We begin this section with two illustrative excerpts of instruction (see Fig.1). These excerpts overlap in their incorporation of translation moves between languages, but contrast in the ways in which moves between mathematical representations are worked with, and helped us to think about the distinction pointed out earlier about less and more mathematically oriented conversion moves.
In the first excerpt, the teacher acknowledged the answer offered by learners, and repeated it in English and then in Sepedi. The teacher then moved the orally offered number into symbolic form by writing in ‘12’ into the number sentence on the board. Translation moves between English and Sepedi and the restatement type of conversion moves between symbolic and oral language number forms we described earlier therefore occurred in the first excerpt. Excerpt 2, drawn from Mirriam’s class, is similar in many ways to Excerpt 1: a translation move is evident in the teacher first stating the number name orally in Sepedi and then in English. There is also a pointing to the symbolic numeral representation of 376 following its oral presen- tation, but here, there is a small, but important distinction in that 376 is featured within a 100-number-chart representation. In South Africa, Ensor et al. (2009) have noted the widespread prevalence of unit counted versions of number, and the limited inclusion of symbolic number system-based representations of number. While the base ten structure is not referred to in the instructional talk, the inclusion of this artifact expands the working here a little further into the conversion move terrain than Excerpt 1. Excerpt 2, therefore, offers a marginally widened repertoire of work with conversion moves in comparison to Nkele’s teaching in the first excerpt.
Excerpt 1: Nkele ML1 & MM2
Within a section of whole class teaching of a sequence of examples of missing addend tasks with total 20, the teacher writes: 8 + ⎕
=20 on the board. The following interaction then ensues:
T: {Pointing to 8 on the board:} Re ka hlakantšha nomoro e le eng go bona masomepedi? [What can we add to this number to make twenty?]. We need to add something to it so that we make masome pedi [tens two/twenty]. {Calls out a learner’s name for a response.}
L1: Lesome pedi [ten
two/twelve]. {Said immediately without counting.}
T: We can add eight and twelve, lesome pedi. [Ten two/twelve]. {T fills in 12 in number sentence}. Do you understand how you counted to get masome pedi [tens two/twenty]?
Ls: Yes.
Excerpt 2: Mirriam ML1 and MM2
{Learners have the 301-400 number charts on their desks.}
T: Put your fingers on makgolo tharo masome šupa tshela [hundreds three, tens seven, six/three hundred and seventy-six/ three hundred and seventy-six].
{Pause.}
T: Point to three hundred and seventy-six.
Say the number and point.
T: Yes, makgolo tharo- masome šupa tshela [hundreds three, tens seven, six/ three hundred and seventy-six].
{Points to 376 on the large number chart on the board}
(restatement move)
Fig. 1 Two different translation moves
The subsequent two excerpts, both drawn from Nkele’s class teaching illustrate a different type of language move to the translation move observed in Excerpts 1 and 2 (see Fig.2). As with the two excerpts above, each of these excerpts is accompanied by a different type of conversion move pointed out in the two previous excerpts.
Excerpts 3 and 4, in our analysis, involve instances of the intentional use of moves between languages for meaning-making. Excerpt 3 begins with the oral ‘reading out’
of the 3+=20 number sentence written on the board. A basic move between symbolic and oral language representations, therefore, occurs early in this excerpt similar to that observed in Excerpt 1. In the context of the incorrect offer of 70, the teacher incorporates an intentional use of moves between languages to highlight the difference between the number names for 17 and 70 in Sepedi. The more marked difference in number names in Sepedi compared to English is used here to support awareness of the number distinctions, and a correct answer is subsequently offered.
The teacher goes on to contrast the ‘lesome’ part (‘one ten’) with the ‘masome’ part
Excerpt 3: Nkele ML2 & MM2
Later, within the same sequence of examples in focus above, Nkele offers this instructional interaction when dealing with the task: 3 +⎕
=20, written on the board:
T: {Pointing to 3+⎕=20}: What can we add to 3 to make twenty? What number is that? {L1 name}
L1: Seventy.
T: What? {Teacher pauses, looks surprised.} Say that again.
L1: Seventy.
T: Say the number in Sepedi so you can hear what you are talking about.
L1: Lesome supa. [Ten seven/seventeen.]
T: Seventeen. Not seventy. If you say seventy then you are talking about masome šupa. [Tens seven/seventy].
There is a difference between masome šupa [tens seven/seventy]
and lesome -šupa [ten
seven/seventeen]. {T writes 17 and 70 on the board and points to each number as she says them}. Lesome- šupa [ten seven]is one ten.
Masome- šupa [tens seven/seventy]
is many tens, seven tens. Do you see it?
Ls: Yes ma’am
T. {Pointing to 3 in the number sentence on the board}. Now count from this number to twenty.
Excerpt 4: Nkele ML2 & MM2
{Teacher takes out three number cards with numerals 100, 30 and 5 written on the cards from a container on her table. She calls three learners to the front and gives each learner one of these number cards: 30 to Jabulani, 5 to Relebogile and 100 to Thabo.
She rearranges the learners from the left to the right as 100, 30 and 5.}
T: Lekgolo- masometharo- hlano [Hundred, tens three, five/one hundred and thirty-five]. What number is this?
{Moves her hand to point to across all three children}
Ls: Lekgolo masometharo- hlano.
[Hundred, tens three, five/one hundred and thirty-five.]
T: All I wanted to explain to you with this activity is the place value of numbers. Place value of numbers.
{Points to 3 in the number 135 on the board}. This is not just 3. Ke masometharo, [‘It is tens three/thirty’], {pointing to 3 in the number 135 on the board}. It stands for 3 tens. {Holds Jabulani who has the number card written
masome/tens}. The one stands for hundred. Ke lekgolo. {Points to 1 in 135 and moves to hold Thabo who has the hundred/ makgolo name tag}.
It is not just one. It is one hundred.
Five stands for units. It is five. Do we understand each other?
(restatement move)
Fig. 2 Two different translanguaging moves
(many tens), through her offer of translations into English of both of these parts.
Subsequently, a unit counting on from 3 to 20 is enacted that also produces ‘17’ as the answer. While concrete counting actions, as a further representational form, are incorporated here, the actions remain at the level of unit counting, with no reference to using the base ten benchmarks that form a transparent part of the ways in which numbers are named in Sepedi.
In Excerpt 4, we see fluid moves between languages with elaborations both in terms of language and moving between oral and symbolic mathematical registers:
when the digit 3 in 135 is pointed to on the board, this is accompanied by an explana- tion that emphasizes: ‘This is not just 3. Ke masometharo, [‘It is tens three/thirty’], {pointing to 3 in the number 135 on the board}. It stands for 3 tens. As with Excerpt 2 above though, what is added here is a reference to how the numbers are structured within the base ten place value system.
In this explanation, the teacher does not only substitute a word or phrase with a corresponding number word or phrase in another language or moves between the basic oral and symbolic representations. She makes connections between learners representing numbers in different ‘positions’, the digits on number cards and the language of the number system using Sepedi and English as she explains how the numbers are structured in the decimal system. This multimodal way of moving between the two languages, involving pre-prepared resources as in Excerpt 2, suggests a planned instruction that goes beyond substitution. In the context of this chapter, we refer to such moves as translanguaging moves which, as noted earlier, are more akin to conversion moves in Duval’s (2006) theory of mathematical moves between registers.