Using visual maths — three ways to help students see their way to success

One of the biggest aha moments I’ve had on this professional learning journey is that maths without visual opportunities is like soup without stock — missing a crucial ingredient. 

Gaining access to visual mathematics opportunities later in life, and learning that two centres of our maths brain are based in visual processing, dramatically transformed my own maths experience both personally and professionally. 

Gradually, teaching and learning that incorporates visual maths is reaching teachers and schools for the benefit of student understanding, confidence and general enjoyment of mathematics both in and out of the classroom.

Using visual maths is not really a teaching method but more of a philosophy that permeates everyday maths teaching and learning. This article shares three teaching and learning habits for supporting conceptual and relational understandings which in turn support student engagement, confidence and joy in the mathematical realm.

Key teaching and learning habit 1: 

Use photos regularly

One of the simplest yet powerful strategies is using mathematically-rich photos to provoke discussion. Not only does this practice allow students to uncover what they already know and can do, it can help them review and make connections across important concepts. Observing and discussing a carefully chosen photo provides authentic opportunities to learn new and existing mathematical language, facts and skills to describe what they see.

A lovely way to describe this practice is 'exercising our maths eyes'. Have you got Maths Eyes? is a UK-based project helping people notice the maths around them in ways that build mathematical confidence, knowledge and skills.

The official website explains: Mathematics that surrounds people in their everyday lives, for the most part, remains “invisible”. If individuals are supported to look at familiar things and begin to see that mathematics is all around them, they can build confidence in their own ability to use mathematics.

Three tips for using photos regularly: 

• Run a week of daily 'Maths Eyes Exercise'. Kick-start a culture of seeing the maths around us (and connecting mathematical language and knowledge to what we see) by displaying one mathematically-rich photo each day for one week. Ask the class an open-ended question or question pair like “What do you notice? What do you wonder?” before funneling into a more focussed question like “How many _____ altogether and how do you know?”. This routine will build students’ observational skill and provide a safe space for contributing and responding to different mathematical ideas. See here for some suggested photos and accompanying questions.

• Create a ‘Maths Eyes’ Wall. Invite students to contribute photos from their life that show mathematical concepts. This could be anything like arrays of items in a grocery store, patterns in nature, geometric designs, etc — the list goes on! Display these on a 'Maths Eyes' wall and invite students to write captions about the maths they notice or are wondering about.

• Use a photo to launch an exploration. Keep your maths eyes open for photos that are mathematically-rich in ways that align closely with concepts you’re required to teach and provide a natural hook to motivate active exploration. For instance, this photo I took of a painted wall in Newcastle has inspired all kinds of hands-on investigations involving multiplication, area, perimeter and symmetry.

Key teaching and learning habit 2: 

Use physical materials regularly

Using physical materials allows students to manipulate and interact with mathematical ideas and provides meaningful opportunities to make connections with, think about and apply abstract concepts. 

General materials like icy-pole sticks, buttons and bottle caps can be used as well as natural materials like twigs, leaves, pebbles and shells. Then there are more maths tools and manipulatives like dice, dominoes, square tiles, multi-link cubes, double-sided counters and pattern blocks (to name just a few).

Exploring possibilities involving fractions using pattern blocks, students use spatial reasoning and experimentation to determine whether it’s possible to build triangles that are exactly ⅔ yellow, exactly ⅔ red, exactly ⅔ blue or exactly ⅔ green.

Opportunities to explore and play with physical materials helps students embody maths concepts, make tangible connections and visualise processes in ways that verbal explanations or written symbols alone cannot achieve. 

Encouraging students to explain their thought processes as they use physical materials also cultivates important mathematical language and reasoning skills. By articulating what they’re doing, students solidify their understanding and learn to communicate their mathematical ideas clearly, which supports overall confidence and engagement.

Three tips for using physical materials regularly: 

• Establish a collaborative Counting Collections routine. Appropriate and adaptable for all grade levels, a Counting Collections provides repeated opportunities for students to exercise everyday arithmetic skills in different ways that support understanding, fluency and confidence.   

• Bring concepts to life with manipulatives. Schools have a duty to ensure that mathematics is accessible and meaningful for all learners. Maths manipulatives play a  critical role in this and considered a must-have resource in schools. For manipulative and task recommendations, check out this article.

• Establish a system of access to materials. Set up an organised and accessible area where students can independently choose from a range of maths materials at any time. For instance, as they see fit to explore and model problems or when they are independently exploring hands-on puzzles or games while you’re running individual or small group maths coaching.

Key teaching and learning habit 3: 

Use pictorial models regularly

Pictorial models are visual representations used to illustrate mathematical ideas. By depicting mathematical relationships, pictorial models make it easier to recognise structures and patterns which support a deeper understanding of numbers and operations, which ultimately supports mathematical prevision, efficiency and confidence.  

Static models (like graphs, diagrams and recordings of student thinking in number talks and other activities) give students a reliable visual reference that helps break down concepts into manageable parts. 

A maths talk image which invites students to think mathematically by identifying relationships, structures and patterns present in multiplication.   

There are also dynamic models (both digital and non-digital) which allow students to manipulate sets of static models and learn through sorting, games, and other types of play and exploration.

For younger students, dot number cards provide dynamic opportunities to apply and develop number skills like subitising, addition and part-part-whole knowledge.

For older students, mini-challenges and puzzles can be used to strengthen new and existing knowledge.

In the ‘Six Rectangles Puzzle’ the challenge is to use all 15 pieces to create six 2 x 6 rectangles. Once solved, students label rectangles using addition or the distributive property of multiplication.

Both static and dynamic models allow students to “see” mathematics, which can help more complex problem-solving become accessible and intuitive. By regularly incorporating both static and dynamic visual representations, teachers can help students grasp complex mathematical ideas in ways that are accessible, engaging and memorable. 

Three tips for using pictorial models regularly: 

• Integrate pictorial models into daily routines. Embed the use of models like ten frames, number lines and data displays in everyday activities. For instance, display a large number line and 120 number chart for everyday reference. At the start of the year, co-create with students useful data displays representing information that has ongoing relevance to the group like class birthdays. In junior primary, use ten-frames to establish a 100 days of school routine.

• Establish a Number Talks routine and incorporate the use of models. Regular Number Talks support students to develop skills and confidence with mental arithmetic by providing time and space (5-15 minutes) to consider and discuss different ways of solving 1-2 arithmetic problems. Part of the teacher’s role is to record a variety of student solutions on the board using tools like number lines and geometric models.  This helps students “see” and clarify the steps involved and make connections between various methods. Seeing models used regularly in this way will support students to draw and use their own models. Learn more about Dot Number Talks here and Number Number Talks here.  

• Encourage Student-Created Models: Ensure regular opportunities for students to draw models that represent their mathematical thinking through different types of activities. For instance, in Counting Collections recording largely represents mental arithmetic processes while open-ended problem explorations can create opportunities to document patterns using models like tables and graphs.

One philosophy, many benefits

Deciding to prioritise visual mathematics adds colour, joy and relevance to the everyday maths experience by connecting students to the material in a way that aligns with the natural ways in which they learn. 

Additionally, by focusing on doing one fundamental thing exceptionally well (like fostering visual understanding in these ways) we inadvertently cover many other educational objectives like building a positive and collaborative maths culture, creating habits of mathematical reasoning, and helping learners see that exploring mathematics can be enjoyed for its own sake.

For further resources supporting positive, playful and powerful  approaches to maths teaching and learning, consider signing up for Maths Play’s monthly newsletter. There are two monthly editions — one for teachers and one for families. Subscribe for free here.

If you have questions or feedback about this article, you can send them to [email protected].