The Focus

The Ventura County Mathematics Council is an affiliate of the California Mathematics Council (CMC) and the National Council of Teachers of Mathematics (NCTM). We serve educators in Ventura County and surrounding areas by conducting meetings, professional development and activities where ideas can be shared and networks formed.

Photo Credit: Felix Cortez

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"Evidence from neuroscience suggests that some of the most effective understandings come about when connections are made between visual/physical and numerical or symbolic representations of ideas. When students relate numbers to visual representations and, more broadly, develop multiple ways to think about mathematical concepts, they become more effective users of those ideas."

(2023 California Mathematics Framework, Chapter 4 pages 23-24).

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Using multiple representations in math instruction is crucial for fostering deeper understanding, enhancing problem-solving skills, and accommodating diverse learning styles. By presenting mathematical concepts in various forms, such as visual, symbolic, verbal, contextual, and physical, educators can cater to the diverse needs of students and provide multiple entry points for comprehension.

Image from: California Math Framework Chapter 4 page 24

Different students grasp concepts more effectively through various representations. Visual learners may benefit from diagrams, graphs, or geometric shapes, while others may prefer symbolic representations such as equations or algebraic expressions. Providing multiple representations ensures that every student can access the material in a way that resonates with them.

Employing multiple representations encourages critical thinking and flexibility in problem-solving. Students who encounter a mathematical problem presented in various forms are prompted to analyze it from different perspectives, leading to a deeper understanding of the underlying concepts. This enhances their ability to transfer knowledge to new situations and solve complex problems creatively.

Integrating multiple representations in math instruction fosters a more inclusive learning environment, supports deeper understanding, and cultivates versatile problem-solving skills essential for success in mathematics and beyond.

Engage students with a math game focusing on multiple representations!

This is a favorite class activity for me and my students! I love it so much- I have created a set of Spoons cards to accompany almost all my math units! Hearing my students' laughter and conversations as they play brings me much joy. My students even ask me often when we will play the game again. Spoons is an excellent way for students to think critically as they identify multiple representations!

And many more topics!

The game's object is to get four cards with the same representation. Place one less spoon as the number of players in the circle's center. The dealer gives each player four cards and places the extra in a stack next to them. The dealer begins by picking up the first card and deciding if he/she/they want to pass it face down to the next person or keep it for themself. If the dealer chooses to keep the card, they must discard a different card from their hand and pass it to the next person face down. Each player must only have four cards in their hand at one time. The following person looks at the card and decides whether to keep it or pass it to the next person. The game continues with each player passing the cards in a circle as they try to obtain four matches. If the dealer runs out of cards, they grab cards from the stack of the last person in the circle's discard pile.

The game continues until a player has four of the exact representations. Once that happens, they grab a spoon and pass the cards as if they were still playing the game. Once one person grabs a spoon, any player can grab one, even if they do not have four representations. The person without a spoon at the end of the round receives a letter. The goal is not to spell the word spoons.

Before the start of the game, I make sure to copy, cut, and laminate a set of cards for each group that will be playing the game. There can be anywhere from 3-5 players in one group. I also make sure to label each set with a color-coded number in case cards happen to fall on the ground or get misplaced.

When I read the newly adopted California Math Framework, I’m struck by how getting away from a one-correct-procedure approach is a through-line. The Framework asks us not to focus on doing the math the “right” way if such a thing ever existed, but to facilitate the sense-making abilities of our students that will lead them to a deeply personal understanding of math. We are asked to focus as much on the process as on the answers. The Framework asks us to “allow” math to be thought about in creative, beautiful ways. When we make multiple representations a mainstay of our teaching practice, we are setting up an environment where we don’t just talk about diversity and creativity of ideas. We’re making different ways of thinking and making sense of things part of the core of how we model thinking. Let’s find time to embrace multiple representations daily as we explore math with our students.