Math is, by definition, a subject about numbers. But at the National Council of Teachers of Mathematics this week, math educators said the subject has its own language, too—and knowing how to speak it is critical to success.
In presentations and discussions at the annual conference, held this year here in Chicago, teachers, administrators, and higher education faculty enumerated the ways that math literacy unlocks students’ understanding.
Some words have one meaning in general conversation, but another meaning in math. Others, like “square,” can have two different math definitions: the geometric figure, and the square of a number, a type of exponent. Math also has its own grammar—decimal points, commas, parentheses, and brackets, for example, all have meanings distinct from those in an English/language arts class.
Internalizing and using this language is one way that students build math knowledge, said Gladis Kersaint, the vice provost for academic affairs at the University of Connecticut, in a presentation on supporting English learners in the subject.
For every lesson, she said, teachers should ask themselves: “What language objective do I have to attend to so that my content objective can be achieved?”
Presenting multiple definitions could help students understand new math concepts
Emily Illig, a 4th and 5th grade math teacher in Chicago schools, said she sometimes has trouble defining abstract concepts with her students.
When Illig can’t demonstrate ideas with physical manipulatives like counters, “there becomes this disconnect,” she said in a Q&A session at the conference. “And then I’m like, well, how do I get them ready for middle school, when things get more abstract?”
In a separate presentation, Matthew Winsor, an associate professor of mathematics at Illinois State University, explained one strategy he used to help English learners grasp complex definitions when he taught high school.
Winsor tasked students with creating “word squares”—index cards divided into four quadrants. In the top left quadrant, students wrote a math term in their own language; in the top right quadrant, they wrote the term in English. Then, they wrote a definition in their own words in the bottom left quadrant—in whichever language they chose—and drew a picture or wrote an example in the bottom right.
Students rely on these tools throughout the course, Winsor said: “I’ve had a student come back to me the year after and say, ‘Hey, Mr. Winsor, I used my word squares today.’”
Other educators used similar strategies with younger students.
In a presentation on supporting struggling elementary schoolers, Laura Drechsel and Carolyn Stadlman, both math educators in the Crystal Lake school district in Illinois, discussed using the Frayer Model, another way to diagram vocabulary words.
In this model, students write the word they’re defining in the middle of a four-quadrant box. In each of the squares, they write 1) the definition in their own words, 2) facts and characteristics of the concept, 3) examples, and 4) non-examples.
“When I first started teaching kindergarten, I thought, ‘I’m going to just learn the curriculum, and that’s going to be OK,’” said Drechsel, now an interventionist. She soon learned, she said, that she needed tools like this model to help shore up some students’ understandings.
English learners will face specific math challenges—but also bring unique founts of knowledge
During Kersaint’s talk on supporting English learners, several teachers identified a hurdle in assessing their newcomer students’ math knowledge.
Many recent arrivals have experienced interruptions to their education in their home countries, they said. It can be hard to distinguish when students truly have gaps in their knowledge from when students know the material but don’t have the math language in English to communicate it.
Math teachers need to be “listening hard” to their English learners, Kersaint said. For example, a student might offer a partial definition in English, a Spanish word, and draw a picture—teachers should consider all of this information as students’ demonstration of their knowledge, she said.
English learners may also bring unique skills to math class, said Larisa Bukalov, a math teacher at Bayside High School in New York City.
When these students learn English, they often get explicit instruction in diagramming sentences—a skill that can help them parse word problems. “If you’re not a native speaker, you’re a lot more sensitive when you try to translate the [words in the] problem to the algebra,” Bukalov said during Q&A after Kersaint’s presentation.
For this reason, she said, she often has English learners model how they solved word problems for the group—positioning them as classroom leaders.
Multiple Definitions and Visual Tools
Emily Illig, a 4th and 5th grade math teacher in Chicago, described the challenges of teaching abstract concepts without adequate visual support. When physical tools like counters aren’t available to demonstrate these ideas, a disconnect can occur, making it difficult for students to bridge the gap as they progress into more abstract, higher-level math. Illig’s experience highlights the need for educators to present multiple definitions and varied examples to illuminate new concepts.
Word Squares: A Creative Approach for English Learners
Matthew Winsor, an associate professor of mathematics at Illinois State University, shared a creative method designed to help high school students—especially English learners—grasp complex mathematical definitions. In his approach, students create “word squares” using index cards divided into four quadrants. In the top left quadrant, they write a math term in their native language; in the top right, they write the equivalent term in English. The bottom left quadrant is reserved for a definition in the student’s own words, while the bottom right is used for a drawing or example that illustrates the term. Winsor recalled that some students return to use these word squares even a year later, demonstrating how this tool helps solidify their understanding over time.
The Frayer Model for Younger Students
For younger learners, educators in the Crystal Lake school district, Laura Drechsel and Carolyn Stadlman, discussed employing the Frayer Model to help struggling elementary school students. This model involves dividing a box into four sections: one for the definition of a term, one for key characteristics, one for examples, and one for non-examples. Drechsel reflected on her early teaching days, noting that she initially thought mastering the curriculum alone would suffice. However, she soon realized that using visual tools like the Frayer Model was crucial in building a deeper, more robust understanding of math concepts.
Addressing the Needs of English Learners
Math educators recognize that English learners face unique challenges in the classroom. Many newcomer students arrive with gaps in their formal education due to interruptions in their home countries. This makes it difficult to discern whether a student truly lacks understanding of a math concept or simply struggles with the specialized language of math in English.
Gladis Kersaint urged teachers to “listen hard” to these students. A single response might combine a partial definition in English, a term in another language like Spanish, and a simple diagram—all of which are valuable indicators of the student’s underlying comprehension. Rather than penalizing these mixed responses, educators are encouraged to view them as a composite demonstration of the student’s knowledge.
Larisa Bukalov, a math teacher at Bayside High School in New York City, pointed out that English learners often bring unique strengths to math class. Having received explicit instruction in diagramming sentences, these students tend to be more methodical when translating word problems into algebraic expressions. Bukalov even invites her English learners to model their problem-solving approaches for the class, positioning them as leaders and role models.
Conclusion
Mathematics is inherently a language—a precise, symbolic language that, when mastered, unlocks the door to understanding complex concepts and real-world problems. As educators discussed at the National Council of Teachers of Mathematics conference, building math literacy is not only about learning numbers and formulas; it’s about developing the ability to communicate mathematical ideas effectively.
