Mr. Garcia’s 3rd grade students are proficient at multiplying and dividing within 100. In this lesson, students will reinforce their understanding of problem-solving strategies by creating their own word problems and writing solutions.
Learning how to solve word problems requires students to understand the context and then develop a strategy to solve the problem. Students build on their ability to organize, create visual representations, and use precise language. You can help your students build on these skills and understand how to use them in specific situations (see UDL Checkpoint 6.2: Support planning and strategy development).
Best Practices with Technology
Step 1: Provide Clear Explanations
- When students face new or relatively complex tasks, it is critical that teachers pause and provide clear explanations of what is expected and of the mathematics being learned. When providing guidance, use short declarative sentences, scaffold the questioning, and give students sufficient time to understand and react (see UDL Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships).
Example of a sequence to support students
- Announce: “Here is exactly what I need you to do.”
- Give initial direction: “I need you all to read the problem silently to yourselves.”
- Continue: “Now I would like you to read the problem again, paying particular attention to the meaning of the numbers.”
- When students are ready, explain: “Now I need you to tell your shoulder partner what the problem is asking you to find.”
- A common and effective strategy to reinforce directions and explanations is to ask students to present the directions or explanations in their own words. Prompt students by asking, “Can you repeat the directions that I just gave to the class?” or “Can you put the explanation that I just gave in your own words and repeat it for the class?”
- Ask students to compare and contrast different approaches and then summarize what you hear. Students should understand what works and what doesn’t work (and why), which methods are more efficient, and how models are different. It is critical that teachers elicit, value, and celebrate approaches that are different but still arrive at the correct solution.
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
- Use a process chart to guide students when presenting them with new problems. Teachers should focus on how each step in the process supports better access to the problem. For example, reading the problem a second time with annotations helps students sort out the core information from the background noise. Visualizing a story can be a powerful strategy that helps students create a picture or diagram of the problem. Estimating or approximating an answer helps students decide if they’re on the right track.
Possible problem-solving process
- Read the problem, and then reread it and highlight key words and numbers.
- Draw a picture of the situation that the problem presents. It may be helpful to first visualize a story or imagine a movie scene.
- Determine the goal of the problem.
- Establish a strategy or write an equation to represent the picture. Estimate an answer, if possible.
- Solve the problem and check the reasonableness of your answer.
- Explain your solution method.
- Create a gallery walk of student solutions to help them evaluate and expand their repertoire of appropriate models. Gallery walks allow for public discussion and easy comparisons of solutions to the same problem. Technology tools, such as Thinking Blocks, can also expand their repertoire with virtual models.
- Encourage students to embrace mistakes and errors, correct them as necessary, and move on with confidence. Note that it is rare to complete a problem from start to finish without a mistake or misstep (see UDL Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships).
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
- Check in with students to ask, “Can you explain to me what you are doing here (pointing to a student’s work)?” to reveal whether or not students are on the right track and if additional guidance is needed.
- When students use pictures, diagrams, charts, expressions, and equations as part of the problem-solving process, it is very helpful to ask questions such as, “How does that diagram relate to the situation in the problem?” or “Can you explain to me why you think that is a good mathematical expression to use for this problem?” (See UDL Checkpoint 3.3: Guide information processing, visualization, and manipulation.)
- Mathematics should be a sense-making process. We don’t really know whether or not it makes sense until we see student work and hear student explanations. The formative assessment question, “Does that make sense to you?” is a powerful strategy for supporting learning.
Supporting a sense-making process
- Do not allow students to simply respond “yes” or “no.” Probe to understand the degree to which the mathematics makes sense to them. Elicit more detailed responses, prompting students by asking, “Why does it make sense?” or “Exactly what doesn’t make sense?”
- • Make the sense-making process explicit when teaching students problem-solving strategies.
- Keep the focus on student thinking and understanding. Ask students, “Does this make sense?” when they have solved a problem. Eventually, students should be expected to do this for themselves before finishing a problem.
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:
A Meta-analysis of the effects of computer technology on school students' mathematics learning
Resource Type: Research
This study examines the impact of computer technology (CT) on mathematics education in K-12 classrooms through a systematic review of existing literature. A meta-analysis of 85 independent effect sizes extracted from 46 primary studies involving a...
A synthesis of empirical research on teaching mathematics to low-achieving students
Resource Type: Research
This meta-analysis synthesizes the research on the effects of interventions designed to improve mathematics achievement of students considered low achieving or at risk for failure. It examined the effects of experimental studies that used five...
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...
Direct instruction in math word problems: Students with learning disabilities
Resource Type: Research
The study investigated the effectiveness of strategy teaching and sequencing practice problems when teaching students with learning disabilities (LD) how to solve word problems. Students were randomly assigned to one of three intervention groups: 1...
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)...
Explicit instruction in mathematics problem solving
Resource Type: Research
This study examined the effectiveness of the explicit translation strategy method that explicitly teaches fourth-graders to translate word story problems into mathematical equation form. The study also explored the impact of providing students with...
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.