### Mathematical achievement

Preliminary analyses were carried out to evaluate assumptions for the *t* test. Those assumptions include: (a) the independence, (b) normality tested using the Shapiro–Wilk test, and (c) homogeneity of variance tested using the Levene Statistic. All assumptions were met.

The Levene Statistic for the pretest scores (*p* > 0.05) indicated that there was not a significant difference in the groups. Independent samples *t* tests were conducted to determine the effect error analysis had on student achievement determined by the difference in the means of the pretest and posttest and of the pretest and delayed posttest. There was no significant difference in the scores from the posttest for the control group (*M* = 8.23, SD = 5.67) and the treatment group (*M* = 9.56, SD = 5.24); *t*(51) = 0.88, *p* = 0.381. However, there was a significant difference in the scores from the delayed posttest for the control group (*M* = 5.96, SD = 4.90) and the treatment group (*M* = 9.41, SD = 4.77); *t*(51) = 2.60, *p* = 0.012. These results suggest that students can initially learn mathematical concepts through a variety of methods. Nevertheless, the retention of the mathematical knowledge is significantly increased when error analysis is added to the students’ lessons, assignments, and quizzes. It is interesting to note that the difference between the means from the pretest to the posttest was higher in the treatment group (*M* = 9.56) versus the control group (*M* = 8.23), implying that even though there was not a significant difference in the means, the treatment group did show a greater improvement.

The Assignment Time Log was completed by only 19% of the students in the treatment group and 38% of the students in the control group. By having such a small percentage of each group participate in tracking the time spent completing homework assignment, the results from the *t* test analysis cannot be used in any generalization. However, the results from the analysis were interesting. The mean time spent doing the assignments for each group was calculated and analyzed using an independent samples *t* test. There was no significant difference in the amount of time students which spent on their homework for the control group (*M* = 168.30, SD = 77.41) and the treatment group (*M* = 165.80, SD = 26.53); *t*(13) = 0.07, *p* = 0.946. These results suggest that the amount of time that students spent on their homework was close to the same whether they had to do error analyses (find the errors, fix them, and justify the steps taken) or solve each exercise in a traditional manner of following correctly worked examples. Although the students did not spend a significantly different amount of time outside of class doing homework, the treatment group did spend more time during class working on quiz corrections and discussing error which could attribute to the retention of knowledge.

### Feedback from participants

All students participating in the current study submitted a signed informed consent form. Students process mathematical procedures better when they are aware of their own errors and knowledge gaps [15]. The theoretical model of using errors that students make themselves and errors that are likely due to the typical knowledge gaps can also be found in works by other researchers such as Kawasaki [14] and VanLehn [29]. Highlighting errors in the students’ own work and in typical errors made by others allowed the participants in the treatment group the opportunity to experience this theoretical model. From their experiences, the participants were able to give feedback to help the researcher delve deeper into what the thoughts were of the use of error analysis in their mathematics classes than any other study provided [1, 4, 7,8,9, 11, 14,15,16,17, 21, 23,24,25,26, 29]. Overall, the teacher and students found the use of error analysis in the equations and inequalities unit to be beneficial. The teacher pointed out that the discussions in class were deeper in the treatment group’s class. When she tried to facilitate meaningful mathematical discourse [18] in the control group class, the students were unable to get to the same level of critical thinking as the treatment group discussions. In the open-ended question at the conclusion of the delayed posttest (“Please provide some feedback on your experience.”), the majority (86%) of the participants from the treatment group indicated that the use of erroneous examples integrated into their lessons was beneficial in helping them recognize their own mistakes and understanding how to correct those mistakes. One student reported, “I realized I was doing the same mistakes and now knew how to fix it”. Several (67%) of the students indicated learning through error analysis made the learning process easier for them. A student commented that “When I figure out the mistake then I understand the concept better, and how to do it, and how not to do it”.

When students find and correct the errors in exercises, while justifying themselves, they are being encouraged to learn to construct viable arguments and critique the reasoning of others [19]. This study found that explaining why an exercise is correct or incorrect fostered transfer and led to better learning outcomes than explaining correct solutions only. However, some of the higher level students struggled with the explanation component. According to the teacher, many of these higher level students who typically do very well on the homework and quizzes scored lower on the unit quizzes and tests than the students expected due to the requirement of explaining the work. In the past, these students had not been justifying their thinking and always got correct answers. Therefore, providing reasons for erroneous examples and justifying their own process were difficult for them.

Often teachers are resistant to the idea of using error analysis in their classroom. Some feel creating erroneous examples and highlighting errors for students to analyze is too time-consuming [28]. The teacher in this study taught both the control and treatment groups, which allowed her the perspective to compare both methods. She stated, “Grading took about the same amount of time whether I gave a score or just highlighted the mistakes”. She noticed that having the students work on their errors from the quizzes and having them find the errors in the assignments and on the board during class time ultimately meant less work for her and more work for the students.

Another reason behind the reluctance to use error analysis is the fact that teachers are uncertain about exposing errors to their students. They are fearful that the discussion of errors could lead their students to make those same errors and obtain incorrect solutions [28]. Yet, most of the students’ feedback stated the discussions in class and the error analyses on the assignments and quizzes helped them in working homework exercises correctly. Specifically, they said figuring out what went wrong in the exercise helped them solve that and other exercises. One student said that error analysis helped them “do better in math on the test, and I actually enjoyed it”. Nevertheless, 2 of the 27 participating students in the treatment group had negative comments about learning through error analysis. One student did not feel that correcting mistakes showed them anything, and it did not reinforce the lesson. The other student stated being exposed to error analysis did, indeed, confuse them. The student kept thinking the erroneous example was a correct answer and was unsure about what they were supposed to do to solve the exercise.

When the researcher asked the teacher if there were any benefits or disadvantages to using error analysis in teaching the equations and inequalities unit, she said that she thoroughly enjoyed teaching using the error analysis method and was planning to implement it in all of her classes in the future. In fact, she found that her “hands were tied” while grading the control group quizzes and facilitating the lessons. She said, “I wanted to have the students find their errors and fix them, so we could have a discussion about what they were doing wrong”. The students also found error analysis to have more benefits than disadvantages. Other than one student whose response was eliminated for not being on topic and the two students with negative comments, the other 24 of the students in the treatment group had positive comments about their experience with error analysis. When students had the opportunity to analyze errors in worked exercises (error analysis) through the assignments and quizzes, they were able to get a deeper understanding of the content and, therefore, retained the information longer than those who only learned through correct examples.

Discussions generated in the treatment group’s classroom afforded the students the opportunity to critically reason through the work of others and to develop possible arguments on what had been done in the erroneous exercise and what approaches might be taken to successfully find a solution to the exercise. It may seem surprising that an error as simple as adding a number when it should have been subtracted could prompt a variety of questions and lead to the students suggesting possible ways to solve and check to see if the solution makes sense. In an erroneous exercise presented to the treatment group, the students were provided with the information that two of the three angles of a triangle were 35° and 45°. The task was to write and solve an equation to find the missing measure. The erroneous exercise solver had created the equation: *x* + 35 + 45 = 180. Next was written *x* + 80 = 180. The solution was *x* = 260°. In the discussion, the class had on this exercise, the conclusion was made that the error occurred when 80 was added to 180 to get a sum of 260. However, the discussion progressed finding different equations and steps that could have been taken to discover the missing angle measure to be 100° and why 260° was an unreasonable solution. Another approach discussed by the students was to recognize that to say the missing angle measure was 260° contradicted with the fact that one angle could not be larger than the sum of the angle measures of a triangle. Analyzing the erroneous exercises gave the students the opportunity of engaging in the activity of “explaining” and “fixing” the errors of the presented exercise as well as their own errors, an activity that fostered the students’ learning.