The Language of Math

The Language of Math

By Summer Erskine, M.Ed., TJS Math Learning Specialist

Throughout my teaching career, parents have often commented on how differently math is taught these days. During the school year, I can count on hearing refrains of, "That's not how I learned it," and, "Why aren't you teaching it the way I remember it?"

One of the reasons math is taught differently today is that learning math is seen as very similar to learning a new language. Many of us have heard that the best way to learn a language is to talk with native speakers in a real-life setting. The same is true for math. It's best learned through doing, experiencing and communicating.

Many parents, however, remember the days when math classes focused on rote memorization. The trouble is, rote procedural knowledge of how to solve a problem only helps in routine situations. It doesn't give students the ability to problem-solve when a situation requires reasoning.


In the quest to help my students become fluent in the language of math, I use the CPA approach, which stands for Concrete, Pictorial, Abstract. When students first encounter math, every concept is abstract. Math concepts remain meaningless until students can attach tangible experiences and images to equations.

CPA is a gradual way of teaching in which each stage builds upon the previous one. It starts by using physical objects to make abstract concepts concrete.


When using CPA, you begin with manipulatives – physical objects students can touch such as counters, pieces of candy, straws or blocks – to connect ideas to something concrete. Using concrete materials, students are able to grasp mathematical concepts. (1)

Imagine it's the first day of kindergarten, and the equation

"2 + 1 = _" is written on the board. The meaning of plus and equal signs is abstract and must first be taught through application. Rather than attempting to decipher these newsymbols, the kindergartener would use manipulatives to show that you can put "2" and "1" together to make a new number.

While that's a simple example, CPA isn't limited to younger students. When concepts like place value are introduced in fifth grade, we use counters called "base ten blocks," which represent values of ones, tens, hundreds and so on, to understand the value of a given number. For example, 423 would have 4 hundreds, 2 tens and 3 ones.

Using concrete objects allows students to develop understanding of these abstract concepts through their experience. The next step in deepening understanding is to use pictorials.


Pictorials, the second stage of the CPA model, involve the use ofimages or drawings to represent math equations. Many times, kids are told to, "just picture it in your head," but to do so, they must have seen the thing they're being asked to imagine.

To illustrate this concept, visualize yourself holding a deck of cards with an ace on top. I moved the ace to the bottom. Were you able to visualize me moving the ace to the bottom? Probably so! Visualization is possible because you have most likely held a deck of cards in your hand. Because of your concrete experience with cards (stage one of CPA), you are able to make a pictorial representation in your head. This is why we start with manipulatives and move to pictorials before taking on the abstract representation of math: symbols.


The final stage of CPA is the abstract, or symbolic, stage. Once we've built a foundation and understanding of mathematical concepts through the use of concrete and pictorial representations, students can solve problems abstractly. They can use symbols to represent their thinking.

This is the stage in which students are best able to speak the language of math, which has been described as "a vehicle for bringing thinking to the surface, classifying ideas, moving ideas forward, revealing misconceptions and making key mathematical connections clear, transferable and memorable." (2)


"The use of speaking, reading, listening and writing in math class helps students... learn to communicate mathematically." (3) Meaningful math discourse involves comparing ideas and methods, constructing viable arguments, critiquing each other's reasoning and helping each other make sense of mathematics.

Math talk supports social skills as well. It reinforces students' ability to reason mathematically and communicate their reasoning, which is the social aspect. A student may be able to process the answer to an equation in his head, but communicating his solution requires deeper reasoning and explanation skills. I once had a student who, when called upon to explain her thinking, would ask, "Can I show you?" instead of attempting to discuss her reasoning. It's hard to explain an idea you're not quite sure about yet. Although challenging, talking it through is truly a gift; a gift of perseverance, mastery and confidence.

This social component of math discourse runs deeper. When a teacher shares a concept, some students will understand it right away, while others may need more information. When one student phrases a concept in a way a confused student understands it, the occurrence will stand out in the confused student's mind. This is due to the social significance of having a peer explain the concept. "The reason is our minds place more priority on memories which involve actual human and social experiences, memories which have emotions tied to them." (2)

In addition, it is much easier for misunderstandings to surface when students are talking through a question. Often in class, a student will stop in the middle of an explanation of her answer because she's able to identify her own mistake. This reinforces reasoning and problem-solving through simple conversation.

The bottom line is this: the number one way to learn a new language is through conversation. It is more interesting, and your students will be more invested in talking to another person than in any other form of learning. (2)


I'll close with a quote from my mother, Connie Erskine, who said, "Math is not memorization, but multisensory. Using, hearing and knowing the language makes it concrete in one's mind."


1. National Council of Teachers of Mathematics


2. Discourse in Math — Don't Just Talk About It

Dean Ballard -

3. The Language of Mathematics

By Faye Bruun, Joan M. Diaz, and Valerie J. Dykes –

*This article originally appeared in the Summer 2018 edition of KeyNotes, an annual publication of The Joy School.

Summer Erskine

About the Author

Summer Erskine grew up in Texas City, Texas with her two sisters and her parents. Summer's passion throughout high school was cheer leading and swimming. Among her many leadership roles, she was the senior class president. She graduated in the top 10% of her class, overcoming her learning difficulties. She attended Texas A&M University and graduated with honors with a Bachelor of Science Education. Summer began teaching in a title-one school in Hearne, Texas. She went on to continue her education at Texas Tech University, earning her Master in Special Education. Since moving to Houston, she has had the opportunity to work with students from pre-k through fifth grade. While she was the Special Education Support Teacher, she was voted as the campus' Teacher of the Year. After nine years of teaching, her passion for students with learning differences led to her journey at The Joy School in 2017. Summer has a passion for helping students find their love for learning, creating a fun and safe environment, guiding students to work through their struggles to reach their full potential and become life-long learners. Summer feels very fortunate to work at The Joy School and truly sees the JOY at this school. In her free time, she enjoys being with people she cares about, including family and friends. Three of those people are her nephews that she couldn't adore more. She also enjoys kickboxing and watching Aggie football.

Degrees held:

Bachelor of Science in Education from Texas A&M University

Masters of Education in Special Education from Texas Tech University

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