My Struggle with Less Than

by Rita-Soledad Fernández Paulino  

I use to believe that mathematics was a universal language. I loved that no matter where I was in the world given an equation like 8x – 7 =  17, I would find the same solution as someone who did not speak English. I stopped believing this once I became a mathematics teacher.

In the classroom, I quickly realized that the ability to perform calculations was pointless if students could not apply the mathematics they learned to solve real-world problems. As I shifted my teaching to focus on addressing the mathematical practices from the CCLS, I was overwhelmed with how many of my students struggled with solving problems due to their limited English vocabulary. Though I understood that I had to teach academic/math related words such as “simplify, evaluate, exponent, slope, radical, parabola, etc. ” I had no idea I would also need to teach students commonly used terms to engage in a discussion, such “thermometer, commute, manufacturer, assembled, cashews, cents, etc.” I realized that teaching students to solve problems embedded in real world scenarios involved making sure students understood all the terms used in these word problems.

As I became mindful of vocabulary words my students would need to understand in order to gain meaning from the context, I became aware of how confusing mathematical terminology could be. Though I had previously taught students how to translate quantitative expressions into verbal sentences, I now viewed the task with new eyes.

When I first taught students how to solve one-step mathematical equations given a verbal sentence, I was aware of how students struggled with remembering how “less than” was similar to “subtracted from” and so I made sure to check for understanding constantly. All students, including my ELLs, were able to articulate the difference and shoe evidence of understanding by correctly writing mathematical sentences given a verbal sentence with the words “less than”. Yet, when I started to teach students how to solve one-step inequalities given a verbal sentence, new confusion regarding the words “less than” arose.

Consider the following math problems and their associated content objectives:.

Objectives

Example Math Problems

Content Objective: I will be able to solve a one-step mathematical equation given a verbal sentence.
  1. 15 less than a number x is 45.
CORRECT INCORRECT
x – 15 = 45

+ 15   + 15

x = 60

15 – x = 45

-15         -15

- x = 30

x = -30

Content Objective: I will be able to solve a one-step inequality given a verbal sentence.

 

  1. 15 is less than the sum of 45 and a number x.
CORRECT INCORRECT
15 < 45 + x 45 + x – 15

 

In the first problem, the phrase “less than” represents a mathematical operation, subtraction. The phrase “less than” is different from “decreased by, reduced by, take away, minus” because the order of the values in the subtraction problem change. Order is crucial in subtraction problems and though math teachers can emphasize that there is no such thing as the commutative property of subtraction, it does not help the struggling English Language Learner. Furthermore, in the second problem, “is less than” represents a relation symbol, <. Students in my classroom were now confusing when “less than” referred to subtraction versus a relation symbol

Examples of Student Work_A.A.6 and A.CED.1 for ELLsIn order to address my students’ confusion, I spoke to a coach who taught me the value of writing measurable language objectives for math lessons. In order ensure students understood when the words “less than” represented subtraction versus a relation symbol, I made sure to address the issue directly when teaching students to solve equations with variables on both sides of an equation. By including both a content objective and language objective in my math lesson, I was able to accurately identify where and how my students were struggling with mastering the content standard. For example, student A demonstrated mastery of both the language objective and content objective. However, student B only demonstrated mastery of the language objective. By teaching to mastery of a language objective in addition to a content standard, student B was able to at least create a mathematical sentence in the first problem of her exit slip. I know now to further support student B, I will need to make sure to provide the student with additional practice on solving equations with one variable on both sides.

I realize now that only by consistently writing language objectives and measuring mastery of those language objectives in my math lessons will I be able to ensure that my ELL students do not perform less than equal to their non-ELL peers.

Examples of Student Work_A.A.6 and A.CED.1 for ELLs

Student Handout_Lesson A.A.6 and A.CED.1 for ELLs

BIO: Soledad teaches Integrated Algebra at New Heights Academy Charter School in New York City. She is a 2009 Math for America Fellow, 2013 Funds for Teachers Fellow and a participant of the “Centering the Teaching of Mathematics on Urban Youth” Professional Learning Team, funded by the National Science Foundation. When she is not in the classroom, she enjoys attending professional development and taking care of her godchildren. She looks forward to remaining a teacher for the rest of her life.