In order to provide empirical insights into teachers’ learning pathways, the section starts with a case study from Cycle 1 showing an example of absence of relevant categories. This case formed the base for a PD activity in later cycles, and also provided an occasion to investigate other teachers’ starting points on the jobs of SUPPORTING language and IDENTIFYING relevant language demands. The section identifies teachers’ typical categorical starting points for these jobs, and the section traces the learning pathway of one group of teachers.
Need for connecting jobs and introducing categories: the case of Martin Schreiber from Cycle 1
One case study from cycle 1 is briefly presented in order to show the importance of the job of IDENTIFYING language demands relevant for mathematics, a typical challenge that teachers face when introducing language responsiveness into their classrooms:
Reporting the case
Martin Schreiber (pseudonym), a monolingual middle school mathematics teacher, chose to participate in the CPD as he recognised the need to respond to the language diversity of his classes. By the second CPD session, he had adopted all intended orientations (see the “Disentangling teacher expertise for language-responsive mathematics teaching” section: He assumed responsibility for students’ diverse language learning, strived for pushing instead of reducing language, and tried to adopt integrative rather than additive strategies).
In order to realise them in his classrooms, Martin Schreiber experimented with the two jobs of DEMANDING and SUPPORTING language. After a teaching unit on the equivalence of fractions (see Fig. 3, 1st step), he posed a writing task with scaffolds in a list of useful phrases (Fig. 3, 2nd step). After conducting writing conferences in small groups, one model text was jointly developed in a whole-class discussion (3rd step) and then transferred to the second procedure of simplifying fractions in seatwork (4th step). He considered his teaching unit successful since all children could formulate a text similar to the one shown in Fig. 3 (4th step).
Martin Schreiber brought this experience and his students’ written products to the third PD session and asked his colleagues to NOTICE the students’ language in the written products. In the first minutes, the group (including Martin) assessed the texts only according to orthography, grammar errors (errors not translated to English in Fig. 3, 2nd step), and use of technical terms. The discussion did not assess whether the content goals had been achieved.
When the PD facilitator asked what the students’ products can tell about their conceptual understanding of equivalence of fractions, one teacher realised, “They do not write about it, only about how to calculate it”. Subsequently, the teachers started to reconsider the texts and articulate that most students only refer to the procedures. Now, other teachers recognised that the third text was an explanation: “‘Divide the fraction’ is not the right word, but the perfect idea for explaining the finer structure of the shaded amount”. It was only after this moment that the discussing teachers realised that the phrase list was only instrumental to reporting on formal procedures, but no help in explaining meanings. With the facilitators’ support, they started to collect other meaning-related phrases such as “finer partitioning of the rectangle.”
Analysing the case with respect to activated categories
The analysis of individually activated concepts-in-actions reveals an explanative account for the situation: in the first step, Martin Schreiber activates the two categories ||procedural knowledge|| and ||conceptual knowledge|| during the introduction of equivalence. Also, when DEMANDING language in the second step, he refers to both forms of knowledge in the writing task and asks for two ||discourse practices||, namely ||reporting procedures||, and ||explaining meanings|| of equivalence. However, in his scaffolding phrase list, he only offers words that support ||reporting procedures||. Consequently, the students only show ||reporting procedures||, and this reduction is maintained through the third and fourth step.
When first discussing the students’ products in the PD session, the teachers (including Martin) only activate categories for the surface of language, namely ||orthography||, ||grammar errors|| and the use of ||technical terms||. These activated categories do not enable them to assess whether the content goals have been achieved, as they cannot yet IDENTIFY the mathematically relevant language demands.
It is the PD facilitator’s question about students’ conceptual understanding of equivalence that prompts the teachers to shift from the surface level of language to the relevant ||discourse practices||. When they assess the students’ texts with respect to the categories of ||discourse practices||, they start to distinguish the two content goals of ||procedural knowledge|| on how to expand fractions and the ||conceptual knowledge|| about the meaning of equivalence of fractions. At this point, other teachers recognise that the third text shows ||explaining meaning||, which is relevant for the content goal of consolidating ||conceptual knowledge||. When the facilitator guides them back to the vocabulary, they start to see the vocabulary in its tight connection to the discourse practices, not as an end in itself but as ||lexical means|| for the ||discourse practices||.
From this analysis, it can be inferred that although Martin Schreiber has treated ||procedural and conceptual knowledge|| before the writing task, his language efforts are restricted to the ||discourse practice of reporting procedures||, for which he provides ||formal vocabulary|| support and which he optimises in the joint discussion. In contrast, addressing the ||conceptual knowledge|| requires the ||discourse practice of explaining meanings||, and this could have only been supported with more meaning-related phrases such as “the same share” and “structuring in a finer way.” But these meaning-related discourse practices and lexical means were not immediately IDENTIFIED as relevant and—although DEMANDED in the writing task—neither SUPPORTED nor NOTICED by the discussing teachers.
The brief analysis of this first case study shows how the different jobs are intertwined: teachers can only NOTICE and SUPPORT the language features that they have IDENTIFIED; for these jobs, they need adequate categories. Otherwise, they stay on the surface level of language (Prediger et al. 2018a). Several similar episodes during Cycle 1 led the researcher to include not only ||discourse level|| as relevant category in the PD, but to explicitly introduce the ||interplay of content goal and discursive practice|| and the ||interplay of discursive practice and lexical means||, as also emphasised by Short (2017). Figure 4 illustrates these identified categories and their connections, with the interplay between categories shown as horizontal arrows.
Design consequences: designing an activity for the PD courses in Cycle 2
With the permission of Martin Schreiber, his classroom documents were used in Cycle 2 of the design experiment to design an activity for further PD sessions in which other teachers could discover the strong connection between the different jobs and the relevance of ||discourse practices|| as the crucial category for IDENTIFYING mathematically relevant language demands and to construct the relevant connections between categories as indicated in Fig. 4. Figure 5 shows the designed activity in its final form (iteratively refined during five mini-cycles in Cycle 2).
The goal of the activity is to enhance teachers’ awareness of the necessary job of IDENTIFYING relevant language demands before SUPPORTING the writing task. Task 1 (in Fig. 5) asks the teachers to write a “dream text” on their own; the intent of this is to focus on the discourse practices and on a better choice of supporting phrases, drawn from the text for Task 2. Task 3 then asks the teachers to contrast their own supporting phrases with the pre-given ones (not yet visible in Tasks 1 and 2). This provides an occasion for teachers to experience a contrast between the pre-given, formal phrases and their own phrases that are hopefully more meaning related. On this base, they can discuss in which way students’ texts are influenced by teachers’ support.
As teachers often articulate their ideas in quite intuitive ways, a whole-group reflection in the end is necessary to connect their first intuitive articulations to the systematic categories of ||content goals||, ||discourse practices|| and ||lexical means|| and make explicit the necessary distinctions and interplays in a graphical organiser such as the one shown in Fig. 4. The necessary distinction of categories is symbolised by the vertical arrows between the lines and the interplays by the horizontal arrows between the columns.
Teachers’ typical starting points in the activity: comparing cases from Cycles 2 and 4
The later cycles were used to investigate teachers’ typical starting points while working on the job of IDENTIFYING language demands and SUPPORTING language in the PD activity in Fig. 5. For this, teachers’ (n = 65) written dream texts and supporting phrases were analysed and systematically compared in order to condense them to typical cases. As a result of this analysis, five typical cases are presented here to exemplify the differences found for the 65 teachers.
Reporting the cases
Figure 6 shows five typical teacher writings for the PD activity in Fig. 5 (all teacher names are pseudonyms, and all were monolingual teachers in multilingual classrooms with some years of teaching experience, and the teachers’ classroom practices in these classrooms were also observed). The left column shows their original writings (translated from German).
Analysis of cases
The right column of Fig. 6 shows the condensed results of the analysis, with the activated categories and interplays in each case mapped onto the landscape of Fig. 4 (in black letters and arrows). The navigation in the landscape then enables the researcher to infer a possible next step in the teacher’s mutual learning pathway for each case (in green letters and arrows).
Case 1: Tom Taylor’s immediate phrase list
Although Tom Taylor was asked to write an explanation, as he expected the students to do, he skipped this step and immediately collected technical terms for the phrase list: “numerator, denominator, expanding, multiply”. Like Martin Schreiber in the and many other teachers, he focused only on the ||word level|| and the ||formal vocabulary||, without taking into account other categories. The rationality behind this approach was extrapolated from many discussions: His supporting phrases comprised the terms on which he had focused in the last decades, i.e. the unfamiliar technical terms that the students need to be encouraged to use. A possible next step in the teachers’ learning pathway refers to the ||interplay of discourse practices and lexical means|| and then later the extension to ||meaning-related vocabulary||.
Case 2: Selma Zellers’ exclusive focus on procedures
Whereas Tom Taylor had not thought about discourse practices and the content goal when collecting the supporting phrases, Selma Zeller deliberately wrote a ||report on the procedure|| and chose corresponding support phrases only from the ||formal vocabulary||. The possible next step in her learning pathway should extend the content goals to ||conceptual knowledge|| and then the ||interplay to the discourse practices||. (For cases such as Selma and Tom, the students’ written text examples were complemented by an explanation of meaning in the activity sheet in Fig. 5.)
Case 3: Olivia Urk’s inconsistency
In contrast, Olivia formulated a comprehensive ||explanation of meaning|| (see Fig. 6). Interestingly, her support phrases still remained on the ||formal level|| and contained phrases that do not appear in her text. Although she connected the explanation to the conceptual learning goal, the support phrases were independent from this. Thus, the possible next step of her learning pathway should address the ||interplay between discourse practices and lexical means|| and the ||distinctions between meaning-related and formal vocabulary||.
Case 4: Sibylle Niehaus’s struggle with articulating explanations
Sibylle was very concerned with ||conceptual understanding|| (and often focuses on it in her classrooms by extensively working with visual models and graphical representations, as seen in classroom video-recordings). However, when asked to write an explanation, she struggled and said, “This task really brings me to my limit”. Like Sibylle, many of the deepest thinking teachers in our sample became aware that they had no specific goal in mind in terms of how they want their students to formulate explanations. They realised their own “speechlessness” and took it as a starting point for the next step in the learning pathway by reflecting independently on making the ||interplay of the discourse practice of explaining meanings and the underlying lexical means|| explicit to students by compared it to ||interplay of reporting procedures and the formal vocabulary||.
Case 5: Elisa Erikson’s mix of discourse practices and language means
Elisa combined both the discourse practices of ||reporting procedures|| and ||explaining meanings|| in her writing and even tried ||justifying the rule||. Accordingly, she collected supporting phrases from ||meaning-related vocabulary|| and ||formal vocabulary||. The next section will show how the discussion with her discussion partner enabled her to independently discover further categories and the relevance of the distinctions and the interplays.
Summing up, the analysis of these five cases shows that most teachers can activate some relevant resources for this activity in their first PD session, but do not yet activate categories required for realising the integrative-instead-of-additive orientation. In this way, their vocabulary support does not correspond to the original mathematical goal of supporting the consolidation of conceptual knowledge. This finding of one-sided starting points was also confirmed during other activities and observations of their classroom practices (Prediger 2019).
Tracing teachers’ typical pathways through the activity: the case of Elisa, Katja, and Sanne
The following episode shows not only teachers’ starting points that are made explicit by activities such as the one in the section but also a possible learning pathway that developed during a small-group discussion of three monolingual teachers, Elisa Erikson (case 5 from the previous section), Sanne Gerster, and Katja Ludwig. To trace their learning pathways with respect to the articulation of categories, the transcripts and the analysis are presented in an alternating pattern, sequence by sequence.
Like Elisa Erikson, her colleagues Sanne Gerster and Katja Ludwig have also written rich texts in which both discourse practices appear in a mixed form. Their phrase lists are also mixed, with both formal and meaning-related vocabulary (as was Elisa’s in Fig. 6). The transcript starts with Task 3 from Fig. 4, when they compare their own phrase lists with the list provided by Martin Schreiber. This comparison initiates their process of making distinctions and connections explicit:
6 Elisa I think I would start a step earlier. What it means.
[After considering the supporting phrases from purely formal vocabulary given by the teacher in Fig. 4.]
10 Sanne Eh? Well, I would have worked much more with the figure. Structuring, in a finer way.
In turn 6, Elisa starts the discussion by activating the distinction ||procedural vs. conceptual knowledge||, which was not visible in her written text (see Fig. 6) but elicited by the communication in the small group. Sanne joins her with similar ideas (in nonprinted turns 7–9). When first approaching the teachers’ given phrase lists, Sanne transfers this distinction to the lexical means and distinguishes (in turn 10) vocabulary that can be used for explaining meanings (→ ||meaning-related vocabulary||) from ||formal vocabulary||.
Having distinguished the content goals and the lexical means, the three teachers make explicit the ||interplay of specific discourse practices and lexical means|| in specific cases each (to increase the readability of the shortened transcripts, additional words marking explicit references were added in brackets):
13 Katja For describing [procedures] this [the formal vocabulary] is quite good.
14 Sanne For describing, yes. But with it, they [the students] do not see that they [the fractions] are equivalent.
15 Katja Yes, that’s right. But this [meaning-related] strategy, making it finer anyway, but it stays the same anyway. But this is much more difficult to put into words, yes. There [for the procedure of expanding] I can construct sentences more easily … Because I can simply formulate it concretely.
In Turn 13, Katja articulates the ||interplay of reporting procedures and formal vocabulary|| for the procedural utterances and generalises to both discourse practices in turn 15. Sanne connects it to the ||interplay of content goals and discourse practices|| in Turn 14 for both discourse practices when emphasising that the discourse practice ||reporting procedures|| does not support the content goal of acquiring ||conceptual knowledge||. Some turns later, they generalise the discovered distinctions and connections:
24 Sanne It just also depends on what is intended with the question. Is its aim how to calculate? Thus how to turn the 2/6 into 6/18. Or what does it mean?
34 Sanne What else do we have [reads out loud Task (3) from the activity sheet printed in Fig. 4]. Compare this phrase list with your list and justify what you refer. We justify. I believe it really depends on how you understand the question. Well, on what you put emphasis … on procedures [or on explaining meanings].
35 Elisa [writes for Task (3)]: Here, only a supporting phrase for the procedures is given, no support for the graphical representation. That’s why we prefer our “longer” list.
Even if still somewhat contextualised and implicit, Sanne articulates the interplay of ||content goals|| and ||discourse practices|| in Turn 24 and the interplay of ||discourse practices|| and ||lexical means|| in Turn 34. Finally, Elisa writes down the distinction of ||reporting procedures vs. explaining meanings|| and the interplay of ||content goals||, ||discourse practices||, and ||lexical means||.
Finally, the plenary discussion in which the landscape (printed in Fig. 4) was explicitly written on the whiteboard enabled the teachers to articulate their ideas using the terms offered by the facilitator.
The graphical summary in Fig. 7 resumes the analysis of these teachers’ pathways through the activity in which they articulate more and more explicitly the distinctions between the procedural and the conceptual content goals, discourse practices, and lexical means and also the interplay between the three categories. Thus, teachers’ learning pathways concerning activated categories can be described as a succession of inventions, refinements and connections of categories.
The fine-grained analysis in the case of Elisa, Katja, and Sanne has provided an insight into how some teachers can develop their positions by making explicit the categories and connections on their own: when they start with some intuitive distinctions of ||procedural knowledge|| and ||conceptual knowledge||, they can discover the connection to different discourse practices and become aware of their distinctions as well as the distinctions of ||vocabulary||. Similar processes have been observed for teachers with starting points such as Sibylle’s.
In contrast, teachers with starting points such as Tom’s, Selma’s and Olivia’s require an external input for reflection before they can develop their thinking. This external input is sometimes given by the small group they work with (like in the case of Selma); if not, it is the facilitator’s job to initiate further steps in the individual learning pathways.