by Diane Kue

Mathematics has existed in every civilization throughout history. Its worldwide presence is why so many people claim it as a *universal language*. Further perpetuation of the claim is present in the classroom. A classroom full of students who may or may not all speak the same language can successfully and individually solve the same problem. After all, doesn’t 1 + 1 = 2 everywhere, no matter the language, country, or culture if the symbolic representation shows that one whole, joined with another one whole, makes two wholes?

However, the idea that math is a universal language because the answer to a problem is always the same no matter what language is spoken can be misleading. **This notion focuses on the answer; it does not acknowledge that the ***processes*** to solve for the answer—or even the ***perspectives*** on the processes—can vary. **When we focus on the processes to solve, we can discover and unveil various perceptions on or ways to look at problem solving. We find the whys behind the hows of problem solving.

For example, the Chinese used negative integers based on a need to calculate owed money in 600 BCE. Their “invention” or “discovery” was based on a cultural, economic need due to open trading. Europeans, on the other hand, did not use negative numbers because of their view that “they darken the very doctrines of the equations” (Maseres, 1758). However, because they could not deny the need for them, they created “laws of arithmetic” to define negative numbers. The Chinese and European approaches differ not just because they use a different process but because their perception of the concept—their perception of the problem to be solved—was different.

Examine the three pictorial examples below of solving the same problem: **12 x 12 = 144** The process used by the solver is distinctly different in each example, yet the answer for all three is the same.

At a glance, none of these pictorial representations share a perspective to solving the same problem. They have distinctively different approaches from a pedagogical perspective as well.** **The ancient Egyptian method represents an understanding of binary to solve successfully (McGann, 2008). The Chinese method reflects an understanding of place value to solve successfully. The last example represents equal grouping to solve successfully and is a method we use in the United States. Although 12 x 12 = 144 appears to be a simple multiplication problem, each culture had/has a different foundational understanding of multiplication and, consequently, varies in their approach and process to solving.

**Thus, stating that mathematics is a universal language deceptively omits how diverse problem-solving can be, even when the final answer is the same.**

As math teachers, we can reflect on our practices to ensure that we encourage students’ various processes to problem solving and embrace their differing perspectives to those processes—even if we do not immediately recognize or understand them. **In understanding that perspective impacts the solving process, sharing perspectives becomes imperative to enriching learners.** In my last article for this blog, I referenced the need for the teacher to gain access to student thinking to address student needs. When students gain a mathematical perspective on problem-solving, the teacher must find a balance between guidance and student-led self-exploration (Huang & Barlow, 2013). Language is adamantly bound to this balance to explain concrete, pictorial, or abstract perspectives. Without it, concepts and ideas cannot be explored, connections cannot be made, and application across disciplines does not occur.

**Activity: Read-Represent-Explain**

*Read-Represent-Explain* is a classroom activity that balances teacher guidance, student perspective, and sharing of process thinking (Mendoza & Beene, 2022). A brief synopsis of this activity is as follows:

- The teacher reads aloud a problem.
- Students represent how they solve the problem concretely, pictorially, and/or abstractly.
- Students explain their process to a partner/small group.

As teachers walk with intention to observe students solving and explaining, they gain access to thinking and perspective and can provide guidance if needed (Motley, 2022). The balance between guidance and self-exploration occurs when students lead discussions and learning, and teachers shift from explaining to occasionally asking questions that encourage further exploration of ideas. Teachers and students also gain new perspectives from one another. By asking questions, they honor one another’s knowledge and further develop content-specific language skills (Ewing, 2021).

Mathematics is universal. However, how a problem is solved can be nuanced and differ based on perspective. With this understanding, we best serve our students when we create learning opportunities in which our students can share and gain problem-solving perspectives.

**Diane Kue, author of Solved: A Teacher’s Guide to Making Word Problems Comprehensible, will be hosting “Teaching Math to English Learners” on February 23, 2023 with Adrian Mendoza, author of Teaching Math to English Learners, and Dr. Jim Ewing, author of Juan Jose, You are Especial.**

**References**

Ewing, J (2021). Math for ELs: A “PATH” for success: 4 mindsets to ensure success for English learners. *TEPSA Leader,* 34(4), 1-4. TEPSA. https://cdn.ymaws.com/membership.tepsa.org/resource/collection/E8A406D5-CC66-40FA-A81E-56D02D6E1A19/fall-leader-2021.pdf

Huang, R., Barlow, A.T. (2013). Matches or discrepancies: Student perceptions and teacher intentions in Chinese mathematics classrooms. In Kaur, B, Clarke, D., Anthony, G., Ohtani, M. (Eds.), *Student voice in mathematics classrooms around the world* (pp. 161-181). Sense Publishers.

McGann, K (Director). (2008). *The story of maths: The language of the universe* [Film]. Open University; BBC.

Mendoza, A, and Beene, T (2022). *Teaching math to English learners*. Seidlitz Education.

Motley, N (2022). *Small moves, big gains*. Seidlitz Education.