Typically, vocabulary instruction is associated as part of reading instruction. However, vocabulary instruction is also a necessary component of mathematics instruction. In fact, state and national standards (e.g., Common Core State Standards) require elementary students to communicate about mathematics to provide clear explanations, construct…

Students, even as young as kindergarten, are natural problem solvers (Carpenter et al., 2017). Even before they learn to count, they can make sense of mathematical problems and real- world situations (National Council of Teachers of Mathematics, 2022). Kindergartners can make sense of complex story problems, including Separate Start Unknown…

This article describes a process called "Backing Up" which is a way to preassess student understanding of a topic and gauge student readiness to move forward in the learning process. This process of backing up begins with using responses to a word problem to identify categories of students' understandings in relation to the expectations…

Many rules taught in mathematics classrooms "expire" when students develop knowledge that is more sophisticated, such as using new number systems. For example, in elementary grades, students are sometimes taught that "addition makes bigger" or "subtraction makes smaller" when learning to compute with whole numbers,…

ELLs need to practice using the language in their speech. Teachers can ask students to restate the definition in their own words and provide opportunities for students to use academic vocabulary in discussions. Chunking (instead of teaching inch in isolation, also teach foot, centimeter, and yard) helps students develop their schema and mentally…

What "is" mathematics? What does it mean to "do" mathematics? The answers to these two questions critically influence the ways in which learners respond to a mathematical problem. On the other hand, if they see questions as having only one correct answer and as needing to be solved in one particular way, they will be prone to…

When the word problems of forty-five sixth-grade students were examined, multiple levels of understanding and misunderstanding multiplication were exposed. The purpose of this article is to demonstrate the potential of problem writing as a technique for assessing the depths of students' mathematical understandings. Discussions include sample…

Now in a second edition, the award-winning "The Trouble with Maths" offers important insights into the often confusing world of numeracy. By looking at learning difficulties in maths from several perspectives, including the language of mathematics, thinking styles and the demands of individual topics, this book offers a complete overview of the…

Every assessment is a literacy test. It matters not whether the content is science, social studies, or mathematics; if students are not able to make sense of the words, their ability to decipher the meaning of the assessment questions is suspect. Comprehending the language of a task becomes even more important as educators strive to move the…

This article shares an assessment tool that uses student-written word problems to provide meaningful information regarding the depth of mathematical understanding. Students' word problems representing 6 [division] 1/2 were classified using the scoring categories described in the NAEP. By categorizing the problems, students' levels of understanding…

Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…

Although English Language Learners (ELLs) quickly acquire basic interpersonal communication skills, most struggle for several years with reading and writing academic content in English. In particular, in English-only mathematics classes, children are likely to have difficulty reading and comprehending text, reading word problems, and giving…

Analyzes multiplication word problems to identify the origin of children's difficulty. Emphasis is given to problems involving the concept of combination. (JN)

The concept of Arithmetic Story Grammar is introduced. Arithmetic Story Grammar maintains that any equation containing a single operand tells, in and of itself, a complete story with a setting, plot, theme, and resolution. Multiple-operand equations differ from their single-operand siblings only in the number of actions undertaken and the number…

During the preliminary stages of classroom-based research, this author conducted a sequence of interviews with six seventh graders regarding their understanding of fractions, decimals, ratios, and proportions. The students represented a variety of ability levels and were all in their first semester of prealgebra. The interviews began with notions…