This study compared student achievement in solving liner algebraic functions using a concrete torepresentationaltoabstract (CRA) sequence of instruction and a repeated abstract explicit instruction model. Six classes of students were randomly...
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Thinking aloud helps your students learn how to reason by focusing on their thinking (see UDL Checkpoint 9.3: Develop selfassessment and reflection). Most students are not used to answering questions that require more than a singleword or shortphrase response. Many don’t know how to talk about mathematics or explain their thinking. These strategies provide a strong starting point for differentiating instruction.
Steps:
Step 1: Provide Clear Explanations

Explain to your students that you're as interested in how they get their answer as you are in whether the answer is correct (see UDL Checkpoint 6.2: Support planning and strategy development). Tell them that you will help them work on “thinking aloud” and explaining their reasoning.

When you demonstrate solving problems, model the process of thinking aloud by explicitly stating what it means to explain your reasoning. Include all the decisions you make, even the very small ones (e.g., which numbers to work with and what operations to use). Employ technology to support note taking and create visualizations of the big ideas as you work to give students a clearer understanding of your thinking process (see UDL Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships).
 Ask guiding questions to help your students focus on their reasoning, not just the solution, even when their answer is correct. Ask students to explain why they chose a process or how they made a calculation, emphasizing that the decision process is a core part of mathematical reasoning. When you highlight the process as part of thinking aloud, you help your students to flex and strengthen their reasoning skills.Examples of guiding questions
 Why did you choose that number/operation?
 How did you find that number?
 Can you explain how you got that answer?
 Who did it differently? How?
 You said _________________. Can you tell us more about what you were thinking then?
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.
Interventions should include instruction on solving word problems that is based on common underlying structures
Teach number and operations using a developmental progression.
Step 2: Give Students Strategies and Models
 Give your students a series of prompts—questions or sentence starters—to guide them through the process of thinking aloud. Make sure you include questions that require them to justify their decisions.Prompts for the thinking aloud process
 I know that...
 I'm trying to figure out...
 One thing I can try is...
 I want to try ... because ...

Have students use models and diagrams to support their thinking and take notes as they go along. These visual supports can help students figure out if their thinking is faulty. The supports also provide you with a way of helping students to pinpoint where and how they have made an error in their thinking. Students can employ technology to improve their visualizations and note taking.
 Ask students to explain their reasoning often. Prompt students to share their thinking when they get correct answers—they can get the correct answer, but for wrong reasons—as well as when they get incorrect answers. All students can benefit from hearing peers’ sound, ontarget reasoning, and all students can learn from peers’ explanations when reasoning goes awry.
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.
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.
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.
Help students understand why procedures for computations with fractions make sense.
Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade
Step 3: Provide Ongoing Formative Assessment

When a student engages in thinking aloud, invite a peer to listen and comment on the content while you concentrate on the student’s use of the strategy (e.g., check to make sure the student shows an understanding of why those steps were needed). With struggling students, a small group activity might work better because it will give the students a chance to hear others before sharing their thinking.

Thinking aloud can help your students learn how to develop a plan to solve a problem (see UDL Checkpoint 6.2: Support planning and strategy development). Have students describe and justify a plan they would use to solve a problem. Then, ask them to implement the plan as they think aloud about why it is effective, where they need to make changes, and why.
 Consider each student’s needs and learning styles when you decide on the 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.
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
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