by Diane Kue
How students solve a problem—whether it is a standalone equation, expression, or a word problem—can be quite perplexing, especially when answers are unexplained. Students’ approaches and strategies vary, and their foundations of knowledge span a wide spectrum. It is probably truer than not that a mathematics teacher has come across the following scenario:
Teacher: How did you solve this problem?
Student: I don’t know. I just did it.
Whether the answer to a problem is correct or not, the pedagogical dilemma in this scenario is gaining access to student thinking so we can address misconceptions and incompleteness as necessary and validate or guide further application. But how do we get our students to use language to convey their mathematical thinking?
Making Sense of the Mathematics
Equations and expressions are generally isolated in mathematics. Word problems are different in that language exposes the mathematics. Before students can solve, they must first make sense of the math. Making sense of the math means understanding the relationship between quantities or performing a quantitative analysis (Clement & Bernhard, 2005). Here is an example:
Tenzin and Ariana steamed tamales for a party. Tenzin steamed 2 dozen tamales. He steamed 4 times as many as Ariana. How many tamales did they steam?
If students are taught to bypass making sense of the math by isolating keywords and performing their corresponding operations, then they will incorrectly solve this problem by identifying times and multiplying by 4. Yes, students can still solve correctly by multiplying as the word “times” suggests, but they must first reason that they are multiplying the inverse of 4, or ¼, not 4 itself. Unfortunately, there is no shortcut to reasoning. To understand what operations to use, the student must first recognize what quantities—entities that can be measured—are needed (Charles, 2011). Here are the quantities in this word problem:
- The number of tamales Tenzin steams
- The number of tamales Ariana steams
- The total number of tamales Tenzin and Ariana steam
After identifying the quantities, students must determine how the quantities relate to each other and plug in values (numbers). This is operational thinking. Sometimes it is easier to visualize than verbalize:
Using the visual as a scaffold for explaining operational thinking, students can link the diagram to their process thinking with language (Driscoll, Nikula, and DePiper, 2006). Here are examples of how this can be expressed verbally:
- Tenzin steamed 2 dozen tamales. A dozen means 12. Two dozen is 2 x 12 or 24 tamales.
- Tenzin steamed 4 times as many tamales as Ariana. This means he steamed a greater amount.
- If Tenzin’s amount is 4 times greater, then Ariana steamed only ¼ the number of tamales Tenzin steamed. This means that Tenzin’s 24 tamales can be divided into 4 equal groups to determine ¼. This is 24 x ¼ or 24 ÷ 4, which equals 6 tamales.
- The total number of tamales steamed is Tenzin’s 24 tamales joined with Ariana’s 6 tamales, or 24 + 6 which equals 30 tamales.
This example explanation will not come naturally or intuitively for students who do not normalize mathematics talk. Making sense of the math, therefore, needs to be taught explicitly. During this practice of making sense of the mathematics, educators can guide student processing (as needed) and language to express and expose thinking (Kue, 2021).
Using Prompts as Tools to Guide Thinking
The first step in making sense of the math is identifying quantities. Thus, prompts focus on exposing what can be counted or measured (Kue, 2021). Here are some generic example prompts to guide thinking that can be applied to the tamale problem:
- What can you visualize, even if you do not know how much of it exists? (These are quantities.)
- I can visualize a lot of tamales.
- I can visualize two people steaming tamales.
- What are values (counted or measured numbers) for what you can visualize?
- I can visualize Tenzin’s 2 dozen tamales.
- What can you not visualize because the information is not directly stated or is missing?
- I cannot visualize how many tamales Ariana steamed.
- I cannot visualize how many tamales there are in total.
- What is something you can visualize that does not have a value but can be solved for a value because of its relationship to something else?
- I cannot visualize how many tamales Ariana steamed, but I know it is related to how many tamales Tenzin steamed.
- I cannot visualize how many tamales were steamed in total, but I know it is the total number of tamales Tenzin and Ariana steamed together.
As prompts guide thinking, students put into practice communicating mathematical thoughts. This is not a single moment or one classroom instruction transformation; it takes continual guidance, modeling, and building upon ideas and understanding. None of this can occur without multiple opportunities for application (Mendoza & Beene, 2022).
Using Sentence Stems to Expose Thinking
As students begin or continue to communicate mathematical thinking, they may require scaffolds to support and expand their capacity to express themselves academically. This is where sentence stems and sentence frames play their vital role.
The following are generic example sentence stems that model operational thinking:
I know the quantities and their values. If I combine their values, I can get the total.
I know the total and the value of one quantity. I must subtract to solve for the value of the second quantity.
I know the number of equal groups of —. I also know the number of — in each group. I can multiply to solve for the product.
I know the total, and I know the number of equal groups of —. I can divide to find the number of — in each group.
I know the total, and I know the number of — in each group. I can divide to find the number of equal groups of —.
In addition to student engagement, it is how the teacher implements the activity that leads to effective learning (Sullivan, Clarke, and O’Shea, 2010). Thus, the above examples are for teachers to use and build upon to help improve instruction, create a more focused content objective, or improve student understanding. As teachers walk around the room and eavesdrop on student conversations, they benefit from insight into student thinking. This active moment of sense-making becomes a teachable moment in which teachers can address misconceptions and incompleteness of thought or guide further application. When students expose their thinking, teachers have the privilege to teach.
Diane Kue, author of Solved: A Teacher’s Guide to Making Word Problems Comprehensible, will be hosting Teaching Math to ELs in February with featured author Adrian Mendoza, author of Teaching Math to English Learners, and Dr. Jim Ewing, author of Juan Jose, You are Especial.
Charles, R (2011). Solving Word Problems: Developing Quantitative Reasoning. Pearson
Clement, L, and Bernhard, J (2005). “A Problem-Solving Alternative to Using Key Words” Mathematics Teaching in the Middle School (10) 7. NCTM
Driscoll, M, Nikula, J, & DePiper, J (2006). Mathematics Thinking and Communications: Access for English Learners. Heinemann
Kue, D (2021). Solved: A Teacher’s Guide to Making Word Problems Comprehensible. Atmosphere Press
Mendoza, A, and Beene, T (2022). Teaching Math to English Learners. Seidlitz Education
Sullivan, P, Clarke, D, & O’Shea, H (2010). “Students’ Opinions about Characteristics of Their Desired Mathematics Lessons” Shaping the Future of Mathematics Education: Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australia. MERGA