Scaffolding Algebra Content and Language: Case Study

Table of Contents

01. Overview and Theoretical Framework

1. Population
2. Problem of Practice
3. Approach

02. Key findings

1. Scaffolds that uphold rigor effectively support students’ engagement in productive struggle:
2. Formative assessment that monitors learning continuously enables teachers to better adapt instruction while facilitating productive struggle:
3. Attention to metalinguistic vocabulary advances students’ learning in productive struggle:

03. Solutions

1. Provide Appropriate Scaffolds
2. Conduct moment-to-moment formative assessment
3. Analyze metalinguistic vocabulary

04. Impact

1. Students extended their time on task during productive struggle to make full use of the available time
2. 85% of students reached correct inferences, and 100% produced descriptions.
3. 90% of participants reported being “very satisfied,” and student performance increased by 10%.

05. Appendix: Video Footage and Student Materials

01 Overview and Theoretical Framework

Supporting a linguistically-diverse student population in implementing a problem-based math curriculum, such as Illustrative Mathematics, sometimes requires teachers to provide additional support beyond what’s given to students whose first language is English (Zwiers et al., 2017). This is especially true because problem-based curriculum materials are structured around “productive struggle” with mathematical tasks. What’s more, problem-based curricula follow an instructional path that differs from more traditional curricula. They tend to begin with an invitation to engage with mathematics, followed by a set of activities to study mathematical concepts and procedures, culminating in the consolidation and application of knowledge (i.e., you do; we do; I do). Successful engagement with productive struggle is indispensable to this learning approach
(Illustrative Mathematics, 2019).

To support multilingual learners, we proposed focusing on three main levers of support: academic language development, appropriate scaffolds, and short-cycle formative assessment. We believe these three levers of support provide a robust theoretical framework (grounded in extensive literature) for assisting students in their early stages of English language development.

1. Population

   The teachers who participated in this analysis are bilingual Algebra teachers at a New York City high school. They are implementing a problem-based curriculum in the second year of adoption (specifically, Illustrative Mathematics) at the school. The students are all in the early stage of English language development, all starting 9th grade with a diverse range of prior formal schooling.

2. Problem of Practice

   The teachers and school leaders identified a key problem of practice: the difficulty in providing scaffolds that appropriately support multilingual learners while maintaining the rigor of the mathematical task and engagement in productive struggle.

   How effective are appropriate scaffolds provided to students during productive struggle in ways that: (1) do not degrade the rigor of the task; (2) maintain student engagement in the task (i.e., productive struggle); (3) promote conceptual understanding; and (4) support their academic language development?

3. Approach

   The fundamental mechanism of these scaffolds is to amplify students’ perceptual thinking by providing a range of cognitive activities. Scaffolds can vary depending on the lesson, from unpacking the academic language of word problems to drawing inferences from the text (i.e., receptive language). This allows students to gain insights into how to decontextualize information into mathematical expressions and later recontextualize the solution. This approach is reflected in the analysis, which also serves as a reflective tool. Other scaffolds support students in generating and producing the language needed to deepen conceptual understanding (i.e., productive language).

02 Key findings

1. Scaffolds that uphold rigor effectively support students’ engagement in productive struggle:

   Scaffolding in problem-based curriculum materials can better support students’ productive struggle by focusing on shifting their perception rather than simply enabling them to complete tasks.

   The concept of scaffolding elicits diverse interpretations among educators, though it is commonly understood as a process where instructors provide temporary, successive levels of support to facilitate students’ attainment of higher-level understanding and skills (Wood et al., 1976). This support is intended for gradual withdrawal as students develop competence and independence. In the context of a problem-based curriculum, however, we have chosen to reconceptualize scaffolding. Here, its scope is limited to instructional supports that empower multilingual learners to engage in productive struggle without compromising their cognitive processes or the inherent rigor of the mathematical task. Consequently, our scaffolds are engineered to induce a shift in learners’ perceptual thinking, aiming to re-engage them with the task from an alternative perspective by leveraging complementary cognitive activities (See Appendix).

2. Formative assessment that monitors learning continuously enables teachers to better adapt instruction while facilitating productive struggle:

   Formative assessment is most effective when understood and implemented as a moment-to-moment, continuously responsive practice embedded within everyday classroom interactions—not as discrete checkpoints or formal tasks.

   Similar to scaffolding, the concept of formative assessment is subject to diverse interpretations among educators. Nonetheless, we adhere to the foundational work of Dylan Wiliam and Marnie Thompson, who define formative assessment as information gleaned from “everything students do, such as conversing in groups, completing seatwork, answering questions, asking questions, working on projects, handing in homework assignments—even sitting silently and looking confused—is a potential source of information about what they do and do not understand” (Thompson & Wiliam, 2008). Drawing inspiration from their research, our focal point is moment-to-moment formative assessment. This perspective precisely encapsulates the dynamic, ongoing, and responsive process that Thompson and Wiliam champion, integrating it seamlessly into daily classroom activities (See Appendix).

3. Attention to metalinguistic vocabulary advances students’ learning in productive struggle:

   Explicit attention to metalinguistic vocabulary in the mathematics classroom supports students’ conceptual understanding. It is crucial for fostering reasoning, deepening conceptual understanding, and strengthening connections between mathematical ideas, and therefore warrants greater emphasis in instruction.

   Drawing from the seminal work of Uccelli and Galloway, who identified seven core academic language skills, we acknowledge the relevance of all to mathematics education. However, we note three as particularly pervasive in this domain: unpacking dense information, organizing analytical text, and understanding metalinguistic vocabulary (Uccelli & Phillips-Galloway, 2017). Our current attention is narrowed to understanding metalinguistic vocabulary. This is defined as vocabulary that refers to reasoning and discussion processes, and its significance lies in its capacity to promote conceptual understanding and establish connections between overarching mathematical ideas—a cornerstone of problem-based curriculum materials. Anecdotally, we observe that mathematics educators often prioritize discipline-specific vocabulary (e.g., proportion, functions, perfect square) while affording considerably less attention to metalinguistic vocabulary (e.g., identify, describe, explain). This disparity serves as the rationale for our selection of this specific area as our focal point.

03 Solutions

1. Provide Appropriate Scaffolds

   Barriers that impede students’ engagement in cognitively-demanding tasks can be effectively addressed by providing scaffolds that preserve and redirect their thinking to comparable cognitive skills, rather than reducing cognitive demand.

   Scaffolding practices may vary depending on the audience and purpose; however, this analysis focuses specifically on designing supports to help students engage in mathematical productive struggle. For the purposes of this work, we define an “appropriate scaffold” as one that preserves students’ thinking rather than taking it away during productive struggle. We built scaffolds that leverage the brain’s natural ability to think (Kandel, 2013, 144). For example, if students encounter a barrier with ‘making sense’ during productive struggle, we might shift their thinking to ‘comparison,’ ‘sequencing,’ or ‘ranking.’ Similarly, if the barrier involves drawing correct ‘inferences,’ we could redirect attention to ‘identifying’ or ‘describing.’ In other words, when students encounter a particular cognitive barrier and this difficulty threatens to halt their progress or lead to disengagement, we provide support that shifts them to a different but comparable cognitive skill, thereby amplifying their perceptual thinking. We found that barriers threatening students’ engagement can be effectively addressed with more thinking, not less.

   Determined to preserve the rigor of the tasks, we synthesized the extensive body of literature on scaffolding into four broad strategies (Anghileri, 2006; Bransford, 2000; Bruner, 1973; Chin-Chung & Chao-Ming, 2002; Lier, 2004; Shannon, 2018; Tharp & Gallimore, 1991; Vygotsky & Cole, 1978; Walqui, 2006; Wood et al., 1976). These four strategies are:

   Bridging: Weaving of new information into existing knowledge. This concept is closely tied to the Zone of Proximal Development (ZPD), as it reflects a scaffolding approach aimed at ‘bridging’ the gap between what a learner can do independently and what they can achieve with support.

   Marking: Amplify the critical features of math and language where conceptual understanding plays a significant role.

   Regulating: The concept of scaffolding involves various aspects, including the regulation of attention, motivation, and engagement.

   Structuring: Cognitive structuring is one of the most comprehensive and intuitive strategies available to teachers, involving the provision of structures for thinking, discourse, and action.

2. Conduct moment-to-moment formative assessment

   Adopting a dynamic, real-time formative assessment approach integrated into everyday classroom activities provides a practical way to gather meaningful evidence of students’ understanding and to guide instruction effectively.

   Teachers of multilingual learners should assume that these students do not enter the math classroom as ‘blank slates,’ but as actively thinking individuals with a wide variety of skills and conceptions. Teaching becomes more effective when it assesses and builds on prior learning, allowing instruction to be adapted to the needs of the learner (Black & Wiliam, 1998). Prior knowledge can be uncovered through activities that provide the learner with opportunities to express their understanding and reasoning. This does not require additional testing. For example, it could involve a single written question designed to elicit a range of explanations, which can then be discussed. This process is often referred to as formative assessment. In this paper, the term is more narrowly defined; it could be conceptualized as a moment-to-moment activity.

   Moment-to-Moment Formative Assessment
   While the phrase “moment-to-moment” might not be a direct quote from their 1998 paper, it perfectly encapsulates the essence of the type of formative assessment Black and Wiliam advocate for: “a dynamic, ongoing, and responsive process embedded directly within the teaching and learning activities of the classroom”.

3. Analyze metalinguistic vocabulary

   Analysis of language demands and explicit teaching of metalinguistic vocabulary—the language that supports reasoning and discussion—helps multilingual learners develop deeper conceptual understanding and make connections between overarching mathematical ideas.

   Academic language presents a challenge for multilingual learners, particularly if they have limited experience and instruction with it. In our experience, explicit attention to academic language is rare and even absent in various math classrooms. Although success in mathematics depends greatly on content knowledge, academic language and literacy skills present major obstacles to the learning of content knowledge (Jetton & Shanahan, 2012, p.172). Granted, there has been a greater collective awareness in the value of academic language among math educators, but most efforts are geared toward discipline-specific vocabulary. Developing academic language proficiency involves a repertoire of skills that are critical for subject-area reading and writing (Hinchman & Appleman, 2017, p.333). Teachers who specialize in mathematics are best positioned to help support students to unpack the academic language of their discipline. In other words, math teachers are responsible for teaching the academic language required for learning mathematics.

   Building on the seminal work of Uccelli and Phillips-Galloway (2016), who identified seven core academic language skills (CALS), we analyzed the language demands presented by math tasks with a focus on metalinguistic vocabulary (i.e., vocabulary that refers to thinking processes). In conducting this analysis, we observed that the demands such vocabulary places on multilingual learners often occur in sequence. For example, consider the following word problems (WP):

   A dining hall had a total of 25 tables. Some are long rectangular tables, and some are round ones. The rectangular tables can seat 8 people. Round tables can seat 6 people. On a busy evening, all 190 seats at the tables are occupied. How many rectangular tables, ‘x’, and round tables, ‘y’, are there in the cafeteria? (Illustrative Mathematics, Alg 1 Unit 2, Lesson 12, Activity 12.3)

   This word problem requires students to process metalinguistic vocabulary in a sequence.

   • Read and make sense of the word problem.
   • Identify the constraints and variables within the problem.
   • Infer the relationship between the quantities encoded in each sentence
   • Write an algebraic description in the form of a system of equations, using the identified information.
   • Solve the system graphically.
   • Interpret the results in the context of the original problem.

   Understanding the language demands that tasks place on students matters, because it enables us to design scaffolds that align with the specific cognitive activities students are being asked to perform. In this example, we chose to build supports around inferences, which proved highly effective in sustaining students’ engagement in productive struggle.

04 Impact

The study employed a mixed-methods approach, emphasizing qualitative data from direct observation and analysis of student artifacts. This combination provided a rich, contextual understanding of the lessons and their impact. The project’s outcomes were measured using multiple methods, primarily qualitative, with a small quantitative component. Here’s a breakdown:

1. Students extended their time on task during productive struggle to make full use of the available time

   Analysis of the video footage indicates that all students successfully engaged in productive struggle and were able to persevere throughout their time on task

   Classroom Observation:
   This is the most prominent method. The lessons were recorded and analyzed for specific teaching moves, with references marked by precise timestamps (e.g., “00:08,” “00:21”) to indicate moments identified through meticulous analysis. The detailed descriptions of ‘Observable Teaching Moves’ and references to scaffolding (e.g., bridging, marking, regulating, and structuring) are direct products of this close observation. This method provided an in-depth understanding of teacher actions, student interactions, and the flow of the lesson in its natural setting. All observations and video footage are available in Appendix A.

2. 85% of students reached correct inferences, and 100% produced descriptions.

   Analysis of student work shows that scaffolds for content and language supported 85% of students in reaching correct inferences and explanations of their work, while 100% of students were able to produce at least some inferences and quality descriptions.

   Document Analysis:
   Artifacts of student work, collected and examined from various activities, supplemented the analysis of video footage. By analyzing student responses to specific questions, we were able to assess the effectiveness of the scaffolds and identify patterns in student understanding and difficulties. This method provides concrete evidence of learning outcomes.

3. 90% of participants reported being “very satisfied,” and student performance increased by 10%.

   Multiple surveys and interviews were analyzed. 90% of participants reported being “very satisfied” and 10% reported “satisfied” that the learning experience was beneficial in building their capacity. Furthermore, analysis of student Regents examination results for participating classes showed a 10% increase in passing rates.

   An Analysis of Teacher Surveys, Interviews, and Consultations and Their Relationship to Student Performance:

   We spent 70 hours conducting professional learning sessions, planning scaffolds, and implementing them in the classrooms together in a continuous cycle of improvement from January 6 to June 30, 2025 (the spring semester). This project also included the larger school-district community, specifically school and district leadership, as well as mathematics teachers from across the district who served as a “think tank” to understand pedagogical challenges and the rationale behind the instructional choices. During the project, multiple surveys and interviews were conducted and analyzed. Additionally, student performance in class and on the state examination was analyzed.

References

Anghileri, J. (2006). Scaffolding Practices That Enhance Mathematics Learning. Journal of Mathematics Teacher Education, 9, 33-52. 10857-006-9005-9

Black, P., & Wiliam, D. (1998, March 1). Principles, Policy & Practice. In Assessment in Education. Routledge. http://www.informaworld.com/smpp/title~content=t713404048

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Chin-Chung, T., & Chao-Ming, H. (2002). Exploring students’ cognitive structures in learning science: A review of relevant methods. Journal of Biological Education, 36(4), 163-169. https://doi.org/10.1080/00219266.2002.9655827

De Bono, E. (1970). Lateral thinking : creativity step by step. HarperCollins.

Hinchman, K. A., & Appleman, D. (Eds.). (2017). Adolescent Literacies: A Handbook of Practice-Based Research. Guilford Publications.

Illustrative Mathematics. (2019). IM 9–12 Math, Algebra 1 [Open Educational Resource]. https://curriculum.illustrativemathematics.org/HS/teachers/1/index.html

Jetton, T. L., & Shanahan, C. (Eds.). (2012). Adolescent Literacy in the Academic Disciplines: General Principles and Practical Strategies. Guilford Publications.

Kandel, E. R. (Ed.). (2013). Principles of Neural Science, Fifth Edition. McGraw-Hill Education.

Lier, L. v. (2004). The Ecology and Semiotics of Language Learning: A Sociocultural Perspective (L. v. Lier, Ed.). Springer Netherlands.

Shannon, A. (2018). Scaffolding Productive Struggle: A Video Case Study (M. Cordero, Ed.). NYC Department of Education.

Tharp, R. G., & Gallimore, R. (1991). Rousing Minds to Life: Teaching, Learning, and Schooling in Social Context. Cambridge University Press.

Uccelli, P., & Phillips-Galloway, E. (2016). Academic Language Across Content Areas: Lessons From an Innovative Assessment and From Students’ Reflections About Language. Journal of Adolescent & Adult Literacy, 60(4), 395-404. 10.1002/jaal.553

Vygotsky, L. S., & Cole, M. (1978). Mind in Society: Development of Higher Psychological Processes (M. Cole, Ed.; M. Cole, Trans.). Harvard University Press.

Walqui, A. (2006). Scaffolding Instruction for English Language Learners: A Conceptual Framework. The International Journal of Bilingual Education and Bilingualism, 9(2), 159-180. http://www.tandfonline.com/action/showCitFormats?doi=10.1080/13670050608668639

Wood, D., Bruner, J. S., & Ross, G. (1976). The Role Of Tutoring In Problem Solving. J. Child Psycho [Psychiat J. Child Psychology], 17, 89-100.

Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017, February 28). Principles for the Design of Mathematics Curricula: Promoting Language and Content Development (2) [UNDERSTANDING LANGUAGE]. Stanford University.

05 Appendix: Video Footage and Student Materials

Note: Activate the English captions for viewers who prefer English.

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