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Imagine that you have posed the following problem to your students, “Find the difference between 18 and 7?” What responses might you expect? You would likely get varied responses depending on students’ interpretation of “difference.” Some students may immediately answer “11” as they understand “difference” to mean “subtraction.” However, other students may respond by listing differences between 18 and 7 such as “18 is bigger than 7,” “18 has 2 digits and 7 has only one,” or “18 is even and 7 is odd” as they interpret “difference” to mean “not similar.”
Problem-Solving in the Mathematics Classroom
Mathematics, particularly as students progress to word problems, becomes very language oriented which can be a stumbling block for those with language processing difficulties, learning disabilities, or for English Language Learners. Students may know how to solve problems if they see symbols because those symbols and their associated procedures have become engrained, rote, and automatic; however, they will not always translate those symbols into words with multiple meanings. In other words, students may have developed “routine expertise” as opposed to “adaptive expertise.” In the former, students can complete problems accurately but without real understanding while in the latter students have multiple strategies for a particular operation they can adapt to the problem situation (Russell, 2010). As the example at the start of this article illustrates, mathematics terms may have multiple meanings; “difference” is one such term and there are many others.
Problem-solving is a weakness for students. Studies in a variety of countries offer strong evidence that many students have limited views of what to do when encountering word problems which is a factor in poor performance on such problems (Verschaffel, Greer, & De Corte, 2007). Instruction, therefore, must emphasize understanding the action and the meaning of the operations in context (Russell, 2010).
George Polya (1945), a famous mathematician, posed a four-step problem solving process:
1. Understand the problem. [What questions or problems are being posed?]
2. Devise a plan. [Can one or more of the strategies below be used?]
Draw a picture, act it out, use a model
Look for a pattern
Guess and check
Make a table or chart
Try a simpler form of the problem
Make an organized list
Write an equation
3. Carry out the plan. [Implementation]
4. Look back. [Does the answer make sense?]
These steps are very useful and can even be posted around the classroom so students have strategies to help solve problems (in mathematics and across other disciplines). This four-step process gives students a plan to problem solve and also provides teachers with critical information about where a child is struggling—which step(s) may be giving the student the most problem(s). Although all steps are very important and necessary to see problems through to completion, the first step, understanding the problem, is critical. Students have to know what the problem is asking of them before they can progress to the other steps. Often it is the mathematical language that is tripping them up.
Building Mathematics Vocabulary Word Walls
Teachers can help students develop their mathematics language skills in context to assist in problem-solving by creating a mathematics vocabulary word wall in their classrooms. Vocabulary word walls are typical for English Language Arts, particularly in elementary school classrooms, but often less so for mathematics. The purpose of the word wall is to identify mathematics terminology that students must understand and use to progress in mathematics. Students need to be familiar with mathematics vocabulary to understand written and spoken instruction. Although word walls are more common in elementary classrooms, they are also effective with secondary students since word walls are a tool for learning core content vocabulary and every grade level has core mathematics vocabulary that students must master in order to understand the concepts.
Here are some suggestions to consider when building mathematics word walls:
Also include symbols/pictures where applicable for students to relate the language of mathematics to the symbols/pictures.
It is preferable to have the words on the wall grouped with like terms, as opposed to in alphabetical order, so students can see and internalize the connections between the words and concepts. These groupings aid with developing conceptual understanding.
Engage the students in creating the word walls by asking them to brainstorm words for each operation or term so they can see the connections or have them add new words as they encounter them in word and other problems.
Make the word wall colorful/visually appealing. Each major term (e.g., terms for the four operations) can be written on a different color index card.
The word wall should be practical and not just something else to be hung up. A goal is for students to utilize it and not have it just be random text around the room. Therefore, it is important that wherever the words are positioned, students can see them.
Depending on the size and space of the classroom, words can be kept up all year or changed as every new unit is introduced.
Consider having a portable version of the word wall to send home to families (either in paper format or electronically) to assist with homework.
Enhancing Mathematics Vocabulary Aids with Problem-Solving Skills
The way in which the mathematics word wall is built and utilized in the classroom will depend on the teacher. However, to strengthen students’ problem-solving skills, particularly those with language processing difficulties, it is important that as problems are introduced in the classroom, the teacher takes the time to relate new terms encountered to terms students already know. To develop a full understanding of each operation and its application as well as relationships among operations, students must encounter a wide range of problem types (Carpenter, Fennema, Franke, Levi, & Empson, 1999). Building up students’ mathematics vocabulary is critical to understanding language-based problems at any grade level. Utilizing a mathematics word wall is one way to assist with developing adaptive expertise so students can approach problems in multiple formats and see them through to completion with a higher degree of autonomy and success.
Carpenter, T. P., Fennema, E., Franke, M., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.
Russell, S. J. (2010). Learning whole-number operations in elementary school classrooms. In D. V. Lambdin & F. K. Lester Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 1-8). Reston, VA: NCTM.
Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557-628). Charlotte, NC: Information Age Publishing, Inc.
Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati’s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at Joan.Gujarati@mville.edu