In the following, two different situations of a Grade 1 mathematics class about relational terms will illuminate language and discursive particularities of the different scenes. In this regard, especially the use of a narrative real-world context about a crocodile in order to visualize the relational terms ‘greater than’ and ‘less than’ demonstrates that the pupils’ language use and their learning process is affected in both, positive and negative ways - e.g. by a lack of clarity concerning the mathematical content. This also becomes visible through the language of the pupils during the two presented scenes.
Scene 1. Class discussion at the start of the lesson: What does Croco like to eat?
The excerpt below is from the opening class discussion in Grade 1 about the topic of the relation terms ‘greater than’ (>) and ‘less than’ (<).The teacher (T) is now initiating a discussion about the formal expression of the terms. With the help of a narrative real-world context about a crocodile with an open mouth (named Croco) and towers of two red cubes and five blue cubes on the board (see Figure 11.1, from the images available at https://www.zaubereinmaleins.de, June 2019) the mathematical content is brought to the pupils.
T: Our little Croco always wants to eat a lot.That’s why its mouth is open that
wide. And now he comes and thinks about. Shall I eat the red ones or the blue ones? What do you think, Ina?
INA: I think red.
T: You think red [Turns Croco with the open mouth to the two red cubesj .Why?
INA: Red is like meat.
FIGURE 11.1 The board before and after the class discussion (left and middle). Simplified visualization of Croco
T: Aha.That would be a consideration. Nabil, what do you think he wants to eat?
T: You say blue is what he wants to eat, why? ... [Nabil does not say anything for
4 seconds). Simply because blue is beautiful. Okay. Rich, what do you think? RICH: Ehm. He wants to eat red because it is like meat and fish.
T: [Turns Croco to the red cubes] Because meat and fish. Mhm. Nagi, what do
NAGI: Blue. Because that is more.
T: That is our little Croco who always wants to eat the most and that’s why he
looks here [Turns the crocodile between the towers and cubes that it looks to the four blue ones, writes a “>’ between the four blue and the red cubes and again places Croco between them]. Can you see this? Because he always wants to eat what is more.
The extract shows that the teacher packs the mathematical content into a narrative real-world context. This seems to lead to pupil answers which are oriented towards the story and less towards the mathematical content.The answers and explanations of Ina and Rich seem to be bound to the story of Croco (as they give the explanation that they argue for red because of the similarity to meat and fish), until Nagi gives a ‘satisfactory’ answer with a short justification about the mathematical insight, which seems to be less oriented to the story. Ina and Rich, by contrast, seem to be ‘caught up’ in the narrative real-world context and try to argue for ‘red’ as it is similar to the color of raw meat and fish. Although it is visible that there are more blue cubes on the board, two children argue for red, which leads to the assumption that they are too fixated on the story.
This is consistent with the assertion of Neth and Voigt (1991), who state that the use of narrative real-world contexts might disguise mathematical facts, and with the statement of Krummheuer (2000) that the mathematical core is co-delivered and implicitly conveyed. Thus, it is up to the pupils to ‘filter out’ and understand the mathematically significant facts of the story. In the analyzed situation, it is never explicitly expressed which mathematical idea Croco and his food are symbols for. Concerning the realistic content, authenticity and personal involvement of the story, it becomes clear that this is too far-fetched. It is neither taken from the everyday world of children, nor is it relevant to the pupils, nor is it conceivable that a crocodile in such a situation would be in a position to decide on its own which color and ‘amount’ of its food it will have. It remains unclear whether the children understand the mathematical concept behind the given story, since only Nagi contributed something to the situation that can be interpreted as primarily mathematical. The rest of the pupils’ utterances seems to be superficially oriented toward the story of Croco and/or the colors of the cubes. The teacher might interpret Nagi’s answer as oriented to the mathematical content and therefore proceeds from the assumption that Nagi (and by her utterance also the rest of the class) might have understood the mathematical content of abstract quantity and the relational terms. The teacher did not support the children to discover the mathematical purposeful concept (they do not have to argue with the numbers of cubes) and the idea of color is seen as one possible way to answer the question of what Croco wants to eat.
Concerning the language-based opportunities and requirements, it becomes apparent that the first two children, Ina and Nabil, were asked by the teacher for an explanation of their given answer. By contrast, the last two children who gave answers (Rich and Nagi) advanced their own explanations. We could imagine that they recognized the specific language-based demands of the situation: Giving an answer is not enough - Instead, some remarks about the ‘why’ are necessary. The tendency to give an explanation to the produced answer seems to be typical for school contexts and academic language. This resembles two sides of the same coin: on the one hand, the opportunities to speak and behave could be very diverse, as it is up to the pupils if the answer with connection to the narrative context or to the mathematical content. On the other hand, there could be limiting expectations as well as (implicit) rules and norms, which are seldom made explicit by the teacher (Schiitte, 2014). In this regard, the suspected wide range of possible answers within the class discussion about Croco is limited by invisible and implicit expectations of the teacher.
Scene 2. Assistance during individual work on the tasks in the workbook: Do you need help, Nabil?
Directly after this class discussion the pupils were asked to complete some exercises about this topic in their workbooks. After four tasks in the style comparable to those on the board - two towers of cubes and in between the pupils had to place the correct sign - in number two there are neither towers of cubes nor a crocodile that wants to eat them (see Figure 11.2).The pupils seem to be supposed to identify the relationship between the numbers and the correct sign on their own.
After several minutes, Nabil is still working on number two, while his seatmate Dani has already finished the task. When Dani sees that Nabil has not finished yet, the following conversation starts:
DANI: I am already done. I can help you.
NABIL: Why do you want to help me?
DANI: [Takes the pencil out of Nabil’s hand and signs on task 6 О 5 in his workbook] Here. See here. Which he would like to eat? Six or five?
NABIL: ... Six.
FIGURE 11.2 Extract of the children’s workbooks
DANI: Yes. So you do it |Writes something in Nabil’s workbook],
NABIL: | Takes his pencil from Dani]
BAILA: [Comes to the desk of Nabil and Dani, Dani stands on Nabil’s left. Baila stands to the right of him. Looks on Nabil’s workbook] ehm ... you are here. Think about it ... [Shows three fingers with her left hand] nine or three. Which is more?
NABIL: | Writes something in his workbook]
BAILA: That’s right.
In this scene, Dani and Baila, who have already finished the tasks in their own workbooks, seem to offer help to Nabil, without asking him if he wants it. Dani’s explanation is similar to that of the teacher as it is bound to the story of Croco. It is more oriented towards the narrative real-world context and less towards the mathematical content (“Which he [Croco] would like to eat?”). It remains questionable whether Dani requires the story about Croco and his food to solve the tasks. The argumentation with Croco does not really fit the situation (as in task number 2 there are no indicators that arise the necessity' to speak about the hungry Croco and his food), thus it can be assumed that she might be aware of the underlying mathematical concept and has already generalized, what Croco and his food are symbols for. Croco and his food might be considered to have a metaphorical presence, creating a link to the classroom discussion and number one in the workbook. Maybe, she is just using the narrative about Croco to visualize the task for Nabil. Dani’s utterances and activities indicate that she is able to make a connection between the mathematical content of relational terms and the comparison of different amounts of numbers and the established narrative real-world context of the hungry crocodile.
Baila approaches Nabil and also ‘helps’ Nabil without asking if he is in need of help. This could be seen as a typical discursive characteristic in educational settings or especially in this class: If there is someone who has not finished yet, pupils who already have finished help. Baila also offers an explanation, which, in comparison to Dani’s, is more formal and oriented to the mathematical content. Since there is no visualized support (like a crocodile or towers of cubes) in number two, she uses her fingers and shows ‘three’ to illustrate something for Nabil. In comparison to Dani, her way of helping Nabil is more concrete, as she uses her fingers to visualize the amount of the numbers to create a connection between the mathematical task in the workbook and her assistance.
Taking a closer look at Nabil and his opportunities to learn, one could conjecture the following: It seems possible that it becomes even more problematic for him when Dani uses the story of Croco for her explanation of the tasks in number two. In this task, neither colors nor towers of cubes, which could be eaten by Croco, can be seen.There are just numbers and a gap in which the correct relational sign (< or >) has to be filled in. Nabil’s answer ‘Six’ could be an indication for his understanding of the mathematical concept. It remains questionable, if he needs the girls’ help, if he requires the story of Croco to solve the task, or if his slow progress is caused by the inappropriate and confusing use of the narrative context.