Three Conceptualisations of Literacy

Literacy is a word with multiple meanings in mathematics education discourses:

• Mathematics education specialists talk about the major aims of mathematics learning as mathematical literacy (Mathematische Bildung in German; see, e.g. OECD [2007] or Jablonka [Reference Jablonka, Bishop, Clements, Keitel, Kilpatrick and Leung2003]), even if they do not explicitly think about the role of language.

• In language education, disciplinary literacies for different subjects are considered as important components of pluriliteracies (Coyle & Meyer, Reference Coyle and Meyer2021, p. 49) and characterised as involving ‘the use of reading, reasoning, investigating, speaking and writing required to learn and form complex content knowledge appropriate to a particular discipline’ (McConachie, Petrosky & Resnick, 2010, p. 16).

• In a similar approach, mathematics education researchers focusing on language for mathematics learning suggested to extend narrow definitions of academic language proficiency to the wider construct of academic literacy in mathematics, involving mathematical proficiency, mathematical practices and mathematical discourse (Moschkovich, Reference Moschkovich2015).

In this chapter, we explain what we consider the core of mathematical literacy and academic literacy in mathematics and how this relates to Coyle and Meyer’s (Reference Coyle and Meyer2021) suggestion to characterise subject-specific literacy by doing, organising, explaining and arguing mathematics (according to Polias, Reference Polias2016).

Many people have experienced doing mathematics only in a very procedural sense, with a strong focus on calculating arithmetic tasks or transforming algebraic equations. In the recent TALIS study, only 18 per cent of German teachers’ explanations referred to deep content, but in Japan, this figure was 55 per cent (Grünkorn et al., Reference Grünkorn, Klieme, Praetorius and Schreyer2020).

The dominantly procedural, inner-mathematical focus on transformational activities is far away from the normative aim of developing mathematical literacy in students, which was conceptualised in consensus between all OECD states as ‘…an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen’ (OECD, 2007, p. 304).

The cited conceptualisation of mathematical literacy mainly emphasises the role of mathematics for describing and organising phenomena, according to which doing mathematics genuinely includes

• structuring context situations so that the mathematical structure underlying the situation can be captured

• mathematising everyday situations by mathematical models such as arithmetic operations or algebraic equations

• inner-mathematical transformations

• interpreting mathematical models or results in the context situation

• validating the mathematisation process and its results (Blum & Borromeo Ferri, Reference Blum and Borromeo Ferri2009).

Narrowing down the conceptualisation to the particular field of algebra, holding mathematical literacy in algebra includes the ability to use algebra as a powerful language for (1) organising phenomena and (2) exploring their mathematical structures (Bednarz, Kieran & Lee, Reference Bednarz, Kieran and Lee1996). Both algebraic activities are considered an integral part of doing mathematics in addition to purely transformational activities.

For the processes of learning algebra in such a mathematical literacy perspective, however, a second kind of literacy is crucial that we distinguish from the mathematical literacy as defined above, namely the academic language practices and academic language means needed to learn algebra and to engage in meaning-making processes for developing conceptual understanding of arithmetic operations and algebraic concepts, such as equations or equivalence of equations (Prediger & Krägeloh, Reference Prediger, Krägeloh, Halai and Clarkson2016; Usiskin, Reference Usiskin, Coxford and Shulte1988).

As many empirical studies on the role of language in developing conceptual understanding in mathematics have shown, two discourse practices are crucial in the processes of developing conceptual understanding (Moschkovich, Reference Moschkovich2015; Prediger, Erath & Opitz, Reference Prediger, Erath, Opitz, Fritz, Haase and Räsänen2019). Specifically,

• not reporting procedures, but

• explaining meanings of mathematical concepts and

• arguing how symbolic and graphical or contextual representations are connected to each other.

For specifying in detail what literacy for learning algebra might entail, we give examples for two of Coyle’s & Meyer’s (Reference Coyle and Meyer2021) five core constructs: (1) target concepts and forms of algebraic reasoning and (2) forms of representations.

Target Concepts in Algebra and Forms of Algebraic Reasoning

Many students and adults think about algebra as a game of meaningless transformations conducted according to transformation rules learnt by heart, as in the following example.

$3x+5=11\iff 3x=11-5=6\iff x=2$

The rules are as follows: we can transform an equation by adding or multiplying the same numbers on both sides; for example, adding -5 on both sides eliminates the 5 on the left side and makes -5 occur on the right side of the second equation. Then, multiplying by 1/3 or dividing by 3 realises the second transformation.

In students’ thinking, transformation rules that are not underpinned with meaning can easily become arbitrary, so typical errors have often been documented, such as the error 3x + 5 = 11 ⇔ 3x = 16 (Malle, Reference Malle1993). Additionally, students who cannot make sense of the meaning of the variables and equations are not able to use equations to structure everyday situations (i.e. to generalise mathematical relations).

Algebra education research has therefore distinguished typical forms of algebraic reasoning, all of them being important in school (Bednarz, Kieran & Lee, Reference Bednarz, Kieran and Lee1996; Kieran, Reference Kieran, Stacey, Chick and Kendal2004):

• transformational activities (i.e. manipulating algebraic expressions and equations according to procedural transformation rules without interpreting the symbolic letters)

• operational activities (i.e. using algebraic expressions and equations mainly for evaluating them for concrete numbers)

• generational and relational activities (i.e. using variables, expressions and equations to express general structures or conditions in everyday situations or inner-mathematical situations)

• meta-level activities (i.e. to justify the transformation rules with respect to the meaning of variables and the relational activities).

Whereas transformational and operational activities mainly focus on procedural aspects of algebra, generational activities and meta-level activities are key for developing conceptual understanding and thereby deeper learning.

For these activities, the four target concepts are variable, equation, solution of equation and equivalence of equation. Table 13.1 presents the algebraic target concepts with their meanings in the different forms of reasoning. Some of them will be explained in further detail in the following sections. Even if not all concepts are explained in detail in the table, the overview shows that the algebraic target concepts form a network of concepts with strong connections that need to be built by students. Furthermore, Table 13.1 depicts that the meta-level activity of justifying the transformation rules includes and connects the three activities of algebraic reasoning.

The list of highly complex meanings sheds a first light as to why the two most important discourse practices are also challenging with respect to subject-specific literacy: explaining meanings of these four algebraic target concepts requires a highly elaborated language and their highly condensed articulation in Table 13.1 is not at all accessible for students in the first approach. That is why algebra education researchers regularly work with multiple representations, as reported in the next subsection.

Multiple Representations for Understanding Equations and Their Equivalence

Understanding the meaning of equations can be supported by the use of multiple representations for algebra concepts, usually the symbolic–algebraic representation, the symbolic–numeric representation, different graphical representations and the textual representation, often of context problems (Friedlander & Tabach, Reference Friedlander, Talbach and Cuoco2001; Kaput, Reference Kaput1998). Figure 13.1 illustrates the representations in a (simplified, thereby not authentic) example.

Figure 13.1 Multiple representations for algebraic equations

The translation from the textual representation of the context problem into the symbolic–algebraic representation is a typical generational activity that requires the interpretation of the variable as unknown (see Table 13.1). Once the algebraic equation is found, it can be solved by transformational activities, that is, by transforming the equation into x = 2.

However, before students are able to find the equation, they need to have developed the conceptual understanding of variables (as unknown numbers) and equations (as symbolising conditions for unknown numbers). These meanings can be constructed by the graphical representation in the bar model and by trial and error for evaluating the equation by several numbers. As the example shows, numerical operations and the bar model are crucial language means for developing the meaning of variables.

The bar model is a variant of the Singapore bar model, which is used in many countries as a graphical model that can support even primary students in solving algebraic context problems without any variables and equations (Fong Ng & Lee, Reference Fong Ng and Lee2009). Malle (Reference Malle1993) has suggested its use (not for avoiding algebraic equations, but) for a deeper learning goal, namely making sense of the transformation rules for algebraic equations: in the generational activity context, two equations are equivalent if they describe the same condition or same situation. In Figure 13.1, the equations 5x + 3x = 11 and 3x = 11 – 5 describe the same bar, and looking at the bar makes their equivalence immediately visible.

In the following sections, we report on design experiments in which we tried to explore Malle’s (Reference Malle1993) bar-model-based approach towards meaning-making for the equivalence of equations. We will show that the instructional approach bears further language challenges as students struggle to argue concisely how the representations are connected.

Methodological Background

In order to determine typical obstacles on students’ pathways towards deep understanding of algebraic equations and their equivalence in the bar-model-based approach, we used a design research methodology (Cobbet al., 2003; Gravemeijer & Cobb, Reference Gravemeijer, Cobb, van den Akker, Gravemeijer, McKenney and Nieveen2006). Based on an initial specification of the mathematical learning content (relevant target concepts, forms of reasoning, relevant representations and their connections according to Table 13.1 and Figures 13.1, 13.2), we designed a learning environment encompassing four sessions of 45 minutes each for the target group of low-achieving tenth graders in remedial mathematics classes aiming to pass their Grade 10 exam in a pre-vocational setting (called ‘Berufsfachschule 1’ in Germany, a Level 2 (pre-)vocational programme in the European framework). In this way, design research iteratively combines the repeatedly refined design of learning environments with the empirical, mostly qualitative investigation of the initiated teaching–learning processes in design experiments so that dual aims can be combined, achieving research-based design products and design-based research results, namely, insights into typical pathways and obstacles. In this process, the language demands occurring in the mathematical learning pathways are empirically identified and inform the construction of language learning goals and language scaffolds for the next design experiment cycles (Prediger & Zindel, Reference Prediger and Zindel2017).

The data-gathering in our design experiments is still ongoing, but revisions between the mini cycles have already been completed. Design experiments have been conducted with four students in pair or individual settings. Since the experiments were conducted amidst the Covid-19 pandemic and resulting school closures, some of them were conducted via video conferences. In total, 675 minutes of video data have already been collected and partially transcribed.

In the current paper, we present five episodes, chosen with respect to their potential of providing insights into typical challenges also found in earlier design research studies on deep algebra learning (Prediger & Krägeloh, Reference Prediger, Krägeloh, Halai and Clarkson2016). Following these episodes, we show how the design of the learning environment was enriched with respect to language.

Episode 1 and 2: Frank’s and Vivien’s Surface Strategies Rather than Deeper Learning

The brief Episodes 1 and 2 stem from the first design experiment session, one with Frank and one with Vivien, two 18-year-old students placed in remedial mathematics courses due to earlier struggles. Both students were born and educated in Germany. Frank’s family only speaks German, Vivien’s family German and Polish.

Figure 13.3 Initial task in first design experiment sessions for connecting the bar model to the equations ‘that belong together’ (together with Frank’s correct handwritten answer)

48 DE leader:

And does this figure have anything to do with it [pointing at the equations]?

49 Frank:

Yes, actually it does, because it makes clear what kind of numbers you can or have to use for it.

50 DE leader:

Oh, actually you only need it [the bar model] to know the numbers or what?

51 Frank:

I think, as well for seeing which number is smaller or bigger.

So, I wouldn’t see anything more in it or take anything out of it.

Although Frank’s equations are correct, he articulates that the bar model only serves as a provider of numbers (Turn 49). This surface strategy does not refer to the mathematical relational structures of the operations, which are also depicted in the bar model. Furthermore, the surface strategy does not unfold what it means that the equations belong together.

Figure 13.4 Vivien’s wrong connection of a bar model to the equation – focus solely on the numbers, not on the relations

74 DE leader:

Can you explain to me what the ‘plus’ indicates in the drawing?

75 Vivien:

I would say again, the line that is in between. You can see that it is definitely a plus. Or it could also be a minus. But it could not be 3 + 12, because I see immediately that x is 9. Thus, a minus. That x is in any case a 9.

76 DE leader:

How did you know where to put the 3, the x, the 12 in the drawing?

77 Vivien:

Because I’ve already seen that on the next one on this sheet [pointing at the bar model on the next task for 3.2 + 4.5 = u], the next drawing of it.

78 DE leader:

But couldn’t you write the 3 in the place where 12 is written or 12 where the x is written?

79 Vivien:

I don’t think so. You would get mixed up. It was the same with the other sheets: the exact number of centimetres was at the bottom and 2 and 3 at the top, so …

70 DE leader:

And where do you see 3 + x in the drawing?

71 Vivien:

Nowhere.

From these two brief Episodes 1 and 2, we learnt that juxtaposing representations is not enough as the design must make sure that students also focus their attention on the additive structures expressed in the bar model and in the equation, rather than only the numbers. The phenomenon has often been documented for younger children but persists in the thinking of low-achieving adolescents who have not learnt to talk about mathematical structures and operations (Prediger, Reference Prediger2019a; Prediger & Zindel, Reference Prediger and Zindel2017).

More in general, the episode resonates with the findings by Amit and Fried (Reference Amit and Fried2005), which state that we need ‘not just the presence of different representations said to be connected but “connectors” as well’ (Amit & Fried, Reference Amit and Fried2005, p. 63).

Episode 3: Vivien’s Limited Language Means for Expressing Multiplicative Structures

As a consequence from the findings in the first and second design experiment sessions, we redesigned the learning environment for the third session by starting with a context problem to support students’ meaning-making processes for the additive and multiplicative structures in the bar model. Figure 13.5 shows the newly introduced task that connects four bar models to the textual representation of context situations with different running programmes.

Figure 13.5 Newly designed Task 0 for connecting different bar models to the context situations (together with Vivien’s handwritten answers)

21 Vivien:

John’s run fits to [lower left bar of New Task] because he states he ‘runs the 21 km every time’.

Thus, the one where 21 is written belongs to Max, not John.

22 DE leader:

That means, we can write John beneath the 21?

23 Vivien:

[Nods] Um, um. And Eva beneath the one next to it [refers to the lower right bar].

Where there is written 3, then the middle line and then 7.

24 DE leader:

And can you explain how you came up with that or why?

25 Vivien:

Because it says, ‘Eva runs three times a week 7 km’.

And then I would say, the 3 stands for ‘three times a week’ and the 7 for ‘7 km’.

Similar to Episodes 1 and 2, in which students only focused their attention on the numbers while connecting the graphical and symbolic representation without taking into consideration the structure of the operation, Vivien uses the same surface focus for connecting textual and graphical representations in this transcript. Rather than expressing the additive structure of 3 and 7 in this bar, she only articulates the numbers, not joint lengths: ‘Where there is written 3, then the middle line and then 7’ (Line 23).

Here, it is worth analysing the lexical details and their connection to deeper thinking: ‘and then’ is the only connective between 3 and 7 that she uses, which does not allow her to distinguish an additive structure from a multiplicative structure.

Some minutes later, the design experiment leader guides her to discover that also the upper right bar (with three segments of 7 km) can be assigned to Eva’s running programme

45 Vivien:

Actually, we could also write Eva for the picture above this one.

46 DE leader:

Why can we write Eva also here?

47 Vivien:

Because, um, it, it is even easier, um, than, this. Because there ‘Eva runs three times a week 7 km’ is written and here [refers to the bar with three segments of 7] are, thus, three sections, and there above, it is 7.

Episode 3 shows that context problems can reveal the didactical potential for supporting students to grasp the multiplicative structure. These first and tentative insights might stay restricted if the students do not get access to any mediating language means; specifically, in Vivien’s case, language means for expressing the key multiplicative structure by expressing the unitising.

Summary of First Empirical Findings

From the qualitative analysis of these three (and many other) episodes in our ongoing design experiments, we conclude that low-achieving students’ processes of meaning construction in the bar model can be substantially hindered by limitations in focusing, identifying and articulating the additive or multiplication structures relevant in the textual, symbolic and graphical representations. It seems these observed adolescents in our sample have already completed ten years of schooling without actively participating in discourse practices involving deeper learning with the bar models, although the bar models were already introduced in Grade 1 (perhaps in slightly different representations).

The connection between thinking the corresponding discourse practices and the language means to articulate them is striking: whereas the sole focus on numbers only requires limited language means, such as deictic means (‘here, there, above’) and one single logical connective (‘and then‘), a deeper conversation about the mathematical structures in each of the representations requires many more connectives for expressing the different meanings of the operations. Although these conceptual (and underlying language-related) challenges are well known from Grades 1–3 (Götze & Baiker, Reference Götze and Baiker2021; Kuhnke, Reference Kuhnke2013) and Grade 5 (Prediger, Reference Prediger2019a), we had not expected to also find them in tenth-grade students with limited access to mathematics and its language. Apparently, if students never get the opportunity to overcome them, the challenges persist.

Identified Mathematically Relevant Language Demands for Meaning-Making

What consequences can we draw from the first design experiments? The bar-model-based approach to meaning-making for the equivalence of algebraic equations (Malle, Reference Malle1993) tries to exploit the connection of graphical, symbolic–algebraic and symbolic–numerical representations (see Figure 13.2). The equations 5 + 3x = 11 and 3x = 11 – 5 are called equivalent as they describe the same bar. Understanding this meaning of equivalence allows students not only to deal with algebra on the surface level but also to reach deeper learning. However, to fully understand and work with this characterisation of equivalence, students need to develop and articulate strong connections between the textual, graphical and symbolic representations (Friedlander & Tabach, Reference Friedlander, Talbach and Cuoco2001; Malle, Reference Malle1993; for language learners in general Moschkovich, Reference Moschkovich2013). Dealing with multiple representations can have different degrees of depth (Kuhnke, Reference Kuhnke2013; Prediger & Wessel, Reference Prediger and Wessel2013):

• only juxtaposing the representations

• only identifying the numbers in each representation

• identifying also the relevant mathematical structures in each representation

• arguing how the relevant mathematical structures correspond between the representations.

The design experiments revealed that, even in Grade 10, low-achieving students do not necessarily master all language demands occurring on students’ pathways towards meaning-making at the deeper levels. In the discursive dimension, these language demands particularly include two discourse practices:

• explaining meanings of mathematical concepts and representations (in particular, additive and multiplicative structures in textual and graphical representations)

• arguing how symbolic and graphical or contextual representations are connected to each other.

In the lexical dimension, these discursive demands require the use of connectives and chunks for expressing additive and multiplicative structures, for the equal sign (connectors for the two expressions in each equation and the two bars in the bar model), the equivalence (connectors for the equations as conditions describing the bars) and the connection of representation (connectors for the graphical, textual and symbolic representation).

From the analysis of the design experiments, we identified a list of essential topic-specific language as summarised in Figure 13.6.

Figure 13.6 Lexical language demands for deeper learning of algebra: Meaning-related connectives and chunks for expressing additive and multiplicative structures

Turning the analytic findings into consequences for the redesign of the learning environment, this list in Figure 13.6 reveals the potential scaffolds needed for students with limited language proficiency on their learning pathway. As Prediger and Krägeloh (Reference Prediger, Krägeloh, Halai and Clarkson2016) have emphasised for other parts of algebra, these meaning-related chunks and connectives play the role of epistemic mediators between students’ everyday language and the technical language referring to the symbolic representation (factor, product, quotient, equivalence, …). The technical language is less relevant for supporting the meaning-making processes than these meaning-related chunks and connectives.

Macro-Scaffolding by Combining Mathematical Content and Language Learning Trajectory

For the new design experiments in Cycle 2, we decided to restructure the learning trajectory as depicted in Figure 13.7. According to the design principle of macro-scaffolding (Gibbons, Reference Gibbons2002; Pöhler & Prediger, Reference Pöhler and Prediger2015), we have now sequenced the mathematical content trajectory and combined it with the corresponding language trajectory: the construct of learning trajectories was introduced in socio-constructivist mathematics education design research in order to structure learning opportunities along hypothetical steps (Gravemeijer, Reference Gravemeijer, Kilpatrick and Sierpinska1998). It can be aligned with Gibbon’s language trajectories since the mathematical content trajectory is also shaped by tasks and support means designed to leverage students’ progress along and between these steps, without assuming unitary pathways, of course.

Figure 13.7 Combining the mathematical content and language learning trajectory towards equivalence of equations

To adapt the macro-scaffolding principle for mathematics education needs, Pöhler and Prediger (Reference Pöhler and Prediger2015) drew upon typical content trajectories as established in the Realistic Mathematics Education approach (Gravemeijer, Reference Gravemeijer, Kilpatrick and Sierpinska1998). The trajectory starts from imaginable context problems that allow students to reinvent mathematical concepts, thereby allowing them to mentally construct their meanings in a guided process of emergent modelling, first as models for context problem situations, which are later used as models of abstract mathematical concepts. Only in a last step of the formal level are purely symbolic transformations introduced.

Level I: Introducing the Bar Model and Arguing How Representations Are Connected

Rather than directly starting at Level II, as in the first design experiment sessions, the restructured content trajectory now starts on Level I with introducing the bar model in context problems and by explicitly directing students to focus on its inherent mathematical structures. Figure 13.8 shows the redesigned task in which the focus is directed more explicitly at the inherent additive and multiplicative structures. As the context problem directly reveals lengths, their representation by line segments is very direct. In later tasks, abstract quantities are to be represented by line segments as well. This is the typical shift from a visual model of context situation to a model of an abstract operation (Gravemeijer, Reference Gravemeijer, Kilpatrick and Sierpinska1998). Here, the equality is represented by three running programmes of equal length in total. Later, it will represent other kinds of equalities (e.g. equally expensive).

Figure 13.8 Level I task with a stronger focus on the structures

Level II: Approaching Equivalence of Numerical Equations and Explaining the Meaning

The major task of Level II is printed in Figure 13.3; in the restructured learning environment, the prompt ‘Do it like Max and switch between the figure and the equations’ is broken down into smaller units with refined prompts:

• Explain how the three equations describe the same bar. How can you see the different operations in the same bar?

• Explain now how these equations belong together.

After this very open, initial explanation task, the next task (shown in Figure 13.9) provides an example explanation that can be adopted by the students. The teachers then discuss how this explanation relates to the students’ own explanations in the earlier task.

Figure 13.9 Unpacking Task 1b in Figure 13.3 and providing scaffolds

Also, the visual scaffold by gray shades in the lower part of Figure 13.9 has been designed to direct students’ attention to the structures and the subtle differences in interpreting the bars.

Level III: Transferring Equivalence from Numerical to Algebraic Equations
and Explaining the Meaning

Level III is a transfer stage when students make the transition from numerical to algebraic equations, involving not only numbers but also variables and signifiers for complete subexpressions (using the analogies depicted in Figure 13.2). In this step, the reasoning with variables is explicitly connected to the reasoning with numbers, a very important focus on the transition from arithmetic to algebra (Kieran, Reference Kieran, Stacey, Chick and Kendal2004; Malle, Reference Malle1993; Warren, Reference Warren2003). Episode 2 above shows that once the additive structures were articulated, the variable was not the central obstacle for Vivien.

The design experiments indicate that when the deeper learning for numerical equations is accomplished successfully, this transfer is not too challenging.

Level IV: Using Equivalence for Solving Equations and Reporting Procedures

In the last step (see Figure 13.10), the developed conceptual understanding is used to introduce the corresponding procedures of transforming equations in order to solve them. In these tasks, the bar model is used to justify why the equation can be transformed. It is the argumentative warrant for the transformation rules, which are generalised in two steps. Task b) provides the explanation scaffolds of Task c).

Figure 13.10 Unpacking Task 1b in Figure 13.3 with visual scaffolds and explanation scaffolds

Episode 4: Scaffolding with Relevant Meaning-Related Chunks

On each of the four levels presented above, scaffolding with meaning-related chunks and connectives (listed in Figure 13.6) is provided to support students in explaining differences. In the following episode, Vivien was asked to find the corresponding equation to the given bar model. Firstly, she assigns the expression 2 × 6 to the model:

81 DE leader:

How can you see now that this corresponds to 2 × 6?

82 Vivien:

With the 2, this middle line, and then 6.

83 DE leader:

Yes, but isn’t it different from the task before [referring to the expression 3 × 4$]$? There we had 3 times 4, for example, and then we also had three groups of four. And do we have two groups of six here now?

84 Vivien:

No, so, then I must apparently have 2 plus 6.

85 DE leader:

Yes. But do you know why?

86 Vivien:

Because in the first one [task before] there are 4 plus 4 plus 4, and here there are only 2 and 6. Because I think if there were ‘times’, it would always be the same [refers to groups of equal size].

87 DE leader:

Exactly, because if we would have times, then we would have here [points to the drawing] also two groups of six. But we don’t have two groups of six. We have a 2 and a 6. And that’s like running again: if we run 2 km and 6 km, we have a total of 8 km.

Episode 5: Frank’s Little Push

At the end of the second session, after working with the addition bar model as well as the multiplication bar model, Frank worked on the task in Figure 13.11, which asks students to verbalise the differences between both. At first glance, Frank does not see any error in Max’s writing. But looking closer, he describes:

76 Frank:

Oh, the 3 is bigger than the 4.

77 Frank:

[Some minutes later while looking at the task] Ah. I would see a plus there [points at the addition bar].

78 DE leader:

How do you see that?

79 Frank:

Because there’s only one 3 and one 4. For me that makes 7 and not 12. But if there’s a 3 twice, or how do you calculate it now? 3 times 4 at the most. Or… I would see a plus. But I don’t know.

80 DE leader:

Can you explain again why you see a plus?

81 Frank:

Because the number isn’t there several times like in the other tasks. In the other tasks, there are several numbers. In the other exercises, when there was a 12, there was a 3, four times, instead of a 3, once. But because there is now a 3 and a 4, it only makes 7 theoretically. And they are the same length as the 12, and that doesn’t make any sense.

This episode indicates a small nudge to push Frank towards a meaningful linking of the underlying mathematical operation in both representations, scaffolded by the teacher’s repeated prompts to articulate more deeply. But Frank is still missing adequate language means to describe more precisely which operation is suitable and, especially, why. It confirms the need of scaffolding language means for expressing the additive and multiplicative structures as planned in the revised teaching–learning arrangement. It has also motivated us to start producing instructional videos in the research project MuM-Video in order to provide models for good, meaning-related explanations.