvantaged SA schools is well known (Fleisch, 2008; Schollar, 2009; Reddy, 2008). According to the Director General, the DoE in SA:

…has measured learner performance in …numeracy at grade 3 and 6…the evaluations have shown that our learners are performing poorly – their …numeracy levels are below the required levels for their age and their grades (DoE 2008).

Furthermore international evaluations have corroborated the national findings and have further criticized that poor performance holds ‘even [for] children whose conditions for learning are far worse than ours [SA]’ (DoE, 2008, p. ?). Other studies noted the higher scores of learners in Botswana and Kenya (Taylor, 2008).

Studies of primary teaching of numeracy include research of teacher questioning at a high cognitive level; teachers beliefs; lesson organization; grouping; extent of whole class teaching and other aspects of classroom practice (Askew and Brown, 1997). Further that effectiveness of teachers was related to ‘connectedness of knowledge and beliefs, enabling teachers to relate their classroom practice to a variety of mathematical ideas and representations (Askew and Brown, 1997). The emphasis in this paper is on how the teacher sets a goal, the learning of a concept, and how the teacher engages in subsequent practices that enable the learning of the concept. Also on how the teacher establishes connectivity across the concept to be learned and the activities done that are different yet connected. Schifter (2001) defines teaching practices as being skilful, patterned regularities that occur in teachers’ classrooms. Molefe and Brodie (2010) distinguish two kinds of practices: mathematical practices and teaching practices. Mathematical practices include symbolizing, representing, justifying and communicating mathematical ideas. Teaching practices involve particular approaches or methods that teachers employ in order to teach. Notwithstanding these distinctions the view taken in this article is that the mathematical practice emanates from the teaching practice and the interest in this paper is to analyse the mathematical practice in terms of the presence of patterned regularities.

This paper presents an analysis of an investigation into one of the causes of poor learning of number in the early years, that of ‘inadequate or inappropriate teaching’ framed by the notion of controlled variation through well-structured mathematical exercises. According to Wright et al a basic question that teachers have to address when planning their lesson is: ‘how to establish the linkages and generalities in the learning, which will allow progress to be made’ (p. 1). The focus in this paper is on examining the ordering of content, and explaining the ways the teacher linked different tasks within the lesson.

Conceptualizing pedagogy for the learning of number

The study is informed by Watson and Mason’s (2006) view of learning mathematics. While Watson and Mason focus on secondary school teachers practice, it is argued in this paper that foundation phase lessons would be enhanced by attention to what change in topic and focus is productive for learning and by controlled rather than random variations in examples selected to be employed to teach the particular number topic. Learning has been defined as sense-making (Watson & Mason, 2006) and as development of mind (Hirst & Peters, 1970). Learning mathematics is conceptualized as an active, problem-solving process, where each child has to construct his or her own mathematical knowledge and develop mathematical concepts as they engage in mathematical activity’ (Wright et al., ) 2006, p. ? where does the quote start?).

Concepts have been themselves the object of analysis (Bergman, 2010; Hirst & Peters, 1970). From the mathematical perspective Skemp defines a concept in terms of being able to classify new data based on having learned the similarities that encompass the concept:

The criterion for having a concept is ….that of behaving in a way indicative of classifying new data according to the similarities which go to form the concept (Skemp,1971, p. 27).

To enable sense-making in mathematics, mathematics education researchers, Watson & Mason, emphasise teaching that is based on ‘a well-structured mathematical exercise’ (p. ?). Mathematical exercises are seen as a collection of procedural questions or tasks that enable the learning of a concept. They further define an exercise as ‘a single object, with individual questions seen as elements in a mathematically and pedagogically structured set’ (p. 91). A well-planned and well structured exercise enables multiple levels of learning or sense-making: it enables the concept to be constructed / conceived by learners; it advances abstract understanding; and it enables generalisation or the applicability of the concept to other situations.

A well-planned mathematical exercise includes the selection and sequencing of learning experiences that provide sensory perceptions structured in ways that enables ‘sense-making’ by learners. Since learners’ perceptions of what is on offer in the mathematics classroom are the central starting point to enable sense-making, teachers ought to give attention to what perceptions learners should have, judged from learner responses, when planning their lessons. The exercises should be selected and sequenced to enable learners to have the desirable sensory perceptions. Therefore, when teachers plan lessons and exercises, consideration ought to be given to ‘what learners might do, what they might see, hear and think, and how they might respond’ as these are initial sensory perceptions that form the basis for meaningful learning. Watson and Mason refer to this practice of conjecturing or predicting what learners will experience and respond to in advance of the lesson, during lesson planning, as a hypothetical learning trajectory (HTL). Since learning is an internal cognitive process, they advise that it’s better to work with what can be observed in a lesson and so prefer hypothetical learner response.

How could sense-making be enabled? Watson and Mason introduce the notion of range of change and dimensions of variation. Marton and Booth (1997) argue that the ‘starting point of making sense of any data is the discernment of variations within it’ (page ?). Therefore ‘tasks that display constrained variation are generally likely to result in progress in ways that unstructured sets of tasks do not’ (Who is saying this??? page ?). This requires the careful structuring of tasks that progressively reveal the regularity through controlled variation of variables that are visible to be perceived. Teachers can therefore aim to constrain the number and nature of the differences, from one example to the next, and thus increase the likelihood that attention will be focussed on mathematically crucial variables. The dimensions of variation ought to be carefully controlled because the aim is to promote perception of the pattern/regularity that may enable generalisation to other similar topics. Marton’s ?? Martin and Booth? notion of ‘dimensions of variation’ offers a way to look at practice in terms of what is available for the learner to notice.

For generalisation of conceptual learning ‘ ? where does this quote end? the learner experiences a shift between attending to relationships within and between elements of current experience (the doing of individual tasks) and perceiving relationships as properties that might be applicable in other situations. Generalisation according to Watson and Mason is the sensing of the possible variation in a relationship and as ‘shifting from seeing relationships as specific to the situation, to seeing them as potential properties of similar situations’ (p. 94). They caution that any task can focus learners attention to the immediate ‘doing’ and while this is important to enable understanding of the object in its own right special steps ought to be taken to promote further engagement towards abstraction, rigor, or conceptualisation beyond that necessary for the current problem.

In this way a mathematical concept is constructed by the learner. So the learner may construct understanding of the number 16 or what the number 16 is by doing individual tasks, such as addition and subtraction to make the number 16. To enable a higher level of learning the specific pairs of numbers that are added or subtracted to make 16 are varied in structured ways so that the patterns becomes perceivable and hence possiblility of generalisation when adding and subtracting other numbers to make 16 or when proceeding to unpack the numerosity of other numbers.

Several different experiences in which a learner may detect similarities, and may hence conceptualise about the similarity, are necessary for such perception to happen. Simon and Tzur (2004) focus on designing sequences of tasks that invite learners to reflect on the effect of their actions in the hope that they will recognise key relationships.

I interpret the above as a paraphrase of what Simon and Tzur says.

BUT IT IS NOT! It is the exact words of Watson and Mason (p. 93) below!

Now I must wonder whether you are merely copying Watson and Mason, and whether, by referencing Simon and Tzur, you have actually consulted Simon and Tzur yourself? If not, this is highly a-scholastic!?

Micro-modelling refers to the processes of trying to see, structure and exploit regularities in experiential data, so that learners are exposed to mathematical structure affording them enhanced possibilities for making their own sense of a collection of questions or an exercise. Exercises should be structured in ways that ‘desirable regularities’ might be perceived by the learner through engagement with the task.

You write the above paragraph as your own words and therefore your own construct!?

BUT IT IS NOT! It is Watson and Mason’s construct, so you must reference them!

It is not even a paraphrase, but is in fact an absolute DIRECT QUOTE from page 93:

This is very poor scholarly work!

Watson and Mason advise that attention be given to what regularities are available to be perceived by learners and which are most likely to be observed in any given task? What conditions make perception of the regularity possible? The regularity is more likely to be discerned ‘ ?? where does this quote end? if its variation is foregrounded against relative invariance of other features. If everything is varying, then regularities are not perceived and nothing will be discerned. Then the regularity ought to be visible so learners see it. Constructing tasks that use variation and change optimally is a design project in which learner responses leads to further refinement and further precision of selection and sequencing of tasks (Gravemeijer, 2004)

Again – you are copying what Watson and Mason says on p. 100, below.

I must wonder whether you yourself have read Gravemeijer!!!!

Given, the advocacy of permissible range of change and structured variation to support conception of patterns and regularities, the lesson was analysed in terms of the range of change of topics within the lesson and in terms of what was fixed, what varied and how they varied within each episode taught. This would enable attention to what regularities were made available to be perceived and discerned by the learner. The episodes in the lessons and the work done on the board were analysed in terms of what aspects were fixed, what was varied and how they varied throughout the lesson.

Data sources

This paper arises from the context of a larger research project of grade 2 numeracy to predominantly Black African learners in 10 government schools in Gauteng? serving historically disadvantaged learners. Using a qualitative approach data was collected through observations of teachers teaching numeracy in grade 2 at the beginning of 2011. Data collection included observation of lessons, writing fieldnotes and video-recording of lessons. Baseline data was collected through lesson observations were carried out in all the Grade 2 classes. Where informed consent for video taping was given, lessons were video recorded with a focus on the teacher. The aim in the baseline study was to start building a picture of the teaching practice in Grade 2 numeracy classrooms.

In this paper, data is drawn from the transcript created from the video record of one Grade 2 teacher’s Numeracy lesson. Without making conclusive claims in this initial paper, the randomness that characterized this lesson appears to be representative of the majority if not all the lessons observed. Other lessons would be analysed according to the analytical framework used to analyse this lesson to see whether this initial claim is representative of a greater number of lessons observed and video recorded. In this transcript, teacher talk, teacher-learner interactions, and across these utterances the objects and representations that were being referred to – sometimes in the form of writing on the board, and at other times, in the form of representations created by learners in their workbooks or with manipulative objects like abacuses or cubes - have been captured. An overview of the enacted lesson is provided below, in the form of a summary followed by a demarcation of episodes distinguished by a specific foci and different from another episode.

Lesson overview

The aim of the lesson was to enable learners to understand what the number 16 is by engaging in addition to make 16, subtraction to make 16 and repeated addition to make 16. The teacher began the lesson with getting all learners to count forward from 1 to 100 and then backward from 100 to 1 and pointing out the numbers on their number chart in each case. The teacher then asked the pupils to use their abacus to work out a word problem: ‘Sipho has 9 sweets. Nomsa has 4 sweets. Mary has 3 sweets. How many sweets are there altogether?’

The teacher then wrote 16 in numerals and words on the board. She asked a learner to draw 16 objects on the board. The learner drew 16 circles. The teacher then drew 16 asterisks on the board in rows – (5 in top line, 4 in second line, 5 in 3rd line and 2 in the last line). She then put up a sheet on the board where she had drawn 16 asterisks in a rectangular format with different numbers of asterisks in each row from the rows on the board. The learners were then asked to find 16 in numbers and words from some cards. The teacher then asked for two numbers that add to 16. Learners answered and the teacher asked the class to verify answers offered by individuals by counting out the numbers on their abacus and adding them up. The correct sums were written up on board.

More context …

I found the this page below in Foundations for Learning Lesson plans Grade 2

While it does not ditract from your argument, I merely ask if the observed lesson is based on the teacher’s own lesson plan, or is this poor teacher actually trying to blindly implement the given official resource??

I wonder, instead of ‘blaming’ this poor teacher for ‘inadequate teaching in the lesson’, should we not blame the poor guidance and resources provided by the DOE??

Lesson Analysis

Identifying episodes

An episode is defined by the introduction of a ‘new’ task, although some tasks are clearly related to each other. Following Mason’s (2004) distinction between task and activity, the episode summary begins with an outline of the task, and then details some of the activity that ensued in classroom work on the task.

Time

Episode no/format

Content outline

0 :00

1/wh class, oral

Task: Forward oral count from 1-100, followed by backward oral count 100-1.

Activity: Learners instructed to follow the numbers on their 100 squares

5:20

2/wh class, oral, some graphic

Task: Word problem up on the board: ‘Sipho has 9 sweets. Nomsa has 4 sweets. Mary has 3 sweets. How many sweets are there altogether?’

Activity: Teacher tells the class that they can use their abacuses. One child, almost immediately can be heard to call out 16. Learners are then told to count out 9, and then 4 and then 3 on their abacus. Most are able to do this, but some appear to struggle. The class is then told, in chorus, to count how many there are altogether. Counting from 1, they get 16.

7:40

3/wh class oral & graphic

Task: Teacher asks for a learner to come up and write 16 on the board.

Activity: A learner does this correctly

8:20

4/wh class, oral, some graphic

Task: Learner asked to draw 16 objects on the board.

Activity: A boy draws 15 circles – 14 on one row and 1 below. In count with class, the bottom circle is counted twice to get 16.

9:45

5/ whole class, oral and graphic

Task: Child asked to count out 16 counters from a box.

Activity: Boy counts out the bottle tops from a bucket on to the floor: 1, 2, 3 …. 12, 13, 16. Class heard to disagree. Boy then counts out another three bottle tops accompanied by the words: 14, 15, 16. There are now 17 bottle tops in the line. Teacher asks him to count out the bottle tops from the end where he started laying out the bottle tops. Boy makes an error in counting – skips a counter at 5, 6, and therefore ends up at 16. Teacher acknowledges with a ‘Very good’.

11:20

6/ whole class, oral and graphic

Task: Teacher writes ‘Number’ with 16 below it on the board. She then writes ‘Word’ up on the board. She asks learner to write ‘the number name’ in words below it.

Activity: A boy writes ‘sixteen’ on board..

13:35

7/ whole class, oral and graphic

Task: Teacher adds a heading ‘Picture’ next to the previous two headings. Below, she draws 16 asterisks on the board (5 in top line, 4 in second line, 5 in 3rd line and 2 in the last line) and then puts up a pre-prepared poster on board with 16 asterisks in a rectangular format, but again with different numbers of asterisks in different rows, 6 in top row and 5 each in 2nd and 3rd rows.

Activity: A learner is called up to count the asterisks – he does this correctly on the 2nd attempt (repeats 14 twice on 1st attempt and gets 15).

15:00

8/whole class, oral and graphic

Task: Two learners are asked to find 16 in numbers and words from some cards with numbers 1-20.

Activity: A learner finds the correct cards and sticks cards on the board.

16:30

9/whole class, oral and graphic

Task:Teacher asks for two numbers that add to 16.

Activity: 8 + 8 offered. Teacher writes 8 + 8 = on the board. Another learner is asked to verify that this makes 16 by counting out 8 + 8 bottletops. Learner arranges these in two rows of 8, then counts all and gets 16. Teacher writes in 16 on her sum on the board, and then asks for ‘another two numbers’ that give 16 when added. 10 + 6 offered by lr.?? Teacher asks class to verify answers offered by individuals by making the two numbers on their abacus and counting. Several learners have made 10 and 6 on abacus, and several heard calling out 16. Teacher indicates that this shows that the two numbers given are ‘correct’. Another two numbers are asked for, and teacher tells the class that there are ‘many’. 9 + 9 offered by a lr. Teacher asks class to make 9 + 9 and check if it gives 16. Some learners are heard to call out 18, although camera is on a child who has counted out 10 on one row of abacus. Teacher tells class that this shows that 9 + 9 ‘is not correct’. Asks class to look for another pair of numbers. 9 + 7 is offered, and class check as before. Most learners are heard to say 16, although a 17 can also be heard. Teacher checks count with one learner, and writes 9 + 7 = 16 on board below the other two correct sums. 15 + 1 offered and checked, written on board. 10 + 1 offered. Checking results in some learners saying it is 11, teacher acknowledging, and asking for another pair. 10 + 6 and 9 + 7 offered again. 9 + 8 offered. 17 given as answer by several learners, and rejected. 13 + 3 offered, and 16 accepted as answer. 7 + 9 offered, and teacher indicates praise ‘Yes, we start with 7 and add 9 to it. Very good.’ Class are not asked to check this. 6 + 10 then offered, and again acknowledged with ‘Very good’. 14 + 2 offered, and class asked to check what this makes. Correct sums following this checking are written up on board. 16 stated and acknowledged by teacher. Only the first 4 correct sums (8+8, 10+6, 9+7, 15+1) have been written on the board; the rest have been acknowledged as correct orally.

27.00

10/whole class, oral and written

Task: Teacher announces shift to work with ‘minus’ – she asks for two numbers that can be taken away to make 16. She reiterates that ‘there are many ways to make the number 16’.

Activity: 17 – 1 is offered. Teacher asks learners to check what this makes on their abacuses. On learner is asked to come up and check using the bottle tops on the floor at the front. Learner proceeds to count out 17 bottle tops, takes away one, and then count out again the number remaining to get 16. Teacher writes 17 – 1 = 16 on the board alongside the column of addition sums from last task. 18 – 2 is offered. Teacher asks class to verify answers offered by individuals by making on their abacus and counting. Many seem able to do this and give correct answer. 20 – 4 offered and accepted following check. 19 – 3 dealt with similarly. Correct sums written up on board.

33:05

11/whole class, oral and written

Task: Teacher shifts class to ‘Now we look for one number, you add it many times to give us 16.’ Asks learners to ‘work it out on your abacus’ and tells them to look also for how many times they have to add this number to give 16. Reiterates that there are many ways to make 16.

Activity: A learner is heard to call out 6. Teacher tells class that she wants to see HOW they have added it. Learner offers 16 as an answer. Learner seems to say ‘1’ as teacher repeats ‘You put 1 to 16’. Teacher rejects this answer: ‘But that’s not what I want. I want one number, one number. You add it several times. One number, you add it several times. And you tell me how many times did you add that number to give you the number 16. That is repeated addition.’ A girl offers 8 2s (which she has made on her abacus). Teacher acknowledges and re-explains to whole class that this learner has got ‘2 eight times’ and this has given her 16, referring to the girl’s abacus, but does not show whole class the arrangement. Teacher writes on board 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. She asks class to make this on their abacuses. Several learners seem to have difficulty with this task – one learner has pulled down ten 2s on her abacus; another has 10, 7 and 1 pulled down; another five 3s and a single bead pulled down. When 16 has been acknowledged as the answer for eight 2s, another number is asked for that will give 16 when added ‘many times’. 4 offered. Teacher asks ‘How many 4s’. Some say 2; others say 4. Teacher accepts 4; she writes 4 + 4 + 4 + 4 on the board, and asks learners to make this arrangement on their abacuses. One learner is seen to make five 4s, but corrects this with reminder from the teacher that they need only four 4s. Others then make four 4s as well and give 16 as answer. Teacher re-states that ‘There are many ways to make 16.’ And that ‘different kinds of methods can be used to make the number 16’. She summarises that ‘The important thing here is for you to know how to write 16 in number, 16 in words – the number name – and how many pictures are we talking about when we talk about the number 16’.

44:50-55.50

12/individual, written

Teacher writes up a set of sums on board for learners to answer individually in their books: 15 + 1 = 16 + 0 = 14 + 2 = 7 + 9 = 18 – 2 = 20 – 4= 19-3=

(Previous lesson’s work in one learner’s book shows multiple ways of making 12 – addition, subtraction and repeated addition). In individual working that follows, some learners appear able to simply write in 16 as answer without use of abacus.

Analyzing variation and invariation in episodes

The episodes were analysed in terms of the key questions: What is fixed? What is varied? And how they vary? The analysis of four episodes, 9, 10, 11 and 12 are tabulated and explained in detail below.

Question/Criteria

Within each episode

Lesson 1

Episode 9

Episode 10

Episode 11

Episode 12

What is fixed?

Two numbers are added

The answer is 16

Two numbers are subtracted The answer is 16

The answer is 16

The answer is 16

What is varied?

The addends differ

The subtrahends differ

The numbers of addends repeated to make 16

The addend to be repeated to make 16

The operation (addition, subtraction and repeated addition)

The addends, subtrahends and addends to be repeated vary

How it varies?

Randomly

Controlled and randomly

Randomly

Randomly

Episode 9 – Addition to make 16

The aim of this episode is for learners to add numbers to make 16. The numbers are elicited from learners, and the teacher writes the pairs on the board. This is followed by learners counting 8 + 8 bottle tops. A variety of pairs of numbers that make 16 is written on the board:

8 + 8 =

10 + 6 =

9 + 7 =

15 + 1 =

What is fixed or invariant in all the examples is that the sum/product is 16.

What is varied? The examples vary across all 4 examples. Different pairs of numbers are added to make 16.

How they vary? ??? They vary randomly, for the most part. It is only in the first sum that the same number is added to make 16. From the first to the second sum there is no relation. From the first to the second, the differences are so great that no pattern may be discerned. From the second to the third there is a pattern beginning to emerge. So this was not planned but was an example selected by chance by the learner. From the third to the 4th there is no pattern as all four numbers change randomly again. While it was pedagogically sound to elicit the sums from the learners and write them in the order in which they were given and then get learners to check the accuracy of the answers by counting on their abacuses – a key step towards writing the numbers in ways that patterns are discernible was not progressed to. The first weakness of this random ordering was that - all possibilities of adding pairs of numbers to make 16 were not shown and secondly – the writing of number pairs on the board so that the pattern is perceivable/visible was not achieved. Had all the possibilities been written – the teacher could ask the learners to arrange the pairs in order, for example, the first number in order from highest to lowest resulting in the second number being in order from lowest to highest, or lead the class through the activity. Writing the sums in order will offer to learners to see the numbers and the regular variations in the numbers.

In this example the teacher could have selected examples to reveal controlled variation where the number change by one or two systematically and in that sequence write all possibilities:

15 + 1 =

14 + 2 =

13 + 3 =

12 + 4 =

11 + 5 =

10 + 6 =

9 + 7 =

8 + 8 =

7 + 9 =

This controlled variation would enable learners to calculate by deriving answers from the pattern without manipulating counters and by predicting that because 15+1 = 16 then 14 + 2 = 16 because??? and so on.

Or

8 + 8 =

10 + 6 =

12 + 4 =

14 + 2 =

16 + 0 =

Similarly, when the pairs vary by 2 systematically, learners would be able to calculate that because 8 + 8 =16; 8 + 2 =10 and 8 – 2 = 6 added will also give 16. Yes!!!!!!!!!! Actually, of course, this a merely Piaget’s conservation of number!

Episode 10 – Subtraction to make 16

In episode 10 of the same lesson, the teacher changes to subtraction of two numbers to make 16. The numbers are elicited from the learners and then written on the board followed by learners counting on their abacus to verify accuracy. The following pairs of numbers were written on the board:

17 – 1 =

18 – 2 =

20 – 4 =

19 – 3 =

A controlled variation has been selected by learners in example 1 and 2. This is by chance. Then example 3 disrupts the pattern. From example 3 to 4 again there is a disruption of pattern. Then a restricted number of examples have been put up on the board that also hinders perception of the pattern. Similar to episode 9, progression to writing the subtraction sums so that the pattern is discernible is not made.

Episode 11 – Repeated addition to make 16

In episode 11 of the same lesson the teacher progresses to repeated addition to make 16. The number to be repeatedly added to make 16 varies and are written on the board as:

2+ 2+ 2+ 2+2+2+2+2 = 16

4 + 4 + 4 + 4 = 16

The variations are controlled in that 2s are added to make 4 which is then added four times to make 16. But the sequence is too short to enable perception of the pattern. The repeated addition of 1s 16 times to make 16, the initial sum is left out. The final sum to complete the pattern 2 groups of 8 make 16 is left out. In episode 11 a restricted controlled variation was achieved by chance?

Episode 12 – Application exercise – of addition, subtraction, repeated addition to make 16

Again, in episode 12 the total of 16 is fixed. The operations of addition, subtraction and repeated addition and the addends and subtrahends vary. The following examples were written on the board:

15 + 1 =

16 + 0 =

14 + 2 =

7 + 9 =

18 – 2 =

20 – 4=

19-3=

In the examples given as written work the variation from one example to the next is firstly random, and secondly too insufficient to enable perception of a pattern.

Discussion

The random variations contribute to lack of deep conceptual learning. The teaching of 3 topics – addition, subtraction and repeated addition to make 16 in a lesson – led to learning at a basic level. The pedagogic practice enabled learners to do basic individual tasks in each of the episodes. The pedagogy makes available to learners to understand how to make 16 at a basic level. In each aspect learners were active in counting counters and in the move from the concrete objects to written symbolization of arithmetic. After counting 9 counters and 7 counters and getting the answer, the teacher wrote the number sentence on the board: 9 + 7 = 16. All possible pairs of numbers that make 16 were not exhausted and written as number sentences. The teacher then moved on to the next topic subtraction to make 16. Here again the teacher elicited from learners the subtraction of numbers to give 16.

The dominance of random variation in the episodes negates opportunities for perception and making sense of the patterns and transfer of learning. The ordering of the examples to enable the regularity to be seen by learners was neither aimed at nor attempted. What was missing was the prior selection of examples to reveal the regularity; the selection of a sufficient number of examples so that the regularity is observed; the ordering of the random examples called by the learners so that the regularity is visible to all, and writing them on the board in order to be visible , so that its ?? available for perception. On the contrary, the dimensions of variation were for ?? too random to enable conception to happen.

The activities were specific to the situation at hand and did not offer opportunity for seeing them as potential properties of similar situations. The tasks focused learners attention to the immediate ‘doing’ and while this was important to enable understanding of 16 in its own right the special steps that Watson and Mason advise to promote further engagement towards abstraction, generalisation, that make derived answers possible or conceptualisation beyond that necessary for the current problem was not progressed to.

Conclusion and implications

The paper considers what might be gained from structuring lessons and classroom practice in ways that focus on a particular concept at a deeper level through different activities selected to enable regularities to be perceived by learners. The argument being made is that possibilities for numeracy learning may be enhanced when, exercises that enables sufficient and deep engagement is considered as a design strategy in planning exercises. This means that a specific aspect, e.g. addition to make 16, is focused on in different ways; concrete manipulation, pictorial, semiotic, and symbolic representations that focus on numerosity of 16. Once addition to make 16 is adequately taught then subtraction ought to be progressed to. The lesson observed illustrated a lack of focused attention to each operation and too rapid shift to other operations within the lesson.

Secondly, the lack of progress to structured variation negates firstly, the perception of regularities that could enable generalization to further operations and exercises. For lesson planning the powerful role that perception plays in the discernment of variation should be considered. The absence of what matters at first – the opportunity to see what looks the same and what looks different denies learners progression to perceiving the regularity and its conceptualization. The possibility to see the invariants and the variations was not provided. Then teachers ought to constrain the number and nature of variations they present to learners and thus increase the likelihood that attention will focused on the regularities. Secondly, random variation negates the possibility of generalization to other similar situations.

Griffin (1989) holds that ‘learning takes place over time as a result of repeated experiences that are connected through personal sense-making’. When the repeated experiences are orientated to enable sense-making there are more possibilities for advanced learning of patterns and sophisticated strategies of number. But where exercises are planned and enacted in ways where structured variations are not available to learners to be seen and perceived due to random variations as observed in the lesson analysed then one can say that the repeated experiences may be one of disconnection that negates personal sense-making. If this is the case then one is actually seeing a regularity in the teachers practice that denies learners access to higher levels of numeracy understanding.

The random variation in the selection of examples negated possibility to perceive the regularity. This analysis is therefore in agreement with Watson and Mason that control over dimensions of variation is a powerful design strategy for producing exercises that encourage learners to know the pattern, to conceptualise and generalize and be enabled to derive answers from previous examples.

While one could say that the teacher observed in the study has made progress with regard to making available to learners a range of basic and particular operations, thought ought to be given to controlled rather than random variation in exercises done in the classroom. Designing lessons that apply structured variations as a design principle informing the selection and sequencing of tasks and activities and enacting such lessons would create possibilities for learners to engage with deep structure, to generalize and to conceptualise beyond the specific situation at hand.

Acknowledgements

This paper forms part of the work in progress within the Wits SA Numeracy Chair project, entitled the Wits Maths Connect – Primary project. It is generously funded by the FirstRand Foundation, Anglo American, Rand Merchant Bank, the Department of Science and Technology and is administered by the NRF - National Research Foundation.

Abstract

Both South African and UK, US and Australian researchers cite inadequate pedagogy as a cause of poor learning and performance in numeracy in the early years. This paper examines a numeracy lesson to show one aspect of inadequate pedagogy. Following a qualitative research approach data collection included observation of lessons, writing fieldnotes and video-recording of lessons. The analysis has been framed by the concept of random and controlled variation derived from variation theory (Watson & Mason, 2006). Variation theory proposes structured variation as a mechanism for supporting learners to observe and discern central features or patterns or regularities in numbers. In relation to key levels of how variation can be structured to support numeracy learning, data is presented from a lesson to suggest that the dominance of random variations in consecutive steps of the pedagogic practice are key aspects of inadequate teaching in the lesson. The random variations, lead to the lack of provision of perceptual experiences that make conceptual learning, generalization and derivation of answers through sense realtions ?? possible. Together, these weaknesses negate firstly possibilities for learners immediate discernment/sense-making, conceptual learning, generalisation and secondly, the grounding necessary for cumulative learning to come. The implications of the analysis for teacher lesson planning are raised.

Random variations and unproductive range of change in a Grade 2 Numeracy lesson

Introduction

There has been an attempt to raise standards of numeracy in first world countries by placing emphasis on educational policy and practice (Wright et. al., 2006). While initial gains were made they have not been sustained as children continue to experience difficulty in numeracy. The causes of such difficulties include individual learner characteristics, inadequate or inappropriate teaching, absence from school and lack of pre-school or home experience with mathematical activities and language (Wright et al., 2006). Within the broader context of ongoing poor performance in numeracy in South Africa, this paper focuses on one aspect of what has been cited as one of the causes of such poor performance – inadequate and inappropriate teaching (Wright et. al., 2006). In this paper the focus is on one aspect of ‘inadequate and inappropriate teaching,’ the orientation towards basic numeracy skills as ends in themselves, rather than as means towards more sophisticated conception of patterns and regularities and generalizations that are transferable to other situations.

The poor performance of foundation phase learners in numeracy in historically disadvantaged SA schools is well known (Fleisch, 2008; Schollar, 2009; Reddy, 2008). According to the Director General, the DoE in SA:

…has measured learner performance in …numeracy at grade 3 and 6…the evaluations have shown that our learners are performing poorly – their …numeracy levels are below the required levels for their age and their grades (DoE 2008).

Furthermore international evaluations have corroborated the national findings and have further criticized that poor performance holds ‘even [for] children whose conditions for learning are far worse than ours [SA]’ (DoE, 2008, p. ?). Other studies noted the higher scores of learners in Botswana and Kenya (Taylor, 2008).

Studies of primary teaching of numeracy include research of teacher questioning at a high cognitive level; teachers beliefs; lesson organization; grouping; extent of whole class teaching and other aspects of classroom practice (Askew and Brown, 1997). Further that effectiveness of teachers was related to ‘connectedness of knowledge and beliefs, enabling teachers to relate their classroom practice to a variety of mathematical ideas and representations (Askew and Brown, 1997). The emphasis in this paper is on how the teacher sets a goal, the learning of a concept, and how the teacher engages in subsequent practices that enable the learning of the concept. Also on how the teacher establishes connectivity across the concept to be learned and the activities done that are different yet connected. Schifter (2001) defines teaching practices as being skilful, patterned regularities that occur in teachers’ classrooms. Molefe and Brodie (2010) distinguish two kinds of practices: mathematical practices and teaching practices. Mathematical practices include symbolizing, representing, justifying and communicating mathematical ideas. Teaching practices involve particular approaches or methods that teachers employ in order to teach. Notwithstanding these distinctions the view taken in this article is that the mathematical practice emanates from the teaching practice and the interest in this paper is to analyse the mathematical practice in terms of the presence of patterned regularities.

This paper presents an analysis of an investigation into one of the causes of poor learning of number in the early years, that of ‘inadequate or inappropriate teaching’ framed by the notion of controlled variation through well-structured mathematical exercises. According to Wright et al a basic question that teachers have to address when planning their lesson is: ‘how to establish the linkages and generalities in the learning, which will allow progress to be made’ (p. 1). The focus in this paper is on examining the ordering of content, and explaining the ways the teacher linked different tasks within the lesson.

Conceptualizing pedagogy for the learning of number

The study is informed by Watson and Mason’s (2006) view of learning mathematics. While Watson and Mason focus on secondary school teachers practice, it is argued in this paper that foundation phase lessons would be enhanced by attention to what change in topic and focus is productive for learning and by controlled rather than random variations in examples selected to be employed to teach the particular number topic. Learning has been defined as sense-making (Watson & Mason, 2006) and as development of mind (Hirst & Peters, 1970). Learning mathematics is conceptualized as an active, problem-solving process, where each child has to construct his or her own mathematical knowledge and develop mathematical concepts as they engage in mathematical activity’ (Wright et al., ) 2006, p. ? where does the quote start?).

Concepts have been themselves the object of analysis (Bergman, 2010; Hirst & Peters, 1970). From the mathematical perspective Skemp defines a concept in terms of being able to classify new data based on having learned the similarities that encompass the concept:

The criterion for having a concept is ….that of behaving in a way indicative of classifying new data according to the similarities which go to form the concept (Skemp,1971, p. 27).

To enable sense-making in mathematics, mathematics education researchers, Watson & Mason, emphasise teaching that is based on ‘a well-structured mathematical exercise’ (p. ?). Mathematical exercises are seen as a collection of procedural questions or tasks that enable the learning of a concept. They further define an exercise as ‘a single object, with individual questions seen as elements in a mathematically and pedagogically structured set’ (p. 91). A well-planned and well structured exercise enables multiple levels of learning or sense-making: it enables the concept to be constructed / conceived by learners; it advances abstract understanding; and it enables generalisation or the applicability of the concept to other situations.

A well-planned mathematical exercise includes the selection and sequencing of learning experiences that provide sensory perceptions structured in ways that enables ‘sense-making’ by learners. Since learners’ perceptions of what is on offer in the mathematics classroom are the central starting point to enable sense-making, teachers ought to give attention to what perceptions learners should have, judged from learner responses, when planning their lessons. The exercises should be selected and sequenced to enable learners to have the desirable sensory perceptions. Therefore, when teachers plan lessons and exercises, consideration ought to be given to ‘what learners might do, what they might see, hear and think, and how they might respond’ as these are initial sensory perceptions that form the basis for meaningful learning. Watson and Mason refer to this practice of conjecturing or predicting what learners will experience and respond to in advance of the lesson, during lesson planning, as a hypothetical learning trajectory (HTL). Since learning is an internal cognitive process, they advise that it’s better to work with what can be observed in a lesson and so prefer hypothetical learner response.

How could sense-making be enabled? Watson and Mason introduce the notion of range of change and dimensions of variation. Marton and Booth (1997) argue that the ‘starting point of making sense of any data is the discernment of variations within it’ (page ?). Therefore ‘tasks that display constrained variation are generally likely to result in progress in ways that unstructured sets of tasks do not’ (Who is saying this??? page ?). This requires the careful structuring of tasks that progressively reveal the regularity through controlled variation of variables that are visible to be perceived. Teachers can therefore aim to constrain the number and nature of the differences, from one example to the next, and thus increase the likelihood that attention will be focussed on mathematically crucial variables. The dimensions of variation ought to be carefully controlled because the aim is to promote perception of the pattern/regularity that may enable generalisation to other similar topics. Marton’s ?? Martin and Booth? notion of ‘dimensions of variation’ offers a way to look at practice in terms of what is available for the learner to notice.

For generalisation of conceptual learning ‘ ? where does this quote end? the learner experiences a shift between attending to relationships within and between elements of current experience (the doing of individual tasks) and perceiving relationships as properties that might be applicable in other situations. Generalisation according to Watson and Mason is the sensing of the possible variation in a relationship and as ‘shifting from seeing relationships as specific to the situation, to seeing them as potential properties of similar situations’ (p. 94). They caution that any task can focus learners attention to the immediate ‘doing’ and while this is important to enable understanding of the object in its own right special steps ought to be taken to promote further engagement towards abstraction, rigor, or conceptualisation beyond that necessary for the current problem.

In this way a mathematical concept is constructed by the learner. So the learner may construct understanding of the number 16 or what the number 16 is by doing individual tasks, such as addition and subtraction to make the number 16. To enable a higher level of learning the specific pairs of numbers that are added or subtracted to make 16 are varied in structured ways so that the patterns becomes perceivable and hence possiblility of generalisation when adding and subtracting other numbers to make 16 or when proceeding to unpack the numerosity of other numbers.

Several different experiences in which a learner may detect similarities, and may hence conceptualise about the similarity, are necessary for such perception to happen. Simon and Tzur (2004) focus on designing sequences of tasks that invite learners to reflect on the effect of their actions in the hope that they will recognise key relationships.

I interpret the above as a paraphrase of what Simon and Tzur says.

BUT IT IS NOT! It is the exact words of Watson and Mason (p. 93) below!

Now I must wonder whether you are merely copying Watson and Mason, and whether, by referencing Simon and Tzur, you have actually consulted Simon and Tzur yourself? If not, this is highly a-scholastic!?

Micro-modelling refers to the processes of trying to see, structure and exploit regularities in experiential data, so that learners are exposed to mathematical structure affording them enhanced possibilities for making their own sense of a collection of questions or an exercise. Exercises should be structured in ways that ‘desirable regularities’ might be perceived by the learner through engagement with the task.

You write the above paragraph as your own words and therefore your own construct!?

BUT IT IS NOT! It is Watson and Mason’s construct, so you must reference them!

It is not even a paraphrase, but is in fact an absolute DIRECT QUOTE from page 93:

This is very poor scholarly work!

Watson and Mason advise that attention be given to what regularities are available to be perceived by learners and which are most likely to be observed in any given task? What conditions make perception of the regularity possible? The regularity is more likely to be discerned ‘ ?? where does this quote end? if its variation is foregrounded against relative invariance of other features. If everything is varying, then regularities are not perceived and nothing will be discerned. Then the regularity ought to be visible so learners see it. Constructing tasks that use variation and change optimally is a design project in which learner responses leads to further refinement and further precision of selection and sequencing of tasks (Gravemeijer, 2004)

Again – you are copying what Watson and Mason says on p. 100, below.

I must wonder whether you yourself have read Gravemeijer!!!!

Given, the advocacy of permissible range of change and structured variation to support conception of patterns and regularities, the lesson was analysed in terms of the range of change of topics within the lesson and in terms of what was fixed, what varied and how they varied within each episode taught. This would enable attention to what regularities were made available to be perceived and discerned by the learner. The episodes in the lessons and the work done on the board were analysed in terms of what aspects were fixed, what was varied and how they varied throughout the lesson.

Data sources

This paper arises from the context of a larger research project of grade 2 numeracy to predominantly Black African learners in 10 government schools in Gauteng? serving historically disadvantaged learners. Using a qualitative approach data was collected through observations of teachers teaching numeracy in grade 2 at the beginning of 2011. Data collection included observation of lessons, writing fieldnotes and video-recording of lessons. Baseline data was collected through lesson observations were carried out in all the Grade 2 classes. Where informed consent for video taping was given, lessons were video recorded with a focus on the teacher. The aim in the baseline study was to start building a picture of the teaching practice in Grade 2 numeracy classrooms.

In this paper, data is drawn from the transcript created from the video record of one Grade 2 teacher’s Numeracy lesson. Without making conclusive claims in this initial paper, the randomness that characterized this lesson appears to be representative of the majority if not all the lessons observed. Other lessons would be analysed according to the analytical framework used to analyse this lesson to see whether this initial claim is representative of a greater number of lessons observed and video recorded. In this transcript, teacher talk, teacher-learner interactions, and across these utterances the objects and representations that were being referred to – sometimes in the form of writing on the board, and at other times, in the form of representations created by learners in their workbooks or with manipulative objects like abacuses or cubes - have been captured. An overview of the enacted lesson is provided below, in the form of a summary followed by a demarcation of episodes distinguished by a specific foci and different from another episode.

Lesson overview

The aim of the lesson was to enable learners to understand what the number 16 is by engaging in addition to make 16, subtraction to make 16 and repeated addition to make 16. The teacher began the lesson with getting all learners to count forward from 1 to 100 and then backward from 100 to 1 and pointing out the numbers on their number chart in each case. The teacher then asked the pupils to use their abacus to work out a word problem: ‘Sipho has 9 sweets. Nomsa has 4 sweets. Mary has 3 sweets. How many sweets are there altogether?’

The teacher then wrote 16 in numerals and words on the board. She asked a learner to draw 16 objects on the board. The learner drew 16 circles. The teacher then drew 16 asterisks on the board in rows – (5 in top line, 4 in second line, 5 in 3rd line and 2 in the last line). She then put up a sheet on the board where she had drawn 16 asterisks in a rectangular format with different numbers of asterisks in each row from the rows on the board. The learners were then asked to find 16 in numbers and words from some cards. The teacher then asked for two numbers that add to 16. Learners answered and the teacher asked the class to verify answers offered by individuals by counting out the numbers on their abacus and adding them up. The correct sums were written up on board.

More context …

I found the this page below in Foundations for Learning Lesson plans Grade 2

While it does not ditract from your argument, I merely ask if the observed lesson is based on the teacher’s own lesson plan, or is this poor teacher actually trying to blindly implement the given official resource??

I wonder, instead of ‘blaming’ this poor teacher for ‘inadequate teaching in the lesson’, should we not blame the poor guidance and resources provided by the DOE??

Lesson Analysis

Identifying episodes

An episode is defined by the introduction of a ‘new’ task, although some tasks are clearly related to each other. Following Mason’s (2004) distinction between task and activity, the episode summary begins with an outline of the task, and then details some of the activity that ensued in classroom work on the task.

Time

Episode no/format

Content outline

0 :00

1/wh class, oral

Task: Forward oral count from 1-100, followed by backward oral count 100-1.

Activity: Learners instructed to follow the numbers on their 100 squares

5:20

2/wh class, oral, some graphic

Task: Word problem up on the board: ‘Sipho has 9 sweets. Nomsa has 4 sweets. Mary has 3 sweets. How many sweets are there altogether?’

Activity: Teacher tells the class that they can use their abacuses. One child, almost immediately can be heard to call out 16. Learners are then told to count out 9, and then 4 and then 3 on their abacus. Most are able to do this, but some appear to struggle. The class is then told, in chorus, to count how many there are altogether. Counting from 1, they get 16.

7:40

3/wh class oral & graphic

Task: Teacher asks for a learner to come up and write 16 on the board.

Activity: A learner does this correctly

8:20

4/wh class, oral, some graphic

Task: Learner asked to draw 16 objects on the board.

Activity: A boy draws 15 circles – 14 on one row and 1 below. In count with class, the bottom circle is counted twice to get 16.

9:45

5/ whole class, oral and graphic

Task: Child asked to count out 16 counters from a box.

Activity: Boy counts out the bottle tops from a bucket on to the floor: 1, 2, 3 …. 12, 13, 16. Class heard to disagree. Boy then counts out another three bottle tops accompanied by the words: 14, 15, 16. There are now 17 bottle tops in the line. Teacher asks him to count out the bottle tops from the end where he started laying out the bottle tops. Boy makes an error in counting – skips a counter at 5, 6, and therefore ends up at 16. Teacher acknowledges with a ‘Very good’.

11:20

6/ whole class, oral and graphic

Task: Teacher writes ‘Number’ with 16 below it on the board. She then writes ‘Word’ up on the board. She asks learner to write ‘the number name’ in words below it.

Activity: A boy writes ‘sixteen’ on board..

13:35

7/ whole class, oral and graphic

Task: Teacher adds a heading ‘Picture’ next to the previous two headings. Below, she draws 16 asterisks on the board (5 in top line, 4 in second line, 5 in 3rd line and 2 in the last line) and then puts up a pre-prepared poster on board with 16 asterisks in a rectangular format, but again with different numbers of asterisks in different rows, 6 in top row and 5 each in 2nd and 3rd rows.

Activity: A learner is called up to count the asterisks – he does this correctly on the 2nd attempt (repeats 14 twice on 1st attempt and gets 15).

15:00

8/whole class, oral and graphic

Task: Two learners are asked to find 16 in numbers and words from some cards with numbers 1-20.

Activity: A learner finds the correct cards and sticks cards on the board.

16:30

9/whole class, oral and graphic

Task:Teacher asks for two numbers that add to 16.

Activity: 8 + 8 offered. Teacher writes 8 + 8 = on the board. Another learner is asked to verify that this makes 16 by counting out 8 + 8 bottletops. Learner arranges these in two rows of 8, then counts all and gets 16. Teacher writes in 16 on her sum on the board, and then asks for ‘another two numbers’ that give 16 when added. 10 + 6 offered by lr.?? Teacher asks class to verify answers offered by individuals by making the two numbers on their abacus and counting. Several learners have made 10 and 6 on abacus, and several heard calling out 16. Teacher indicates that this shows that the two numbers given are ‘correct’. Another two numbers are asked for, and teacher tells the class that there are ‘many’. 9 + 9 offered by a lr. Teacher asks class to make 9 + 9 and check if it gives 16. Some learners are heard to call out 18, although camera is on a child who has counted out 10 on one row of abacus. Teacher tells class that this shows that 9 + 9 ‘is not correct’. Asks class to look for another pair of numbers. 9 + 7 is offered, and class check as before. Most learners are heard to say 16, although a 17 can also be heard. Teacher checks count with one learner, and writes 9 + 7 = 16 on board below the other two correct sums. 15 + 1 offered and checked, written on board. 10 + 1 offered. Checking results in some learners saying it is 11, teacher acknowledging, and asking for another pair. 10 + 6 and 9 + 7 offered again. 9 + 8 offered. 17 given as answer by several learners, and rejected. 13 + 3 offered, and 16 accepted as answer. 7 + 9 offered, and teacher indicates praise ‘Yes, we start with 7 and add 9 to it. Very good.’ Class are not asked to check this. 6 + 10 then offered, and again acknowledged with ‘Very good’. 14 + 2 offered, and class asked to check what this makes. Correct sums following this checking are written up on board. 16 stated and acknowledged by teacher. Only the first 4 correct sums (8+8, 10+6, 9+7, 15+1) have been written on the board; the rest have been acknowledged as correct orally.

27.00

10/whole class, oral and written

Task: Teacher announces shift to work with ‘minus’ – she asks for two numbers that can be taken away to make 16. She reiterates that ‘there are many ways to make the number 16’.

Activity: 17 – 1 is offered. Teacher asks learners to check what this makes on their abacuses. On learner is asked to come up and check using the bottle tops on the floor at the front. Learner proceeds to count out 17 bottle tops, takes away one, and then count out again the number remaining to get 16. Teacher writes 17 – 1 = 16 on the board alongside the column of addition sums from last task. 18 – 2 is offered. Teacher asks class to verify answers offered by individuals by making on their abacus and counting. Many seem able to do this and give correct answer. 20 – 4 offered and accepted following check. 19 – 3 dealt with similarly. Correct sums written up on board.

33:05

11/whole class, oral and written

Task: Teacher shifts class to ‘Now we look for one number, you add it many times to give us 16.’ Asks learners to ‘work it out on your abacus’ and tells them to look also for how many times they have to add this number to give 16. Reiterates that there are many ways to make 16.

Activity: A learner is heard to call out 6. Teacher tells class that she wants to see HOW they have added it. Learner offers 16 as an answer. Learner seems to say ‘1’ as teacher repeats ‘You put 1 to 16’. Teacher rejects this answer: ‘But that’s not what I want. I want one number, one number. You add it several times. One number, you add it several times. And you tell me how many times did you add that number to give you the number 16. That is repeated addition.’ A girl offers 8 2s (which she has made on her abacus). Teacher acknowledges and re-explains to whole class that this learner has got ‘2 eight times’ and this has given her 16, referring to the girl’s abacus, but does not show whole class the arrangement. Teacher writes on board 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. She asks class to make this on their abacuses. Several learners seem to have difficulty with this task – one learner has pulled down ten 2s on her abacus; another has 10, 7 and 1 pulled down; another five 3s and a single bead pulled down. When 16 has been acknowledged as the answer for eight 2s, another number is asked for that will give 16 when added ‘many times’. 4 offered. Teacher asks ‘How many 4s’. Some say 2; others say 4. Teacher accepts 4; she writes 4 + 4 + 4 + 4 on the board, and asks learners to make this arrangement on their abacuses. One learner is seen to make five 4s, but corrects this with reminder from the teacher that they need only four 4s. Others then make four 4s as well and give 16 as answer. Teacher re-states that ‘There are many ways to make 16.’ And that ‘different kinds of methods can be used to make the number 16’. She summarises that ‘The important thing here is for you to know how to write 16 in number, 16 in words – the number name – and how many pictures are we talking about when we talk about the number 16’.

44:50-55.50

12/individual, written

Teacher writes up a set of sums on board for learners to answer individually in their books: 15 + 1 = 16 + 0 = 14 + 2 = 7 + 9 = 18 – 2 = 20 – 4= 19-3=

(Previous lesson’s work in one learner’s book shows multiple ways of making 12 – addition, subtraction and repeated addition). In individual working that follows, some learners appear able to simply write in 16 as answer without use of abacus.

Analyzing variation and invariation in episodes

The episodes were analysed in terms of the key questions: What is fixed? What is varied? And how they vary? The analysis of four episodes, 9, 10, 11 and 12 are tabulated and explained in detail below.

Question/Criteria

Within each episode

Lesson 1

Episode 9

Episode 10

Episode 11

Episode 12

What is fixed?

Two numbers are added

The answer is 16

Two numbers are subtracted The answer is 16

The answer is 16

The answer is 16

What is varied?

The addends differ

The subtrahends differ

The numbers of addends repeated to make 16

The addend to be repeated to make 16

The operation (addition, subtraction and repeated addition)

The addends, subtrahends and addends to be repeated vary

How it varies?

Randomly

Controlled and randomly

Randomly

Randomly

Episode 9 – Addition to make 16

The aim of this episode is for learners to add numbers to make 16. The numbers are elicited from learners, and the teacher writes the pairs on the board. This is followed by learners counting 8 + 8 bottle tops. A variety of pairs of numbers that make 16 is written on the board:

8 + 8 =

10 + 6 =

9 + 7 =

15 + 1 =

What is fixed or invariant in all the examples is that the sum/product is 16.

What is varied? The examples vary across all 4 examples. Different pairs of numbers are added to make 16.

How they vary? ??? They vary randomly, for the most part. It is only in the first sum that the same number is added to make 16. From the first to the second sum there is no relation. From the first to the second, the differences are so great that no pattern may be discerned. From the second to the third there is a pattern beginning to emerge. So this was not planned but was an example selected by chance by the learner. From the third to the 4th there is no pattern as all four numbers change randomly again. While it was pedagogically sound to elicit the sums from the learners and write them in the order in which they were given and then get learners to check the accuracy of the answers by counting on their abacuses – a key step towards writing the numbers in ways that patterns are discernible was not progressed to. The first weakness of this random ordering was that - all possibilities of adding pairs of numbers to make 16 were not shown and secondly – the writing of number pairs on the board so that the pattern is perceivable/visible was not achieved. Had all the possibilities been written – the teacher could ask the learners to arrange the pairs in order, for example, the first number in order from highest to lowest resulting in the second number being in order from lowest to highest, or lead the class through the activity. Writing the sums in order will offer to learners to see the numbers and the regular variations in the numbers.

In this example the teacher could have selected examples to reveal controlled variation where the number change by one or two systematically and in that sequence write all possibilities:

15 + 1 =

14 + 2 =

13 + 3 =

12 + 4 =

11 + 5 =

10 + 6 =

9 + 7 =

8 + 8 =

7 + 9 =

This controlled variation would enable learners to calculate by deriving answers from the pattern without manipulating counters and by predicting that because 15+1 = 16 then 14 + 2 = 16 because??? and so on.

Or

8 + 8 =

10 + 6 =

12 + 4 =

14 + 2 =

16 + 0 =

Similarly, when the pairs vary by 2 systematically, learners would be able to calculate that because 8 + 8 =16; 8 + 2 =10 and 8 – 2 = 6 added will also give 16. Yes!!!!!!!!!! Actually, of course, this a merely Piaget’s conservation of number!

Episode 10 – Subtraction to make 16

In episode 10 of the same lesson, the teacher changes to subtraction of two numbers to make 16. The numbers are elicited from the learners and then written on the board followed by learners counting on their abacus to verify accuracy. The following pairs of numbers were written on the board:

17 – 1 =

18 – 2 =

20 – 4 =

19 – 3 =

A controlled variation has been selected by learners in example 1 and 2. This is by chance. Then example 3 disrupts the pattern. From example 3 to 4 again there is a disruption of pattern. Then a restricted number of examples have been put up on the board that also hinders perception of the pattern. Similar to episode 9, progression to writing the subtraction sums so that the pattern is discernible is not made.

Episode 11 – Repeated addition to make 16

In episode 11 of the same lesson the teacher progresses to repeated addition to make 16. The number to be repeatedly added to make 16 varies and are written on the board as:

2+ 2+ 2+ 2+2+2+2+2 = 16

4 + 4 + 4 + 4 = 16

The variations are controlled in that 2s are added to make 4 which is then added four times to make 16. But the sequence is too short to enable perception of the pattern. The repeated addition of 1s 16 times to make 16, the initial sum is left out. The final sum to complete the pattern 2 groups of 8 make 16 is left out. In episode 11 a restricted controlled variation was achieved by chance?

Episode 12 – Application exercise – of addition, subtraction, repeated addition to make 16

Again, in episode 12 the total of 16 is fixed. The operations of addition, subtraction and repeated addition and the addends and subtrahends vary. The following examples were written on the board:

15 + 1 =

16 + 0 =

14 + 2 =

7 + 9 =

18 – 2 =

20 – 4=

19-3=

In the examples given as written work the variation from one example to the next is firstly random, and secondly too insufficient to enable perception of a pattern.

Discussion

The random variations contribute to lack of deep conceptual learning. The teaching of 3 topics – addition, subtraction and repeated addition to make 16 in a lesson – led to learning at a basic level. The pedagogic practice enabled learners to do basic individual tasks in each of the episodes. The pedagogy makes available to learners to understand how to make 16 at a basic level. In each aspect learners were active in counting counters and in the move from the concrete objects to written symbolization of arithmetic. After counting 9 counters and 7 counters and getting the answer, the teacher wrote the number sentence on the board: 9 + 7 = 16. All possible pairs of numbers that make 16 were not exhausted and written as number sentences. The teacher then moved on to the next topic subtraction to make 16. Here again the teacher elicited from learners the subtraction of numbers to give 16.

The dominance of random variation in the episodes negates opportunities for perception and making sense of the patterns and transfer of learning. The ordering of the examples to enable the regularity to be seen by learners was neither aimed at nor attempted. What was missing was the prior selection of examples to reveal the regularity; the selection of a sufficient number of examples so that the regularity is observed; the ordering of the random examples called by the learners so that the regularity is visible to all, and writing them on the board in order to be visible , so that its ?? available for perception. On the contrary, the dimensions of variation were for ?? too random to enable conception to happen.

The activities were specific to the situation at hand and did not offer opportunity for seeing them as potential properties of similar situations. The tasks focused learners attention to the immediate ‘doing’ and while this was important to enable understanding of 16 in its own right the special steps that Watson and Mason advise to promote further engagement towards abstraction, generalisation, that make derived answers possible or conceptualisation beyond that necessary for the current problem was not progressed to.

Conclusion and implications

The paper considers what might be gained from structuring lessons and classroom practice in ways that focus on a particular concept at a deeper level through different activities selected to enable regularities to be perceived by learners. The argument being made is that possibilities for numeracy learning may be enhanced when, exercises that enables sufficient and deep engagement is considered as a design strategy in planning exercises. This means that a specific aspect, e.g. addition to make 16, is focused on in different ways; concrete manipulation, pictorial, semiotic, and symbolic representations that focus on numerosity of 16. Once addition to make 16 is adequately taught then subtraction ought to be progressed to. The lesson observed illustrated a lack of focused attention to each operation and too rapid shift to other operations within the lesson.

Secondly, the lack of progress to structured variation negates firstly, the perception of regularities that could enable generalization to further operations and exercises. For lesson planning the powerful role that perception plays in the discernment of variation should be considered. The absence of what matters at first – the opportunity to see what looks the same and what looks different denies learners progression to perceiving the regularity and its conceptualization. The possibility to see the invariants and the variations was not provided. Then teachers ought to constrain the number and nature of variations they present to learners and thus increase the likelihood that attention will focused on the regularities. Secondly, random variation negates the possibility of generalization to other similar situations.

Griffin (1989) holds that ‘learning takes place over time as a result of repeated experiences that are connected through personal sense-making’. When the repeated experiences are orientated to enable sense-making there are more possibilities for advanced learning of patterns and sophisticated strategies of number. But where exercises are planned and enacted in ways where structured variations are not available to learners to be seen and perceived due to random variations as observed in the lesson analysed then one can say that the repeated experiences may be one of disconnection that negates personal sense-making. If this is the case then one is actually seeing a regularity in the teachers practice that denies learners access to higher levels of numeracy understanding.

The random variation in the selection of examples negated possibility to perceive the regularity. This analysis is therefore in agreement with Watson and Mason that control over dimensions of variation is a powerful design strategy for producing exercises that encourage learners to know the pattern, to conceptualise and generalize and be enabled to derive answers from previous examples.

While one could say that the teacher observed in the study has made progress with regard to making available to learners a range of basic and particular operations, thought ought to be given to controlled rather than random variation in exercises done in the classroom. Designing lessons that apply structured variations as a design principle informing the selection and sequencing of tasks and activities and enacting such lessons would create possibilities for learners to engage with deep structure, to generalize and to conceptualise beyond the specific situation at hand.

Acknowledgements

This paper forms part of the work in progress within the Wits SA Numeracy Chair project, entitled the Wits Maths Connect – Primary project. It is generously funded by the FirstRand Foundation, Anglo American, Rand Merchant Bank, the Department of Science and Technology and is administered by the NRF - National Research Foundation.