Build confidence and skills in independent math practice with embedded instruction and worked examples

Authors: Paul J. Riccomini, Ph.D., Stephanie Morano, M.S.,Jiwon Hwang, M.Ed., The Pennsylvania State University

IBuild confidence and skills in independent math practice with embedded instruction and worked examplesndependent math practice doesn’t need to be frustrating and confusing.   With careful design, independent math practice can build students’ confidence and proficiency and help them move from novice learners to expert learners.

The “interleaved worked solution strategy” is a new approach to providing students with instructional guidance during independent practice and/or homework. Students examine worked examples before independently working on conceptually similar problems.

A growing body of research demonstrates that the interleaved worked solution strategy is an effective way to support independent practice and ultimately improve student outcomes.[1]  “Interleaving” motivates students to pay more attention to worked examples rather than skimming over them because each worked example serves as a model for the work they are about to do.


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Sandwiching worked examples between problems students must solve on their own helps students recognize what they do not understand. A recent computer programming study[2] found that students learned significantly less from worked examples when the examples were provided all at once than when they were interleaved with programming problems the students had to solve.

One common mistake teachers make in assigning independent practice activities (IPA) or homework is giving students work that  requires them to learn a new skill or concept on their own. Related missteps include asking students to work with a concept or skill they have not yet mastered without providing support. This results in incorrect practice, student frustration and/or uncompleted assignments. Research indicates students should be working at around a 90% level of accuracy before independent practice is effective.

In another study,  8th and 9th-grade algebra students[3] in the treatment (interleaved) condition solved problems faster and performed better on the post-test than students in the control condition who were asked to solve all 8 problems on their own.

How to use interleaved strategy

The interleaved strategy can be used during a class period or incorporated into homework assignments. In class, a teacher could use the interleaved strategy by first leading a discussion covering the steps of a worked example, then giving small groups one similar problem to solve together. Next, the teacher could have one of the groups post its work and ask members of other groups to explain the steps of this worked example.

In homework assignments, teachers can instruct students to study the worked-out solution prior to solving a similar problem on their own. Interleaving is often assigned after a new concept is introduced because it supports students in developing a deeper and more flexible understanding of accurate problem-solving procedures.

The research is mixed on the effects of including annotations, or explanations of worked example steps[4], but teachers may want to include explanations they feel are vital based on the content and their students’ needs. As students’ proficiency increases, teachers should increase the ratio of independent problems to worked examples and embed faded worked examples that provide only the first few steps of worked examples.

Researcher Recommendations

Based on our own research-to-practice implementation experiences and other researcher recommendations, we make the following recommendations:

  • Spend instructional time teaching students about the interleaved strategy. It’s especially important to teach low-achieving students how to study a worked-out solution and apply what they learn in solving the subsequent problem. This is not necessarily a natural transition for struggling students.
  • Find ways to embed solutions in homework and encourage students to spend time studying the embedded solutions.
  • Vary the interleaving of worked solutions and problems and fade out worked solutions once students are familiar with the format. For example, a homework assignment with 10 problems might follow this format…problems 2, 4, 6, 8, and 10 are independent (no guidance); problems 1, 3, and 5 are completely worked out; and problems 7 and 9 are partially solved requiring the students to complete the solution. Research indicates students improve their performance more when teachers gradually increase the variability between worked examples and independent problems while keeping problems conceptually similar.[5]
  • In class, provide students with opportunities to explain how the solution is reached in a worked example.This can be done in think-pair share activities, peer tutoring with reciprocal teaching, small-group discussion, and/or whole class discussion. The focus in these activities is not the solution, but rather the explanation of the solution.
  • Consider using the interleaved strategy in your study guides.

What is too much or too little support

Independent practice with little or no built-in support reflects constructivist instructional methods such as problem-based learning (PBL) and inquiry learning (IL). Independent practice activities conforming to the constructivist approach often do not provide scaffolding. Teachers who use these approaches fear that providing too much support will hinder students’ learning and skill development.

Research proves that unsupported independent practice is misguided, especially for low-achieving students. Practicing incorrectly deepens faulty thinking and makes student errors and misconceptions difficult to correct or unlearn. Students can quickly get frustrated and fail to complete assignments if the only additional guidance available is through their notes or textbooks. A large number of studies demonstrate that students perform significantly better when given direct and explicit instructional guidance like that provided by the interleaved strategy. 

Practice Guide: Organizing Instruction and Study to Improve Student Learning (NCER 2007-2004),Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education.  

Contact info for authors: Paul Riccomini, ; Stephanie Morano, ; Jiwon Hwang, .


4 Responses to “Build confidence and skills in independent math practice with embedded instruction and worked examples”

  1. Mary Ellen Gavin

    This is so true. It teaches the rationale for learning these deeper ways of thinking !

  2. Logan

    please add the test results to your support, i don’t think that many people will believe this if you don’t. Otherwise i agree


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