Mr. Pierson is starting a new geometry unit with his Grade 4 class. In this lesson, the class will develop definitions for geometry terms, new vocabulary, and symbols in order to create a common mathematical language for future lessons.
Math requires students to understand new terminology because common words and phrases can have math-specific meanings. Many words (such as “plus” and “equals”) are so common and important that we use symbols rather than words. Teaching precise mathematical language will help your students to think more carefully about their ideas and the ideas of their peers. The strategies below provide a strong foundation for differentiating instruction.
Best Practices with Technology
Step 1: Provide Clear Explanations
-
When introducing new vocabulary, give your students examples that fit the definitions, as well as non-examples that don’t quite fit. Ask the students to decide whether a given item is an example or non-example. Point out that the precision of the definition will help them to decide if an item is an example or non-example.
Examples of talking explicitly- Example 1: Begin by defining a square as a four-sided figure, and then ask students to give you a non-example of a square that fits this definition. Continue to build the definition with them until it is precise enough to account for all cases of squares but will not allow for any other shape.
- Example 2: Define a polygon as a closed figure with straight sides. Now show students some figures that almost—but don’t quite—match the definition. For example, show them an open figure with straight sides or a closed semicircle. Ask students to point out how the definition fits and how it does not.
- Have students create an online glossary of unit-related words. Have students work in pairs initially, and then use a follow-up class discussion to help students shape an accurate class definition with examples and illustrations.
- Introduce new vocabulary words through explanations, examples, and illustrations. For terms that students don’t already know, devote some time to thinking about and discussing their meanings. For terms that refer to objects, students can use tech tools to explore and experiment with examples, even before learning the definitions (e.g., using a dynamic geometry program when learning about different types of polygons).
View evidence behind this recommendation
IES Recommendations
Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
Evidence:
Strong
Interventions should include instruction on solving word problems that is based on common underlying structures
Evidence:
Strong
Teach number and operations using a developmental progression.
Evidence:
Moderate
Step 2: Give Students Strategies and Models
- When supplying examples and non-examples, be sure to vary unimportant aspects such as size, shape, and spatial orientation. For example, when you define a rhombus for your students, make sure to show them squares. If they don’t see squares when first learning what a rhombus is, they may conclude that squares are not rhombuses at all, rather than realizing that they are special rhombuses (see UDL Checkpoint 3.4: Maximize transfer and generalization.
- Have each student create and regularly update his or her own glossary. If students stumble over the meaning of a word when providing an explanation, refer them to their glossaries rather than defining the word for them again.
- Keep a “word wall”—in a spot that all students can see at all times—that displays important vocabulary for a particular unit.
Creating a word wall
- Add new terms to the wall as you define them and students add them to their glossaries.
- Point students to the word wall so that they can choose the proper terms when describing their ideas and solutions to problems, or when they have questions.
- When students use imprecise language, have them find a more precise term on the wall and explain why it is a “more correct” term.
View evidence behind this recommendation
IES Recommendations
Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
Evidence:
Moderate
Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.
Evidence:
Moderate
Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade
Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
Evidence:
Strong
Help students understand why procedures for computations with fractions make sense.
Evidence:
Moderate
Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade
Step 3: Provide Ongoing Formative Assessment
- Give your students opportunities to talk about their mathematical thinking and listen to how they use mathematical terminology. Ask your students to paraphrase what you or other students have said. When students use informal language, point to the word wall and ask them to rephrase using the mathematical term or ask a classmate to rephrase.
- Employ a two-step strategy when a student doesn’t understand something you (or another student) have said that includes math terms. First, make sure the student understands the terms by asking about their meaning. If you can clarify the terms, prompt the student to use the definition to paraphrase the statement or idea in question.
- Consider each student’s needs and learning styles when you decide what actions you will take to move students closer to the learning goals. Whatever actions you take, give students time to ask you questions, share their thinking, and respond to the feedback you provide.
Moving students closer to learning goals
- For some students, it might work best for you to provide feedback individually.
- Other students might benefit from you sharing feedback with them as a group.
- Some or all of your students might need you to review part of the lesson.
View evidence behind this recommendation
IES Recommendations
Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review
Evidence:
Strong
Discover Lessons in Action:
Watch Our Quick View Technology Videos:
Quick View: Virtual Manipulatives
(Get More Info)
Resource Type: Quick View
In the past, most classrooms only had physical manipulatives. Today, more classrooms have improved access to computers and the Internet. As a result, virtual manipulatives are becoming more common. These can be useful tools for students and can help...
Review the Research:
Developing effective fractions instruction for kindergarten through 8th grade: A practice guide
Resource Type: Research
This practice guide presents five recommendations to help educators improve students? understanding of fractions. It features strategies to develop young children?s understanding of early fraction concepts and ways to help older children understand...
Effectiveness of reading and mathematics software products: Findings from two student cohorts—Executive Summary (NCEE 2009-4042)
Resource Type: Research
A 2003 design effort by ED working with educational technology and research experts recommended focusing the study on software products used to support reading and math instruction. The study team set up a competitive process and worked with ED to...
Effects of mathematical word problem solving by students at risk or with mild disabilities
Resource Type: Research
The purpose of this study is two-fold. First, the researchers sought to assess the effectiveness of two instructional strategies, an explicit schema-based strategy and a traditional basal strategy, on one-step addition and subtraction word problems...
Effects of reciprocal peer tutoring on mathematics and school adjusttment: A component analysis
Resource Type: Research
This study investigated the differential impact of the structured peer tutoring and group reward components within a reciprocal peer-tutoring (RPT) intervention to enhance mathematics performance for students at risk for failure. Participating...
Effects of self-explanation as a metacognitive strategy for solving mathematical word programs
Resource Type: Research
The study examined the effects of a metacognitive strategy (i.e., self-explanation) on elementary school students’ ability to solve word problems. Students were assigned to one of three conditions: (1) self-explanation, (2) self-learning, and (3)...
Research on Math Practices
Resource Type: Bibliography
The PowerUp WHAT WORKS Team at the Center for Technology Implementation, has conducted extensive reviews of the literature on mathematics teaching and learning. View our bibliography for a collection of references on evidence-based math practices.