By: Rebecca Adler

Rebecca Adler is a PhD candidate at Vanderbilt University in the department of Psychology and Human Development. Her research focuses on student self-regulation and motivation, particularly in the context of mathematics. She is currently working on her dissertation on self-regulated studying and hopes to continue conducting research bridging theory and practice. When not working, Rebecca enjoys singing, volleyball, and painting. 

This post was originally published on the cogbites website (here).

(Cover photo by Lum3n on Pexels)

Think about the last time you had to solve a challenging math problem. Perhaps you felt overwhelmed and didn’t know where to start. We know that there is substantial evidence suggesting that students should learn things like facts and foreign languages by practice testing themselves (1). But how best do we learn mathematics, where students need both procedural knowledge (i.e., knowledge of the steps necessary to solve a problem) and conceptual knowledge (i.e., knowledge of domain principles)? That is, what study strategies are best for learning mathematics? 

One study strategy that is helpful for learning math is to study a worked example. A worked example, like the one seen in the figure below, is a problem with the solution steps already worked out. Worked examples, paired with traditional problem-solving, have been found to be effective in helping students learn to solve problems across multiple domains, including mathematics, physics, engineering, and computer science. Sometimes, worked examples are accompanied by  prompts for students to self-explain the steps involved. 

Why do worked examples help math learning? It first helps to understand cognitive load theory, which states that people can only think about a limited amount of things at a time (2). When people have too many things to focus on, they may have difficulty focusing on the relevant information needed for learning. Studying a worked example helps overcome this difficulty by highlighting the relevant information. 

In a typical experiment investigating worked examples, students undergo a learning phase and a test phase. Depending on the student’s assigned condition, the learning phase either consists of alternating worked examples with traditional problem-solving, or just problem-solving (like a typical worksheet your math teacher may have given you). Then, there is a test on the math material covered in the learning phase. In general, those who studied worked examples perform better on the test than those who just problem-solved. 

Although there are many experiments showing that worked examples are an effective technique for math learning, researchers Barbieri, Miller-Cotto, Clerjuste, and Chawla (3) wanted to know just how effective worked examples actually are. To answer this question, they conducted a meta-analysis, a technique that statistically summarizes the results from multiple research studies on the same topic. Specifically, they analyzed results from 55 studies that investigated the effects of worked examples in math learning.

How effective are worked examples for math learning?

Averaged across the 55 studies, Barbieri and colleagues found that worked examples benefitted math learning relative to control conditions involving just problem-solving. Specifically, they found that worked examples can lead to a medium benefit relative to simple problem-solving.

In addition to looking at the overall effectiveness of worked examples, Barbieri and her colleagues explored whether the benefits of worked examples depended on certain factors. The first factor they looked at was people’s prior knowledge. Were students given a problem-type that students had received instruction about, or was the problem completely novel? The second factor the researchers investigated was characteristics of the examples. For example, what type of worked example were students studying – correct worked examples only, or were they also (or only) studying incorrect worked examples? Finally, the researchers investigated whether prompting students to self-explain correct or incorrect steps was more beneficial than not.

Does prior knowledge influence the benefits of worked examples?

Those with less prior knowledge of the to-be-learned material may benefit more than those with more prior knowledge from worked examples, as worked examples are especially useful in developing knowledge of procedures. Although the authors didn’t find that the benefit of worked examples consistently depended on people’s prior knowledge, there was a mix of evidence. That is, some studies found a larger benefit for higher prior knowledge, whereas others found a larger benefit for lower prior knowledge. Other studies found no interaction. Thus, more research is needed to understand who benefits most from worked examples.

Is studying incorrect worked examples more beneficial than studying correct examples?

An incorrect worked example is a worked example that displays a common mistake students make while solving the problem and is labeled as incorrect (4). Studying incorrect worked examples is beneficial for learning as it draws attention “to the particular features in a problem that make the procedure inappropriate” (4, p. 25). Do students perform better after studying correct examples only, incorrect examples only, or some combination of correct and incorrect examples? The researchers found that studying the correct steps was more beneficial than studying incorrect examples (alone or with correct ones). It’s possible, though, that incorrect worked examples are particularly beneficial for those with low prior knowledge—those more likely to make the demonstrated mistake. 

Are worked examples more effective if they include self-explanation prompts?

An example prompt for self-explanation is “Why did they multiply the two probabilities?” The researchers found that prompting self-explanation in worked examples reduced the benefit of worked examples, though more work is needed to explore which prompts are beneficial and which are not. This result demonstrates that sometimes beliefs we have about research findings do not always match the statistics– emphasizing the importance of testing our assumptions. 

Overall, worked examples are an effective way to improve math performance. The performance benefit seems greatest for correct worked examples, and did not depend on prior knowledge, but the benefits of worked examples can be hindered when students are asked to explain the example. 

So, what does this mean for you? If you’re a student and you’re struggling to understand a concept or procedure in math, you might want to study a worked example. Textbooks often have worked examples, or your teacher may have included some in their lecture content. I suggest incorporating worked examples into your studying for two reasons. First, they can improve your math performance. As we learned in this post, studying worked examples leads to better performance than problem solving. But studying worked examples has another advantage: it can help you gauge (or monitor) your knowledge. We tend to overestimate what we know, which can lead us astray when studying – by spending too little time on material we need to learn. Fortunately, a few studies have found that studying worked examples reduces this overconfidence (5, 6), which can help you direct your study efforts more appropriately. In summary, studying worked examples can make you a more efficient and effective learner! 

References:

(1) Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58.

(2) Paas, F., van Gog, T., & Sweller, J. (2010). Cognitive load theory: New conceptualizations, specifications, and integrated research perspectives. Educational Psychology Review, 22(2), 115–121.

(3) Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A Meta-analysis of the Worked Examples Effect on Mathematics Performance. Educational Psychology Review, 35(1), 11.

(4) Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.

(5) Baars, M., van Gog, T., de Bruin, A., & Paas, F. (2014). Effects of problem solving after worked example study on primary school children’s monitoring accuracy: Improving JOL accuracy. Applied Cognitive Psychology, 28(3), 382–391.

(6) Baars, M., van Gog, T., de Bruin, A., & Paas, F. (2017). Effects of problem solving after worked example study on secondary school children’s monitoring accuracy. Educational Psychology, 37(7), 810–834.