The recovery intervention program

Wright et al. (2012) view mathematics instruction in terms of progressive mathematization, described as the “development of mathematical sophistication over time” (p. 15). Progressive mathematization is characterized into 8 themes as applicable to the domains of SEAL and CPV. These themes are summarised below:

Theme A: Structuring numbers

The domains of Conceptual Place Value (CPV) and Addition and Subtraction to 100 (SEAL), structure the additive relations between numbers 1 – 100, specifically organised around ones and tens.

Theme B: Extending the range of numbers

Within each domain there is a progression in the range of numbers. As students become fluent in smaller numbers bigger numbers are immediately introduced for the specific task. Numbers used during lessons were adapted according to the response from and level indicated by the learners. Extending the number range was an effective way to supply more advanced learners with mathematical challenges.

Theme C: Decimalizing towards base-ten thinking

Numbers are organized into ones, tens, hundreds etc. in order to develop a skilful habit of organizing numbers and calculations. The focus within SEAL and CPV is to develop flexible addition and subtraction techniques of tens and hundreds and mental strategies based on base-ten thinking.

64 Theme D: Unitizing and not counting by ones

The aim of both the chosen domains (SEAL and CPV) is to develop non-counting strategies through developing understanding of ten as a special unit. Although various strategies are recommended, the number of strategies used during the recovery phase of this study was limited because of the relatively short recovery period (exams started in November followed by extended summer vacation) and the fact that it was clear early on that most learners had difficulty grasping the basic strategies. Additional strategies were therefore only suggested to stronger learners.

As mentioned in Chapter 2, the following strategies used by Wright et al. (2012, pp. 99-107) were addressed during the recovery phase: Split, Jump and Jump to the decuple (Jump to the 10).

Jump strategy: Begin from one number, jump the tens and then jump the ones (or ones first and then tens)

e.g. Addition: 37 + 22 →37 + 10 → 47 + 10 → 57 + 2 → 59 Subtraction: 53 -11 → 53 – 10 → 43 – 1 → 42

Split strategy: The tens and ones are split, then added/subtracted separately and then recombined e.g. Addition: 37 + 22 → 30 + 20 = 50 7 + 2 = 9 → 50 + 9 = 59 Subtraction: 53 -11 → 50 – 10 = 40 3 – 1 = 2 → 40 + 2 = 42

Jump to the decuple (Jump to the 10): Begin from one number, jump to the nearest decuple, jump the tens, then jump the remaining ones

e.g. Addition: 37 + 25 → 37 + 3 → 40 + 10 → 50 + 10 → 60 + 2 → 62 Subtraction: 53 – 19 → 53 – 3 → 50 – 10 → 40 – 6 → 34

65 “Research suggests that most successful students use jump strategies, whereas most low- attainers use split strategies” (Wright et al., 2007, p.849). This finding by Wright et al. is in line with my findings as noted in Chapter 5.

Theme E: Distancing a setting of materials

Distancing the setting is considered an important recovery strategy in Wright et al.’s work. The setting (e.g. the use of bundling sticks) can be progressively distanced in the following stages:

1. The bundling sticks are visible

2. The bundling sticks are shown briefly then screened 3. The task is posed verbally with bundling sticks screened 4. The task is posed verbally with no bundling sticks

I found that it was possible within the group setup to distance the setting for those learners who were ready to progress and, at the same time, supply materials for other learners where needed. The use of materials, or absence thereof, was an effective way of enabling differentiation of support provided to individual learners within each group.

Theme F: Notating

Wright et al. (2012) support the supposition that mathematical notation and mathematical concepts are learned in tandem. Learners are therefore encouraged during the recovery program to use notation to express their thinking. I noted in my research journal that some learners found the notation strategies confusing. Fig. 18, for example, shows one pair of learners’ attempt at using a number line to notate 54 – 37. They preferred the informal line/arrow notation illustrated in Fig. 19 below.

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Fig. 18: An example of a number line used by a pair of learners for notational purposes

Fig. 19: An example of informal line/arrow notation to show the calculation of 54 - 37

Although various notation strategies were used (e.g. number lines, arrow charts etc.) I realised that the complexity thereof should be monitored for individual learners. Some learners preferred the use of arrow charts and others favoured number lines. It was also important to keep in mind that “notation is used to record the mental strategy rather than providing a means of solving the task” (Wright et al., 2007, p.849).

Theme G: Formalizing

Formalizing entails a progression in notation from informal to the standardised use of symbols, columns and signs. Within the short period of recovery most of the learners were not yet ready for formalization of notational strategies. During group sessions we therefore only used the two informal strategies (number lines and line/arrow charts) illustrated in Fig. 18 and Fig. 19 above. We did not address formalizing of strategies.

67 Theme H: Generalizing

After particular example tasks are established, Wright et al. (2012) illustrate that other tasks could be solved in similar ways. So for example in the recovery I would deliberately choose numbers to allow learners to extend the range of numbers used in various ways, e.g. 34 + 10 = 44 and 44 + 10 = 54 can be extended to 134 + 10 = 144 and even to 340 + 100 = 440. This was done by using materials like bundling sticks or simply by mental exercises during games and the use of flash cards (See Appendices I to P for all eight the recovery session plans in this regard).