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How Representing Ideas Visually Helps All Students Make Sense of Mathematics

How Representing Ideas Visually Helps All Students Make Sense of Mathematics CTR-1
What is annotation?
Annotation is an instructional move a teacher makes during full group discussions. It helps to keep the focus on mathematical thinking, orient students to each other’s ideas, and support students who struggle to access and attend to auditory input. These are essential aspects of equitable teaching practices.   

Simply put, annotation is a visual representation of a student’s verbalized mathematical thinking. It helps students make sense of what their classmates are saying by connecting what they hear to what they see. Annotation is not a one-time event: it builds as student thinking and the class discussion unfolds.  

For a variety of reasons, annotation also supports and engages a wide range of students. That’s why it is considered a high-leverage teaching strategy.
Learn more about Routines for Reasoning and Teaching for Thinking
Annotation in action
The image below shows an annotation example for one student’s strategy explanation:  

“I noticed the addition and subtraction signs, so I grouped the subtractions together first, and knew I could add afterward. When I subtracted, I got 4, 3, 2, and 1. I put them together by making 5s.”

Teaching For Thinking Annotation Blog Graphic jam
Note the following in this annotation:

  • the green parentheses imply how the student chunked the expression into smaller parts
  • the purple arrows show the attention to the addition operation
  • the red lines show how the student changed the form of the expression in order to simplify the addition process (by making use of the familiar sums of five)

You can imagine all that students may glean from the annotation during the discussion. We have found that this understanding persists and grows even after the annotation is finished.

How does annotation support multilingual learners?
Annotation provides a critical visual support for multilingual learners. Multilingual learners often struggle to make sense of the ideas and a high volume of complex academic language if their only access point is auditory processing.

However, when the teacher uses color, symbols, and words to create visual residue of the conversation, students can connect the language to the ideas as they emerge. This helps multilingual learners (and all students) develop both mathematical thinking and language simultaneously.

How does it support students with learning disabilities?
Annotation provides an additional processing modality by connecting the verbal to the visual—whether that visual is written or gestured.  

It provides focus for students who struggle to orient visually by drawing their attention to important features and providing organization for representations that may seem abstract or overwhelming. In addition, it creates residue for students whose attention or attendance is interrupted during a discussion


Tips for getting started with annotation

  • Listen first, then gesture to ensure you understand what students are saying. Support students as they listen, then begin annotating.
  • Use color—but not too many colors. Color must provide organization rather than distraction.
  • Use shading and symbols purposefully (arrows, dotted lines, grouping symbols) to highlight and connect mathematical ideas.
  • Include words that are critical to the discussion for students to reference later but avoid scribing students’ ideas verbatim.
  • Less is more. You can always add onto the annotation as the conversation continues.

Teaching for Thinking and Routines for Reasoning (2)Read Teaching for Thinking and Routines for Reasoning for more on leveraging reasoning routines and essential strategies like annotation to support a range of math learners.

For more resources on ensuring all students develop as powerful thinkers, check out FosteringMathPractices.com.


Grace Kelemanik
(@GraceKelemanik) has more than 30 years of mathematics education experience. A frequent presenter at national conferences, her work focuses on urban education, special populations, and teacher training. 



Amy Lucenta (@AmyLucenta) has extensive K–12 mathematics experience with all students, including a focus on special populations. She is a frequent professional development provider who helps teachers implement the Standards for Mathematical Practice.


Topics: Mathematics, Amy Lucenta, Grace Kelemanik, Math, Teaching for Thinking

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