The #1 question educators ask me is “How do I support the students who struggle in my math class?” It can be incredibly challenging to create a math classroom where the grade-level content is accessible for ALL students. This is especially true when we have a wide range of needs in our classrooms. You may have 24 students (if you are lucky) in your math classroom and 6 of those students have an identified learning difficulty, 5 students may struggle to comprehend word problems because of reading struggles, 5 students need enrichment, and 12 students have gaps in their learning that need to be addressed. Where do we begin? It isn’t feasible to create individual plans and tasks for every student, yet the need to differentiate is clear.

Representations are the answer

NCTM (2014) and Van De Walle (2004) identified 5 representations that should be used to create accessible mathematics. Let’ dive into each then connect the representations around math content.

Visual Model (Picture)

How often have we encouraged students to draw a picture representing a real-world problem? Visual images, whether an actual drawing of a situation or a visual model such as a number line, ten frames, tape diagram, base 10 blocks, or 100s chart, help students ‘see’ the mathematics. A visual model also significantly supports students who might struggle to create visual images in their heads as they are solving a problem.  It is appropriate for the teacher to provide students with visual models that can serve as tools for students, however, it is critical for students to do the drawing rather than the teacher. It does not help students when you draw a picture for them. You might be able to improve the picture but that isn’t important. Rather the student’s understanding of the math is the importance of a picture. 

Real World Models

Students must see how mathematics is connected to the world around them. Create interesting, engaging math stories to hook students into the math content. 

A ‘naked’ math problem such as 7.1 x 24 = ? does not excite most students. But a story about how the class is planning a party where there is going to be a surprise bag for all 24 students in the class. The party bag costs $7.10 per bag so let’s figure out how much we will need to raise in our upcoming fundraiser. 

A myth we often encounter is the idea that until students master the ‘naked’, procedural skills they aren’t able to apply the math to word problems. This is FALSE! We want students to engage and get excited about doing math and real-world situations that interest them are a way to tap into their curiosity.

Manipulatives/Concrete Models

Manipulatives and concrete models such as algebra tiles, tape diagrams, fraction bars, base 10 blocks, counters, unifix cubes, or shapes make all the difference for all students. Manipulatives allow the student to build a conceptual understanding of the math. Numbers are abstract concepts and all learners can benefit from representing abstract numbers with concrete models that bring meaning to math. Keep in mind we never outgrow the need for concrete models! Concrete and visual models should be an integral part of every math classroom, whether it is Desmos graphs in Algebra 1 or algebra tiles used to build an understanding of why we solve algebraic equations using inverse operations. Polypad is a fantastic virtual option for using manipulatives if you don’t have access to concrete models, however, touching and feeling the concrete representation is a crucial part of brain development.

Verbal (Talking Math)

Students need to verbalize their mathematical thinking to each other consistently and frequently. We want students to explain why their strategy works best for them and we want them to be able to explain the reasoning of other students. Questions might include:

• Susie, can you explain in your own words how Shawn solved this task?
• Omar, can you share your strategy with your group?
• How is your strategy the same or different than Jack’s strategy?
• Zoe, can you show us with your base 10 blocks how you solved this task?
• Do you agree or disagree and why?

Many students who struggle in math are more able to explain and verbalize their thinking rather than writing it. Mathematical thinking does not need to come in the form of complete, written sentences.

Written Symbols (Numeric)

We spend a large majority of our time in math writing and manipulating written numbers, yet conceptual understanding is so much more important than being able to ‘get an answer’. As we think about how much time in our math class is dedicated to filling out a worksheet or writing equations on a whiteboard, we will realize our current math teaching and learning model is too reliant on numeric representations.  Solving ‘naked math’ problems should consist of one-fifth of our math instructional time. The other four-fifths should be time spent connecting the other four representations.

Supporting Struggling Students Through Representations

The answer to supporting students who struggle in math is simple! We must provide opportunities for many representations to be used within our math lessons. 

Example: 4th-grade lesson on adding fractions with unlike denominators.

• Launch the lesson with a real-world problem that models the math.  (Jade and Alex are students in the classroom)

Jade and Alex are running a lemonade stand. They both made their own lemonade mix, but they realized they needed to combine their batches to have enough for the day!

• Jade made 3/4 of a pitcher of lemonade.
• Alex made 2/3 of a pitcher of lemonade.

They want to combine their lemonade into one pitcher.

How much lemonade will they have in total? 

• Ask students to work with a partner to develop their thinking about the task and strategies for solving the task. Students could also work together in a vertical space in random groups of 3 to support student math discourse.
• Encourage students to draw a picture of the pitchers of lemonade to determine how they might approach the problem. What operation might they use to solve the task?
• Provide all students with fraction bars where they can manipulate the fraction models to determine how they can add the pieces together by using benchmark fractions or common denominators.

• After students can explain their strategies to their group, ask groups of students to share their thinking with the rest of the class to build a conceptual understanding of the concept using a variety of strategies.
• Ask students to write a numeric equation of the task to represent the computations they used to solve the task.

When students are given all 5 representations during a lesson they have so many opportunities to access the math in a way that is meaningful to them. This does not require individualizing the lesson for each student it merely requires encouragement from the teacher to engage in the math using all 5 representations which creates accessible mathematics for all students.  It is not an easy button but it is the answer to making math equitably accessible to every student in your classroom.