Easy and difficult maths problems – and why language matters

Easy and difficult maths problems – and why language matters

by | Emerging Researchers' Group, NW 24 Mathematics Education Research

As soon as we receive a task, we judge it, whether mentally or verbally. Is it interesting, boring, easy, difficult, or worthwhile? In school, teachers and students handle a large number of tasks every day, so it is unsurprising that they do the same.

For example, when  looking at the maths problems below, can you immediately tell which ones you find easy and which you would find more difficult to solve?

Three maths problems are shown. 350 + 120 769 + 858 444 - 111

If you were then asked to sort each addition problem according to its difficulty into the following table, could you solve the task, or would you hesitate?

A piece of paper with a line drawn down the middle. On one side of the line is a simple drawing that resembles a feather or perhaps a stalk of wheat. On the other side of the line is a simple drawing of two circles joined by a line, that resembles a dumbbell.

This sorting task is commonly used in German primary schools. Let’s take a closer look at it, and at the linguistic level of its illustrations. At first glance, the teacher may think this is a simple task, but when considered more closely, it is clear that understanding the task depends on the level of students’ experience, especially those new to the language of instruction, i.e., non-native speakers.  

In this blog post, I’d like to expand on the situational and linguistic factors that influence how a student may perceive the difficulty of a maths problem – beyond their basic mathematical abilities. 

Are there objective or subjective categories in mathematics teaching?

According to mathematics educators, the difficulty of math problems depends on how much effort students have to put into solving each problem, which again depends on their mathematical abilities (Rathgeb-Schnierer & Green 2015), and even diagnoses such as dyscalculia and other learning disabilities.

With this in mind, let’s look again at the maths problems above, how students might perceive them, and how educators might react to students who struggled with the task. Did you find them easy or difficult to solve?

350 + 120 – this task should be easy, given that you understand the decimal place value system. You can add the tens and the hundreds separately without having to carry, and each number is comparatively small.

References in the illustration: Torbeyns&Verschaffel (2013), Baroody (1984), 

Situational factors that affect mathematics understanding

Your judgement of a task’s difficulty might also depend on other factors. Maybe you spent the last weeks working only on subtraction problems, so they seem a little easier right now. Maybe your favourite number is eight, so you enjoy calculating with numbers that contain an eight, which makes the second problem the easiest for you. Perhaps the person next to you received the worksheet earlier, and after watching them calculate, none of the problems are difficult for you anymore. Maybe you want to impress your teacher by showing that you can solve all of the problems, even though they are difficult.

Your mathematical knowledge and familiarity with certain calculation strategies is only one factor that might affect how you solve maths problems. Equally important is your current situation and your relationship with the people you solve the problems with and for.

Whether you use the table to sort the problems as expected also depends on whether the task and the symbols were explained properly.

Linguistic factors that affect mathematics understanding

Whether the table above left you at a loss probably depends on how familiar you are with the German translations of ‘easy’ and ‘difficult’: ‘Leicht’ and ‘schwer’.

These words are often illustrated using drawings in the picture above of a feather and a dumbbell. These symbols are based on a second meaning of the German terms. ‘Leicht’ can also mean ‘light’ in the sense of weight, as illustrated by the feather. ‘Schwer’ can mean ‘heavy’, as illustrated by a dumbbell.

The task is to put the problems you find easy in the left column with the feather and the problems you find difficult in the right column with the dumbbell.

This example of a task observed in a German primary school classroom shows the relevance of language in mathematics classes – and how unexpected it can sometimes be. Without the linguistic explanation, there is no way of logically concluding which column to use for the easy and which for the difficult problems. Trying to find a logical connection between the terms and the symbols might even be misleading.

For example, you might consider the difficulty of a task to be its complexity and apply this to the illustrations. In that case, you might find the dumbbell symbol (which only consists of three lines) a lot less complex than the feather symbol. Consequently, you would sort the easy and difficult problems opposite to the task’s intention. Your only chance to use the table as intended would be to guess. But if you guessed wrong (and you have a 50% chance to do so), that would not be indicative of either your linguistic or mathematical abilities.

Especially when working with a student who is new to the language of instruction, we teachers, teacher assistants, or peers may be quick to assume students’ mistakes are due to a lack of linguistic understanding. We might even believe they didn’t receive proper mathematics education in their previous school.

Conclusion

There is more than mathematics and mathematics learning going on in mathematics lessons. We are often not aware of linguistic aspects, even in tasks and phrases we use every day, that are not explained sufficiently, even by translating.

All students face the challenge of handling these linguistic and situational aspects, but they may be especially confusing to those who have only recently joined a class in a new country, a new language, or a new culture. 

Can you think of an example of ambiguous terms or tasks in your native language? Do you also use the differentiation between ‘easy’ and ‘difficult’ maths problems in your schools? 

Key Messages

  • Beyond the actual learning of mathematics, situational and linguistic aspects are relevant when students are working on a task
  • Any of these aspects can influence whether a maths problem is perceived by the student as easy or difficult. Not understanding the maths problem, therefore, does not unambiguously point to a level of mathematical ability
  • Certain aspects of language can be open to interpretation, such as when using images. We need to be aware of this and take more care when using everyday tasks and phrases that might cause confusion.

Alexandra Dannenberg

Research Assistant and PhD candidate, Kassel University, Germany

Alexandra Dannenberg is currently a research assistant and PhD candidate at Kassel University, Germany, and the graduate school InterFach. She studied primary education in the subjects mathematics, German as a first language, and natural & social sciences. During her studies, she discovered her interest in education research and began her current position in primary education research shortly after graduating. In her doctoral studies, she focuses on the relevance of language in primary school mathematics classroom interactions. Her general research interests are power relations in education, educational disparities, and institutional discrimination.

Other blog posts on similar topics:

References and Further Reading

Baroody (1984): Children’s Difficulties in Subtraction: Some Causes and Cures. In: The Arithmetic Teacher, 32, 3, pp. 14-19. https://www.jstor.org/stable/748349?seq=1&cid=pdf-reference#references_tab_contents

Rathgeb-Schnierer & Green (2015): Cognitive flexibility and reasoning patterns in American and German elementary students when sorting addition and subtraction problems. In: Proceedings of CERME 9 – Ninth Congress of the European Society for Research in Mathematics Education, pp. 339-345. https://hal.science/hal-01281858/document

Torbeyns&Verschaffel (2013): Efficient and flexible strategy use on multidigit sums: a choice/no-choice study. In: Research in Mathematics Education, 15, 2, pp. 129-140. https://doi.org/10.1080/14794802.2013.797745

Fostering Creativity in the Classroom  – developing a cross-curricular module in ITE

Fostering Creativity in the Classroom  – developing a cross-curricular module in ITE

by | ESAI, NW 16 ICT in Education and Training

Outside of the core curricular content that makes up initial teacher education (ITE) programmes, there are increasing callsfor input on a variety of pedagogical, social, cultural, and competence-based issues that impact future teaching practice (MacPhail et al., 2022). Most higher education institutions possess expertise in a range of innovative areas, but the practicalities of timetables, student availability, and academic structures often mean students must choose one or two five-credit, level nine modules from a range of electives. This blog outlines an effort to combat this through an integrated module on the Professional Masters in Education PME (Post-Primary) at Dublin City University, Ireland, which combines Digital Competencies, English as an Additional Language, and Drama-based learning under the umbrella of ‘Fostering Creativity in the Classroom’.

Fostering creativity in the classroom  – developing the module

We know that cross-curricular and integrated teaching can facilitate students in making creative connections and solving complex problems (Harris & de Bruin, 2017). Motivated by this, we began examining our content, values and teaching approaches and quickly realised a common thread of creativity ran through our work. Our module, ‘Fostering Creativity in the Classroom’ places explicit focus on the role of the teacher in fostering creativity and innovation in the post-primary classroom. Using a multidisciplinary approach, we allowed students to explore and experience a range of creative, collaborative and playful approaches to fostering creativity in teaching and learning. This was achieved through lectures, workshops, and a range of strategies from the Digital Learning(e.g. digital storytelling), Drama (e.g. soundscapes), and Linguistic Responsiveness (e.g. multilingualism) domains.

Underpinned by theories of creativity in education (e.g. Gilhooly & Gilhooly, 2021), our students, who are pre-service teachers, worked together to experiment with creative approaches, reflect on their experiences, and plan their practical implementation in the future. In order to draw the different strands together under the theme of creativity, we designed an innovative assignment. The assignments tasked students (in small groups) with creating a digital story on the theme of ‘fostering creativity and innovation in the post-primary classroom’. Their target audience was future PME students and practising teachers. Videos considered how digital media, drama and linguistically responsive strategies can enhance practice and encourage pupil creativity. Groups reflected on the strategies explored during the module and considered their application across curricular subjects. Videos were to be presented as a cohesive narrative or story and include a variety of audio-visual content.

Our reflections

Reflecting on the process, we were pleased it did not result in merely fitting our ‘pieces’ together, but in creating something unique that was enriched by our individual curricular areas. As academic staff, collaborating on the design and delivery provided us with opportunities to learn from each other’s curriculum design and facilitation approaches while demonstrating to students the connections that exist between subject areas. Our challenges were primarily around articulating our vision and structuring the delivery. While, as academic staff, our initial vision for the module was clear, our individual ‘flavours’ of that vision came through at first. It wasn’t until the second iteration of the module that we began to speak in one voice. The structure of the module delivery was another aspect that we found challenging initially and that we improved over time. In the first iteration of the module, we split the content into ‘blocks’, where each team member delivered their content in sequence after one another. We found that this meant students saw the module as three separate parts, and while that made sense in terms of the coherence of each aspect, it took away from the overall flow and interconnected nature of the work we were trying to achieve. Rectifying this was more than the simple act of moving lectures from one week to another. Instead, it necessitated the alteration of certain aspects of content so they more naturally connected to the other areas of study. We also spent more time ‘in’ each other’s lectures in order to display a unified voice.

Students’ impressions

Feedback from students on the experience of participating in this integrated module contained positives and potential areas for improvement. Students commented on the module’s ambitious and forward-thinking nature, saying it was ‘pretty ambitious’ and ‘very relevant in modern education’. They noted that they learned a lot from each strand and, perhaps most importantly, they learned more from how the strands linked together. Comments included: ‘Lovely to have different strands (E.g., Digital Media, Drama-based learning) each week. Helped the creativity’ and ‘It helps shift your focus from the ways in which you were taught at school and to focus on all of the possibilities that exist for enhancing your lessons and making them more meaningful’.

 On the other hand, students also found areas challenging. For example, they found that the three strands meant there was a lot to take in. Comments included ‘There was a lot of information in each module[strand] and not enough time to get to grips with all of it’. Some also found it difficult to see how everything aligned under the umbrella of developing pupils’ creativity in the classroom. Comments included ‘personally felt that there was a bit of a disconnect between the three strands’ and that they ‘did not find the necessarily all aligned under the umbrella of creativity’.

Fin

Our efforts to combine three elective strands into one coherent module were not without their challenges. However, as lecturers, we found that the process not only allowed us to examine connections across curricular areas but facilitated the development of a more nuanced version of ‘creativity’ than we had delivered before. Students also recognised the value of our integrated approach, which is encouraging. However, their comments also provide scope for improvements in the future. For example, further work may be needed to increase the connection between strands so that students see the module as a cohesive approach to develop pupils’ creativity in the classroom.

Key Messages

  • The practicalities of ITE programmes often make the provision of additional pedagogical, social, cultural, and competence-based initiatives challenging
  • We document the development of a cross-curricular, integrated module “Fostering Creativity in the Classroom”
  • The module integrates Digital Learning, Drama-based Learning, and Linguistic Responsiveness
  • The process provided us, as academic staff, with the opportunity to enrich our individual curricular areas and practice by learning from each other’s design and facilitation approaches.
  • Students found the module to be ambitious and forward-thinking, and learned more from how the curricular areas fitted together to ‘foster creativity in the classroom’.
Dr Peter Tiernan

Dr Peter Tiernan

Associate Professor in Digital Learning and Research Convenor for the School of STEM Education, Innovation and Global Studies in the Institute of Education at Dublin City University.

Peter is an Associate Professor in Digital Learning and Research Convenor for the School of STEM Education, Innovation and Global Studies in the Institute of Education at Dublin City University. He lectures in the areas of digital learning, digital literacy and entrepreneurship education. His current research focuses on digital literacy at post-primary and further education level as well as entrepreneurship education for third level lecturers and pre-service teachers.

Peter was shortlisted for the DCU President’s Award for Excellence in Teaching and Learning in 2021.

Find Peter on Twitter.

Dr Fiona Gallacher

Dr Fiona Gallacher

Assistant professor in the School of Applied Language and Intercultural Studies (SALIS) at Dublin City University

Fiona Gallagher is an assistant professor in the School of Applied Language and Intercultural Studies (SALIS) at Dublin City University. Before this, she worked as a teacher and CELTA teacher educator in Sudan, Italy, Spain, Ireland, the US, Australia and Portugal.  Her research interests lie primarily in the fields of second language acquisition, TESOL and bi/multilingual education with particular reference to: L1 use in language learning and teaching; translanguaging and plurilingual pedagogies; and teaching and learning in the linguistically and culturally diverse primary and secondary school classroom. 

She has published widely in her field, both as the author/co-author of various EFL textbooks and teacher guides and in high-ranking peer reviewed journals and edited volumes. 

Dr Irene White

Dr Irene White

Assistant professor in English and Drama Education in the School of Human Development at the Institute of Education, Dublin City University.

Dr Irene White is an Assistant Professor in English and Drama Education in the School of Human Development at the Institute of Education, Dublin City University. She is the Programme Chair of the Professional Master of Education and teaches across a range of initial teacher education programmes. Irene taught English and Drama at the post-primary level for twelve years, during which time she was a mentor for initial teacher education students and a State Exams Commission examiner for the Leaving Certificate Applied Programme.

Irene’s research straddles the arts and education sectors, with a particular focus on creative mindsets, creative learning environments and creative activity for health and wellbeing. Her PhD examined creativity in participatory arts initiatives and articulated a Participatory Arts for Creativity in Education (PACE) model, an applied participatory arts model aimed at fostering creativity in education.

Irene is Chair of the Board of Directors for Upstate Theatre Project, a community-engaged participatory arts organisation funded by the Arts Council of Ireland. Her work in the field of participatory arts includes her role as artist and director with Upstate Theatre Project on The Crossover Project, a cross-border, cross-community participative drama programme, and her work with students from the Steinhardt School of Culture, Education and Human Development, New York University on the study abroad programme. Irene has also worked with Smashing Times Theatre and Film Company on the ‘Acting for the Future’ programme using drama and theatre performance to promote positive mental health and the ‘Acting for Change’ programme using drama to explore cultural diversity and identity and promote anti-racism, anti-sectarianism and equality.

Other blog posts on similar topics:

References and Further Reading

Gilhooly, K. J., & Gilhooly, M. L. M. (2021). Aging and creativity. Academic Press.

Harris, A., & de Bruin, L. (2017). Steam education: Fostering creativity in and beyond secondary schools. Australian Art Education, 38(1), 54–75.

MacPhail, A., Seleznyov, S., O’Donnell, C., & Czerniawski, G. (2022). Supporting the Continuum of Teacher Education Through Policy and Practice: The Inter-Relationships Between Initial, Induction, and Continuing Professional Development. In Reconstructing the Work of Teacher Educators: Finding Spaces in Policy Through Agentic Approaches—Insights from a Research Collective (pp. 135–154). Nature.

Language scaffolding as a complex craft

Language scaffolding as a complex craft

by | NW 31 LEd – Network on Language and Education, VOR (Netherlands)

Worldwide, an ever-increasing number of students undertakes (part of) their education in a second language.[1] And although Dutch exceptionalism would sometimes like us to believe otherwise, this development is also visible in the Netherlands.[2] Language–focused subject education is tailored to these multilingual educational contexts, as it supports specific knowledge and skills, as well as language development. [3][4] Nevertheless, a focus on language is not always an integral part of subject teachers’ teaching in multilingual environments. [5] Yet, very few studies have focused on the kinds of support that are provided.

Our research examines the types of language support teachers in Dutch bilingual secondary education do provide and the reasons they have for providing it. In these multilingual settings, the languages of instruction are Dutch and English. [6]

The Dutch Network of Bilingual Schools currently has more than 130 members [7] and most of the schools follow a curriculum that is similar to non-bilingual schools. The national curriculum guidelines include elements of citizenship education in subjects such as geography. Comparing practices across citizenship-related subjects [8] [9] allows for comparisons across schools and classrooms.

We asked eight teachers across four schools about the ways they support learning through English in their classrooms. We observed three lessons by the first seven teachers, and two lessons by the last one. After observing the lessons, we used Stimulated Recall Interviews to discuss the observed instances of language scaffolding to ask about their motivations for providing these types of support. We are currently trying to make sense of this data by using the concepts of whole-class scaffolding, language levels and scaffolding motivations.

 

We believe that these two examples make three important points. First, teachers do support students’ writing assessments. Second, when they do, it is not only for language or content reasons, but also for reasons related to disciplinary or broader academic literacy. Third, and perhaps most importantly, we need to view teaching as a complex craft where teachers do sometimes engage beyond the word level to help their students to create complex texts.

Perhaps they do not engage in these supports constantly. Perhaps it would be better if they did it more often. Perhaps we should just appreciate the language supports teachers manage to provide in demanding classroom environments where language support is one of the many important educational and psychological student needs that require attention at various moments. We believe that rather than dividing teaching up into checkboxes and deliverables, we should keep investigating the things teachers do well so that we can understand them better and spread good practices.

What is scaffolding?

Scaffolding is a rather contentious concept. [10] [11] Within this research, scaffolding is understood as a process by which a teacher responds to students’ needs by gradually adding support. By focusing on the type of scaffolding that helps students to continue with a difficult task individually, we are able to look at language support in both a focused and a broad way. [12]

The language part of language scaffolding is covered by language levels, as teachers modify their language support according to the varying needs of the students. In this way, we can see whether the scaffold is geared towards the word, sentence or text level, and also whether students are helped in more active or passive skills. [13]

Finally, we were interested in scaffolding motivations, or the reasons teachers have to engage in language scaffolding, whether it is to support students’ language or content development or disciplinary literacy. [14] Whereas content development refers to assisting students in coping with the subject matter, disciplinary literacy is about ‘learning to think like a historian, mathematician, […]’ [15], or in this case, like a social scientist.

A common problem for teachers in contexts is bridging the gap between the knowledge that students access through reading in class, and the type of knowledge they must display in written assessments. [16] After the work of Maton and others, we use the word unpacking to refer to the process by which a teacher helps students to access knowledge. Repacking is used to describe the process by which a teacher helps a student to display this knowledge in writing. [17]

 

Maton, 2013, p. 14

As in previous research, [18] rather than supporting students in producing difficult texts, our teachers mostly support the ways students access knowledge and help them to unpack difficult concepts.

 

Maton, 2013, p. 14

Examples of Repacking

Although this is an interesting finding in itself, some teachers do repack. And they do so not just for language or content reasons. Let me illustrate this with two examples, both from geography classes.

When asked about their language scaffolding practices, the first teacher describes a situation where they help students to answer questions.

‘… the kids write their answers on the board, and I go, okay: this is a peer review and how can we look at improving …the style of answering, the method of answer.

Where is the piece of information, where is the point of information we want, where is the next point? We are looking, we are coming back and highlighting: this is what the question is asking, this is the question word, this is the question phrase, this is the term, have we answered and seen these two question terms in your answer, build it up, and they write.’  

 

We believe that this illustrates that this teacher is engaging in repacking as they help the students to formulate answers to difficult questions. And when asked about the reasons for engaging in this type of language scaffolding, they state that it is about ‘building up a habit’ and ‘how they understand my subject’.

In the second instance, another teacher describes the scaffold they provide for writing an essay.

 

‘I mainly use the interaction with the class to decide what goes up on the board. So the points, the explanations, and the examples are put on the board…

 And for that, I took a number of slides I got from an English teacher with all kinds of linking words and asked them: ‘how can you link these sentences?’

So first, I showed them on the screen what the class already knew, these are also words that you can use to make this link, and also what should be in your final conclusion, what should be in there?’

 

And this teacher also does not see this instruction as something that focuses solely on language or content: 

‘I don’t necessarily think it’s subject specific, I really think it’s quite general. I think most subjects use that [point-example-explain, ed.] structure. I do make my assignments so that this can be integrated so that I can help with this.’

Key Messages

  • Language supports are important for a variety of educational contexts
  • Teaching is a complex craft where teachers attend to many important educational and psychological student needs
  • A framework combining whole-class scaffolding, language levels, and scaffolding motivations can provide insights into the ways language scaffolds unfold in classrooms
  • Teachers support students’ writing assessments beyond language or content, including disciplinary or broader academic literacy
  • Finding out what teachers do, rather than what they should be doing can contribute to spreading good practices
Errol Ertugruloglu

Errol Ertugruloglu

Ph.D. student at Leiden University, Leiden, the Netherlands.

Errol Ertugruloglu is a Ph.D. student at Leiden University Graduate School of Teaching (ICLON). His background is Political Science (BSc. and MSc.). His doctoral research investigates the ways subject teachers support language in multilingual educational settings in the Netherlands. Among other things, he is interested in multilingual education, migration, and classroom practices.

References and Further Reading

[1] Jessica G. Briggs, Julie Dearden, and Ernesto Macaro, “English Medium Instruction: Comparing Teacher Beliefs in Secondary and Tertiary Education,” Studies in Second Language Learning and Teaching 8, no. 3 (August 27, 2018): 673–96, https://doi.org/10.14746/ssllt.2018.8.3.7

[2] Emmanuelle Le Pichon-Vorstman and Sergio Baauw, “EDINA, Education of International Newly Arrived Migrant Pupils,” European Journal of Applied Linguistics 7, no. 1 (February 28, 2019): 145–56, https://doi.org/10.1515/eujal-2018-0021.

[3] Adam Tyner and Sarah Kabourek, “Social Studies Instruction and Reading Comprehension: Evidence from the Early Childhood Longitudinal Study” (Thomas B, September 2020), https://eric.ed.gov/?id=ED609934.

[4] Joana Duarte, “Translanguaging in Mainstream Education: A Sociocultural Approach,” International Journal of Bilingual Education and Bilingualism 22, no. 2 (February 17, 2019): 150–64, https://doi.org/10.1080/13670050.2016.1231774.

[5] Huub Oattes et al., “Content and Language Integrated Learning in Dutch Bilingual Education: How Dutch History Teachers Focus on Second Language Teaching,” Dutch Journal of Applied Linguistics 7, no. 2 (December 31, 2018): 156–76, https://doi.org/10.1075/dujal.18003.oat.

[6] Tessa Mearns and Rick de Graaff, “Bilingual Education and CLIL in the Netherlands: The Paradigm and the Pedagogy,” Dutch Journal of Applied Linguistics 7, no. 2 (December 31, 2018): 122–28, https://doi.org/10.1075/dujal.00002.int.

[7] Nuffic, “Alle Tto-Scholen in Nederland,” accessed January 24, 2023, https://www.nuffic.nl/onderwerpen/tweetalig-onderwijs/alle-tto-scholen-in-nederland.

[8] Wolfram Schulz et al., IEA International Civic and Citizenship Education Study 2016 Assessment Framework (Cham: Springer International Publishing, 2016), https://doi.org/10.1007/978-3-319-39357-5 (p. 97).

[9] Margarita Ivanova Jeliazkova, “Citizenship Education: Social Science Teachers’ Views in Three European Countries” (PhD, Enschede, The Netherlands, University of Twente, 2015), https://doi.org/10.3990/1.9789036540056.

[10] Janneke van de Pol, Monique Volman, and Jos Beishuizen, “Scaffolding in Teacher–Student Interaction: A Decade of Research,” Educational Psychology Review 22, no. 3 (September 1, 2010): 271–96, https://doi.org/10.1007/s10648-010-9127-6.

[11] Esmaeel Hamidi and Rafat Bagherzadeh, “The Logical Problem of Scaffolding in Second Language Acquisition,” Asian-Pacific Journal of Second and Foreign Language Education 3, no. 1 (December 2018): 19, https://doi.org/10.1186/s40862-018-0059-x.

[13] Jantien Smit, Henriëtte A. A. van Eerde, and Arthur Bakker, “A Conceptualisation of Whole Class Scaffolding,” British Educational Research Journal 39, no. 5 (October 2013): 817–34, https://doi.org/10.1002/berj.3007.

[13] Y. Y. Lo and A. M. Y. Lin, “Designing Assessment Tasks with Language Awareness: Balancing Cognitive and Linguistic Demands,” 2014, http://hub.hku.hk/handle/10722/213641.

[14] E Ertugruloglu, T Mearns, and W Admiraal, “Scaffolding What, Why and How? A Critical Thematic Review Study of Descriptions, Goals, and Means of Language Scaffolding in Bilingual Contexts,” accessed January 24, 2023, Manuscript submitted for publication.

[15]Steven Z. Athanases and Luciana C. de Oliveira, “Scaffolding Versus Routine Support for Latina/o Youth in an Urban School: Tensions in Building Toward Disciplinary Literacy,” Journal of Literacy Research 46, no. 2 (June 2014): 263–99, https://doi.org/10.1177/1086296X14535328.

[16] J.R. Martin, “Embedded Literacy: Knowledge as Meaning,” Linguistics and Education 24, no. 1 (April 2013): 23–37, https://doi.org/10.1016/j.linged.2012.11.006.

[17]Karl Maton, “Making Semantic Waves: A Key to Cumulative Knowledge-Building,” Linguistics and Education 24, no. 1 (April 2013): 8–22, https://doi.org/10.1016/j.linged.2012.11.005.

[18]Yuen Yi Lo, Angel M. Y. Lin, and Yiqi Liu, “Exploring Content and Language Co-Construction in CLIL with Semantic Waves,” International Journal of Bilingual Education and Bilingualism, August 30, 2020, 1–22, https://doi.org/10.1080/13670050.2020.1810203.