About Learning with Examples

The general culture of the math classroom in the United States is that errors are this taboo thing that are meant to be gotten rid of as soon as they're made. We're changing that and making the error the focal point of the instruction. We're giving the students a chance to engage with errors without the stigma of it having been their own error. The goal is for students to realize that errors can be a productive part of the learning process, and that by making errors in the classroom, it doesn't mean a student is “not a math person,” but that making mistakes is all part of the process and leads us to become better math learners.

Dr. Julie Booth, Temple University

Why this approach?

Research shows that students often enter a math class holding misconceptions that could derail new learning. Teachers, then, need to identify and target misconceptions and build up accurate conceptual knowledge, all while providing students with instruction on the procedural skills that are required by the standards and by standardized testing.

Studies show that when students engage with both correct and incorrect worked examples and explain the associated mathematics, real progress occurs to help dislodge sometimes stubborn misconceptions.

Learning via Self-Explanation

Self-explanation means explaining information to oneself as one reads, solves problems, or attempts to learn. When individuals self-explain, they integrate knowledge (from prior knowledge and/or instruction), infer to fill their knowledge gaps, and make explicit new knowledge and connections. This is all beneficial for learning.

MathByExamples Characters: Multiplication of Fractions

Explaining Correct Solutions

In some exercises, students are told that the solution is correct and are asked to explain why it is correct. Research suggests it is important to require students to explain not only what was done in the example but also why it was correct. This encourages students to go beyond procedural explanations (e.g., "It's right because he divided both sides by 3") to produce more principle-based, conceptual explanations leading to improved conceptual understanding.

MathByExamples Characters: Multiplication Facts

Learning via Explanation of Incorrect Solutions

In half of the exercises, students receive an incorrect problem solution and are asked to explain why it is incorrect. This type of activity is believed to improve learning because it helps students to reject their own similar, incorrect procedures. The idea that errors can be effective learning tools is not new: Studies show that asking students to explain both incorrect and correct solutions leads to greater learning. Worked examples can also help motivate students, especially those who are underrepresented in the mathematically proficient.

MathByExamples Characters: Adding with Decimals

AlgebraByExample and MathByExample Display

Why we extended the “ByExample” approach to grades 4 & 5.

Analysis of SERP's AlgebraByExample materials revealed that many misconceptions have their roots in earlier grades. Addressing elementary math misconceptions before middle school is doubly important because early student failure can decrease motivation, hindering students’ future success even further. Traditional practice activities do not address common misconceptions about the problems to solve, allowing misconceptions to linger long after elementary school. Addressing this early is even more important as standards for rigor rise. Better-prepared students will free teachers in higher-level math courses to focus on grade-level content, making it more likely that students will succeed in higher-level math.

Read more about Developing and Testing the “ByExample” Approach on the AlgebraByExample website.

Development of MathByExample was led by Julie Booth (Temple University) through a SERP collaboration with several school districts. Major contributors to program development include: Kelly McGinn and Laura Young (Temple University), Allie Huyghe, Matthew Ellinger, Emily Schwartz, Avery Jones, and David Dudley (SERP). Special thanks! to the teachers, administrators, and students in our partner districts—Baltimore City Schools, Public Schools of Beloit, Public Schools of Brookline, Fort Madison Community School District, Oak Park Elementary District 97, and Penns Valley Area School District—who were essential to the project’s success, providing feedback at critical points and inviting us into their classrooms along the way!

The collaboration has been supported to conduct this work by the Institute of Education Sciences, U.S. Department of Education, through Grant R305A150456 to Strategic Education Research Partnership Institute. The information provided does not represent views of the funders.