There is a national crisis where student scores on state assessments are significantly declining. In 2019, less than half of eighth graders nationwide scored proficient on the tests (Barbieri et al., 2023). This leads to the point where effective strategies are crucial for teachers to use for students to understand foundational mathematical concepts, starting from elementary school. Using worked examples has been commonly used and shown positive outcomes. This leads to the research to be conducted to identify how worked and solved examples are affecting students in terms of assisting students to become increasingly proficient in math as a subject. Several studies have found that there are varying levels of evidence to support the effectiveness of this instructional tool known as example-based learning (Barbieri et al., 2023).

Conducted studies and research support the initiative for worked examples, saying that they’re more effective than students completing problem-solving practice questions alone. Using worked examples after knowledge acquisition is the most beneficial time in the students' learning process for implementation (Barbieri et al., 2023). This reinforces students’ newly found schemas instead of only solving practice problems that may not help to improve student understanding in the early stages. Studies have also shown that students with low prior knowledge are the student demographic to improve the most from correctly solved word problems (Barbieri et al., 2023).

Worked examples are a newer model of the Cognitive Load Theory. CLT explains that the human brain has a limited capacity in what it can process and be able to perform remembered tasks (Barbieri et al., 2023). The cognitive load for novice learners is much smaller; therefore, they must be required to remember only relevant aspects of solving math problems (Barbieri et al., 2023). With students using worked examples as a supplemental resource, they won’t need to remember tedious processes to get to a solution. Instead, students’ cognitive efforts can focus on remembering and performing relevant tasks to solve problems during their early stages of grasping new concepts (Barbieri et al., 2023). Worked examples are easily scaffolded— as students move away from the novice stage, they’ve acquired a larger cognitive load to be able to process the more methodical steps to solving problems independently, without the worked examples as a tool (Barbieri et al., 2023).  

While there has been a productive amount of research to evaluate the effectiveness of worked examples, the research studies also emphasize how this is an evolving math concept. Therefore, more needs to be done, continuously, to further identify the value in worked examples to increase student understanding.

What are Worked Examples?

Worked examples in math are step-by-step demonstrations of solved problems to help guide students in similar problems. These can be either correct or incorrect examples. However, the effectiveness of each kind can yield a different outcome— in our research, we utilized correctly solved worked examples. Research has shown that correctly solved worked examples have better outcomes for students than incorrectly solved worked examples in terms of student learning and understanding (Barbieri et al., 2023). These worked examples can be implemented in classrooms during the initial knowledge acquisition. The initial knowledge acquisition is the phase where teachers introduce a new topic or concept for children to interact and start their learning, as well as understanding. Worked examples can also be used as supplementary resources after initial teaching, during student practice. To conduct our research, students used the worked examples for supplemental support after initial teaching, as students begin solving multi-digit division problems involving integers, fractions, and decimals. 

Participants: Forty-six (46) fifth graders from a local elementary school, who were randomly assigned to one (1) of two (2) treatments 

Treatment 1 (control group): a small group using unsolved problems with explicit instruction

Treatment 2: small group using solved problems with explicit instruction 

Teachers used a recorder to have students say their names into during the first session of intervention. After the first session, teachers reviewed one (1) worked example from the last lesson by using a graphic organizer, along with stating the learning goals for the day. Teachers modeled one problem on the graphic organizer while using "think-aloud" statements to show how to solve the specific problem. Teachers showed a multiplication sheet to show students how to use the sheet as a tool when students are solving problems on their own. Teachers also used self-explanation prompts for students to demonstrate their thinking on how the problem is solved after the teacher used modeling instruction. 

Self-monitoring systems were used to understand how students felt about the problem. The systems involved green, yellow, and red lights. 

Green Light: The student understood the problem and was ready to do individual work on similar problems.

Yellow Light: The student understood most of the problem, but they need another solved example before completing individual work on similar problems. 

Red Light: The student is confused and needs more practice with more worked examples provided by the teacher. The teacher will walk the student through while having the student explain the steps within the process. 

After understanding where students felt they were at, teachers wrote a similar problem on the whiteboard with the same divisor to have students solve on their individual sheets at their own pace. When students were done with the problem, they worked on five (5) problems independently. After, teachers saved the worked example for the next lesson review. 

Teachers would go through the same steps previously mentioned, but the modeled worked examples are erased, no multiplication fact sheets were provided, and no self-explanation prompts were used when meeting with the control groups. 

Within treatment two (2), teachers used similar intervention procedures: 

• students stating their names

• Introducing the session goals 

• modeling a 3 by 2 digit division problem 

• students working independently 

Teachers modeled without self-guided questions and removed the problem from the student's view before allowing students to practice independently. This prevented students from referring back to the solved examples as they were working independently.  

There weren’t any significant differences in the results regarding solved and worked examples. However, statistics discovered that using students has improved their performance. Results from this study have shown that the explicit instructional techniques possibly contribute to growth in learning mathematical concepts, along with using worked examples. From this takeaway, using explicit instructional techniques is crucial for children as students to understand mathematical concepts. Utilizing solved examples along with explicit instruction can be helpful for students as well. This research study allows for effective insight regarding the effectiveness of worked examples for division. 

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.

What is Career Readiness? (2015). Default. 

https://www.naceweb.org/career-readiness/competencies/career-readiness-defined/#competencies

Research Compliance Protocol: The protocol is protocol #02046.