Contents
 1 Visual representations and narrative structures can help students build their mathematics literacy, but they may need guidance to understand and use these tools.
Visual representations and narrative structures can help students build their mathematics literacy, but they may need guidance to understand and use these tools.
“Mathematicians use language to make meanings and to share understandings” (Marks & Mousley, 1990, p. 118). As such, it’s important that educators better merge language and content instruction for the benefit of all, and especially for English learners (Moschkovich, 2015). The latest content and practice standards for mathematics (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010) promote the increased use of academic language and discourse in mathematics. And the WIDA Consortium’s (2020) standards for English learners recognize four key language uses — or genres — that are present across disciplines, to varying degrees: narrative, explain, argue, and inform. In addition, we know that visuals can be powerful tools for learning and express for multilingual learners (Driscoll, Nikula, & Neumayer DePiper, 2016; Gibbons, 2015; Markos & Himmel, 2016).
Students need to develop control of language genres in mathematics, and we (the teachers) must embrace these genres to expand students’ communication skills in mathematics. Literacy in mathematics involves “multiple aspects of mathematical proficiency, multiple symbol systems (written text, numbers, graphs, tables, etc.), and multiple modes of communication” (Moschkovich, 2015, p. 45). Students will need to navigate various levels of terminology (Beck et al, 2002); unique grammatical patterns (Schlepper, 2007); and symbolic notation and graphs or visual displays (Lemke, 2003). The problem in Figure 1, which appeared on a grade 10 highstakes assessment in Massachusetts, illustrates the importance of language knowledge.
Note the ways in which language operates in this multiplechoice item — the type of question that is typically viewed as “easy.” There is an introductory sentence that provides context for the problem, but the adjective rare is inconsequential. We then recognize that a person named Grayson has 23 of something and Rosette has two, and then we see that the next sentence contains the word add. The bullet points then indicate how much will be added to each person’s amount and, to complicate the matter, how often.
The question within this word problem is where language becomes the most crucial: How many books will Grayson and Rosette each own when their collections have the same number of books? Now a student must glean that both amounts are changing at the same time and that their job is to funnel these two scenarios into one numerical response, as noted by the answer choices (i.e., determine at what point the two amounts will be equal to one another — the concept of systems of equations). Students can use a number of approaches to determine the answer: graphing both functions, generating and manipulating two equations with two unknowns, or perhaps using open number lines.
Despite the various computational approaches available to a student, the heart of this relatively “simple” problem is an understanding of language and what it implies in the context of mathematics. As such, teaching the language of mathematics must be a priority for all instructors of the subject.
The importance of visuals
In 2000, the National Council of Teachers of Mathematics (NCTM) released principles and standards for the teaching and learning of mathematics that included five process standards: Problem Solving, Reasoning and Proof, Communication, Connections, and Representations. According to NCTM (2000):
Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understandings to one’s self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations through modeling. (p. 67)
A decade later Common Core Standards for Mathematics (NGA & CCSSO, 2010) expanded upon the process standards and introduced the Standards for Mathematical Practice (SMPs). The fourth standard continued the focus on visuals, stating that mathematically proficient students should be able “to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas” (p. 7). With this statement, the SMPs expand the responsibility for using visuals in mathematics beyond the teacher to include the student. In other words, visuals should not only convey meaning to students, but should be a way for students to convey meaning to others.
We help small children understand what they see by counting items around them. Then as we begin more formal mathematics instruction, we use stories.
Embracing visuals within mathematics can help all students (with sight), but these tools can be especially crucial to English learners’ understanding and engagement in the classroom (Blankman, 2021). The use of visuals accesses students’ visualspatial intelligence (Gardner, 2011). It also aligns with principles of Universal Design for Learning (UDL), which encourages the use of multiple means of representation (CAST, 2018). Jo Boaler and her colleagues (2016) report that students’ differences in knowledge dissipated with the inclusion of games and number lines (i.e., visuals). As such, they offered three recommendations:
 Encourage visual approaches and replace memorization/calculation as the primary markers of “intelligence.”
 Encourage finger usage because “Successful mathematics users have well developed finger representations in their brains that they use into adulthood” (p. 5).
 Make teaching and learning more visual, as all concepts can be visualized interpretively and expressively.
Finally, the WIDA (2020) framework repeatedly offers advice related to the use of visuals. For example, not only should teachers “list visually supported key words, crossdisciplinary or technical language, and their meanings” (p. 248), but instructors should also “allow multiple options for students to share their thinking and create their own representations of ideas, including by using other languages, drawing, or using manipulatives.” Again, visuals are not just for receiving information, but also for expressing information (Swain, 2005).
Narrative for making meaning
We use narrative to make sense of the world. We help small children understand what they see by counting items around them. Then as we begin more formal mathematics instruction, we use stories: Johnny had two apples, Sue had three apples. How many apples do they have altogether? This process continues through school experiences, involving more advanced concepts and more reallife scenarios:
 1st grade: Jane has nine dolls, and her friend Amy has five dolls. How many more dolls does Jane have than Amy? (Teaching Expertise, n.d.)
 6th grade: Stefan and Cooper order pasta for $6.15, salad for $5.00, and two glasses of lemonade for $1.35 each. The tax is $1.10. How much change should they get from $15.00? (IXL Learning, 2023)
 Algebra II: John worked five less than twice as many hours as Jane did. How many hours did each work if together they worked 97 hours? (WinstonSalem/Forsyth County Schools, 2023)
The purpose of narrative
Beverly Derewianka (2020) describes the purpose of narrative as entertaining a reader or holding the reader’s interest, providing a full description of one’s experience, or even teaching a lesson. Michael Schiro (1997) offers several reasons to integrate narrative within mathematics instruction:
 To help children learn concepts and skills.
 To provide a meaningful context for learning.
 To facilitate the development and use of language and communication.
 To help children learn problem solving, reasoning and thinking.
 To provide a richer view of the nature of math.
 To facilitate improved attitudes towards math
A narrative text can be fiction or nonfiction and be found in the forms of mysteries, science fiction, romances, horror stories, adventure, parables, fables, myths, historical narratives, and other genres. In mathematics, in particular, word problems are the most common forms of narratives; however, there can also be children’s literature, stories, and autobiographies that incorporate math. Presenting mathematics “in the more ‘playful’ manner of the story” enables students to see the usefulness of the subject and “to see how the various language domains — reading, writing, speaking, listening, drawing — complement and support one another” (Terrell & DeBay, 2020, p. 30).
Mathematics is full of unique, and sometimes complex, grammatical patterns.
Organization of text
A narrative text typically possesses three major components: an introduction or orientation, a complication that emerges from a series of events, and a resolution of the complication (Derewianka, 2020).
In the case of mathematics word problems, the resolution often is not provided. Instead, the reader is given a framework to produce a resolution — typically in the form of a question. For example, in the examples previously discussed, the readers are asked, How many more dolls does Jane have than Amy? (Teaching Expertise, n.d.); How much change should they get from $15.00? (IXL Learning, 2023); and How many hours did each work if together they worked 97 hours? (WS/FCS, 2023). It’s up to the students to create the ending of the stories.
Using narrative in the math classroom
Remember that mathematics is full of unique, and sometimes complex, grammatical patterns (Schleppegrell, 2007). As a colleague once commented, “No one talks like John worked five less than twice as many hours as Jane did!” So it’s important for teachers to guide students through the narrative so they can arrive at the correct conclusion.
Checking for understanding
The first suggestion for teaching narrative in math is to check for understanding. To do this, teachers must incorporate comprehension tasks. One preferred approach is the threeread strategy in which students read a problem three times, each time searching for a different kind of information to make sense of a problem and determine a path for determining a resolution:

 What is the problem about? (i.e., the context)
 What am I trying to find out? (i.e., the question)
 What important information is given? (i.e., key information)
This strategy can be used at all grade levels and at any level of complexity. It allows students to slow their processing time and really consider what a task is about before beginning computations. Plus, it allows the teacher to monitor students’ comprehension of the language and content of the problem.
To expand on the protocol, students might answer a fourth question:

 How can you begin to approach this problem?
At this point, students can consider processes, algorithms, and tools that will enable them to arrive at a solution (i.e., resolution) on their own, or the teacher can incorporate other protocols or graphic organizers.
Diagramming
Another approach to teaching with narrative is the incorporation of diagramming. Diagrams enable students to interact with the language before they are expected to produce it themselves. Sometimes, through diagrams, they can deduce facts about the problem that are less obvious in the full text. Figure 2 shows several diagramming styles that are common in mathematics.
The sequential diagram indicates order or a process wherein one step leads to another. In math, the order of operations is typically presented in this linear fashion:
Grouping → Exponents → Multiplication/ Division → Addition/ Subtraction.
In the case of word problems, students might subdivide the problem into the steps of a narrative so they can better understand the process that is occurring (see Figure 3).
The cyclical visual in Figure 2 can be incorporated to illustrate the process of “guessing and checking,” in which the student chooses a possible solution for an equation and tests the solution. If the value makes the equation true, it is a solution. If not, they are back at the beginning, where they choose a new possibility and test again. Categorization diagrams might help with classifying quadrilaterals according to the number of parallel sides and the measures of opposite or adjacent sides. Other graphic organizers assist students in their languageoriented decoding and production as well. For instance, the concept map can help with brainstorming, and the Venn diagram can help with comparing and contrasting concepts, graphs, and other mathematical information.
Diagrams enable students to interact with the language before they are expected to produce it themselves.
Incorporating technology
Technology can assist students with diagramming and understanding and expressing concepts visually. For example, see how one teacher brought together narrative, diagramming, and technology to help students solve the Window Problem (Lawrence & Hennessey, 2002):
Peerless Window Company puts together square windows from three kinds of units: a corner pane, a center pane, and edge pane. The production manager at Peerless needs to decide how many of each type of unit to make so the company can avoid wasting units or overordering.
Your task: Look for patterns and generate equations that will allow Peerless’ production manager to calculate the number of each type of pane he will need to order for any size square window (n x n).
When I used this exercise with my elementary mathematics methods students, I went through the threeread strategy with the class for the first two reads and then had the students complete the third read within the Desmos Activity Builder program. The activity builder included visualizations of the different types of panes and opportunities for students to answer questions about their problemsolving process in writing, in audio form, or with visuals:
 What tools would be helpful in solving this problem?
 Can you draw a 11 window? Why or why not?
The students then used the geometry tool in the program to diagram what is occurring in the problem (see Figure 4).
Students reflected on their drawings and the patterns they observed, created and graphed equations based on their drawings, and reflected on what they observed in these graphs.
In a debriefing about working through the problem, students overwhelmingly reported that the threeread strategy helped them figure out what was occurring in the task and that diagramming the problem was essential to their being able to complete the task.
Toward more inclusive mathematics
All students are capable of learning the various language features and uses involved in mathematics, with the appropriate scaffolding and instruction. As Jim Cummins (1997) notes, providing language support is not the sole responsibility of multilingual teaching specialists. All teachers in every content area must consider best practices that will enable multilingual students and others to fully engage and succeed in their academic ventures. Many of the strategies used to support multilingual students can benefit Englishonly students as well. All students may, at times, need guidance navigating the language of mathematics.
As we embrace our responsibility to support students’ understanding of the language of mathematics, all learners will acquire content knowledge and language proficiency they need to flourish.
References
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