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• Published: 30 January 2018

Teaching and learning mathematics through error analysis

• Sheryl J. Rushton 1  

Fields Mathematics Education Journal volume  3 , Article number:  4 ( 2018 ) Cite this article

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For decades, mathematics education pedagogy has relied most heavily on teachers, demonstrating correctly worked example exercises as models for students to follow while practicing their own exercises. In more recent years, incorrect exercises have been introduced for the purpose of student-conducted error analysis. Combining the use of correctly worked exercises with error analysis has led researchers to posit increased mathematical understanding. Combining the use of correctly worked exercises with error analysis has led researchers to posit increased mathematical understanding.

A mixed method design was used to investigate the use of error analysis in a seventh-grade mathematics unit on equations and inequalities. Quantitative data were used to establish statistical significance of the effectiveness of using error analysis and qualitative methods were used to understand participants’ experience with error analysis.

The results determined that there was no significant difference in posttest scores. However, there was a significant difference in delayed posttest scores.

In general, the teacher and students found the use of error analysis to be beneficial in the learning process.

For decades, mathematics education pedagogy has relied most heavily on teachers demonstrating correctly worked example exercises as models for students to follow while practicing their own exercises [ 3 ]. In more recent years, incorrect exercises have been introduced for the purpose of student-conducted error analysis [ 17 ]. Conducting error analysis aligns with the Standards of Mathematical Practice [ 18 , 19 ] and the Mathematics Teaching Practices [ 18 ]. Researchers posit a result of increased mathematical understanding when these practices are used with a combination of correctly and erroneously worked exercises [ 1 , 4 , 8 , 11 , 15 , 16 , 18 , 19 , 23 ].

Review of literature

Correctly worked examples consist of a problem statement with the steps taken to reach a solution along with the final result and are an effective method for the initial acquisitions of procedural skills and knowledge [ 1 , 11 , 26 ]. Cognitive load theory [ 1 , 11 , 25 ] explains the challenge of stimulating the cognitive process without overloading the student with too much essential and extraneous information that will limit the working memory and leave a restricted capacity for learning. Correctly worked examples focus the student’s attention on the correct solution procedure which helps to avoid the need to search their prior knowledge for solution methods. Correctly worked examples free the students from performance demands and allow them to concentrate on gaining new knowledge [ 1 , 11 , 16 ].

Error analysis is an instructional strategy that holds promise of helping students to retain their learning [ 16 ]. Error analysis consists of being presented a problem statement with the steps taken to reach a solution in which one or more of the steps are incorrect, often called erroneous examples [ 17 ]. Students analyze and explain the errors and then complete the exercise correctly providing reasoning for their own solution. Error analysis leads students to enact two Standards of Mathematical Practice, namely, (a) make sense of problems and persevere in solving them and (b) attend to precision [ 19 ].

Another of the Standards of Mathematical Practice suggests that students learn to construct viable arguments and comment on the reasoning of others [ 19 ]. According to Große and Renkl [ 11 ], students who attempted to establish a rationale for the steps of the solution learned more than those who did not search for an explanation. Teachers can assist in this practice by facilitating meaningful mathematical discourse [ 18 ]. “Arguments do not have to be lengthy, they simply need to be clear, specific, and contain data or reasoning to back up the thinking” [ 20 ]. Those data and reasons could be in the form of charts, diagrams, tables, drawings, examples, or word explanations.

Researchers [ 7 , 21 ] found the process of explaining and justifying solutions for both correct and erroneous examples to be more beneficial for achieving learning outcomes than explaining and justifying solutions to correctly worked examples only. They also found that explaining why an exercise is correct or incorrect fostered transfer and led to better learning outcomes than explaining correct solutions only. According to Silver et al. [ 22 ], students are able to form understanding by classifying procedures into categories of correct examples and erroneous examples. The students then test their initial categories against further correct and erroneous examples to finally generate a set of attributes that defines the concept. Exposing students to both correctly worked examples and error analysis is especially beneficial when a mathematical concept is often done incorrectly or is easily confused [ 11 ].

Große and Renkl [ 11 ] suggested in their study involving university students in Germany that since errors are inherent in human life, introducing errors in the learning process encourages students to reflect on what they know and then be able to create clearer and more complete explanations of the solutions. The presentation of “incorrect knowledge can induce cognitive conflicts which prompt the learner to build up a coherent knowledge structure” [ 11 ]. Presenting a cognitive conflict through erroneously worked exercises triggers learning episodes through reflection and explanations, which leads to deeper understanding [ 29 ]. Error analysis “can foster a deeper and more complete understanding of mathematical content, as well as of the nature of mathematics itself” [ 4 ].

Several studies have been conducted on the use of error analysis in mathematical units [ 1 , 16 , 17 ]. The study conducted for this article differed from these previous studies in mathematical content, number of teachers and students involved in the study, and their use of a computer or online component. The most impactful differences between the error analysis studies conducted in the past and this article’s study are the length of time between the posttest and the delayed posttest and the use of qualitative data to add depth to the findings. The previous studies found students who conducted error analysis work did not perform significantly different on the posttest than students who received a more traditional approach to learning mathematics. However, the students who conducted error analysis outperformed the control group in each of the studies on delayed posttests that were given 1–2 weeks after the initial posttest.

Loibl and Rummel [ 15 ] discovered that high school students became aware of their knowledge gaps in a general manner by attempting an exercise and failing. Instruction comparing the erroneous work with correctly worked exercises filled the learning gaps. Gadgil et al. [ 9 ] conducted a study in which students who compared flawed work to expertly done work were more likely to repair their own errors than students who only explained the expertly done work. This discovery was further supported by other researchers [8, 14, 24]. Each of these researchers found students ranging from elementary mathematics to university undergraduate medical school who, when given correctly worked examples and erroneous examples, learned more than students who only examined correctly worked examples. This was especially true when the erroneous examples were similar to the kinds of errors that they had committed [ 14 ]. Stark et al. [ 24 ] added that it is important for students to receive sufficient scaffolding in correctly worked examples before and alongside of the erroneous examples.

The purpose of this study was to explore whether seventh-grade mathematics students could learn better from the use of both correctly worked examples and error analysis than from the more traditional instructional approach of solving their exercises in which the students are instructed with only correctly worked examples. The study furthered previous research on the subject of learning from the use of both correctly worked examples and error analysis by also investigating the feedback from the teacher’s and students’ experiences with error analysis. The following questions were answered in this study:

What was the difference in mathematical achievement when error analysis was included in students’ lessons and assignments versus a traditional approach of learning through correct examples only?

What kind of benefits or disadvantages did the students and teacher observe when error analysis was included in students’ lessons and assignments versus a traditional approach of learning through correct examples only?

A mixed method design was used to investigate the use of error analysis in a seventh-grade mathematics unit on equations and inequalities. Quantitative data were used to establish statistical significance of the effectiveness of using error analysis and qualitative methods were used to understand participants’ experience with error analysis [ 6 , 27 ].

Participants

Two-seventh-grade mathematics classes at an International Baccalaureate (IB) school in a suburban charter school in Northern Utah made up the control and treatment groups using a convenience grouping. One class of 26 students was the control group and one class of 27 students was the treatment group.

The same teacher taught both the groups, so a comparison could be made from the teacher’s point of view of how the students learned and participated in the two different groups. At the beginning of the study, the teacher was willing to give error analysis a try in her classroom; however, she was not enthusiastic about using this strategy. She could not visualize how error analysis could work on a daily basis. By the end of the study, the teacher became very enthusiastic about using error analysis in her seventh grade mathematics classes.

The total group of participants involved 29 males and 24 females. About 92% of the participants were Caucasian and the other 8% were of varying ethnicities. Seventeen percent of the student body was on free or reduced lunch. Approximately 10% of the students had individual education plans (IEP).

A pretest and posttest were created to contain questions that would test for mathematical understanding on equations and inequalities using Glencoe Math: Your Common Core Edition CCSS [ 5 ] as a resource. The pretest was reused as the delayed posttest. Homework assignments were created for both the control group and the treatment group from the Glencoe Math: Your Common Core Edition CCSS textbook. However, the researcher rewrote two to three of the homework exercises as erroneous examples for the treatment group to find the error and fix the exercise with justifications (see Figs.  1 , 2 ). Students from both groups used an Assignment Time Log to track the amount of time which they spent on their homework assignments.

Example of the rewritten homework exercises as equation erroneous examples

Example of the rewritten homework exercises as inequality erroneous examples

Both the control and the treatment groups were given the same pretest for an equations and inequality unit. The teacher taught both the control and treatment groups the information for the new concepts in the same manner. The majority of the instruction was done using the direct instruction strategy. The students in both groups were allowed to work collaboratively in pairs or small groups to complete the assignments after instruction had been given. During the time she allotted for answering questions from the previous assignment, she would only show the control group the exercises worked correctly. However, for the treatment group, the teacher would write errors which she found in the students’ work on the board. She would then either pair up the students or create small groups and have the student discuss what errors they noticed and how they would fix them. Often, the teacher brought the class together as a whole to discuss what they discovered and how they could learn from it.

The treatment group was given a homework assignment with the same exercises as the control group, but including the erroneous examples. Students in both the control and treatment groups were given the Assignment Time Log to keep a record of how much time was spent completing each homework assignment.

At the end of each week, both groups took the same quiz. The quizzes for the control group received a grade, and the quiz was returned without any further attention. If a student asked how to do an exercise, the teacher only showed the correct example. The teacher graded the quizzes for the treatment group using the strategy found in the Teaching Channel’s video “Highlighting Mistakes: A Grading Strategy” [ 2 ]. She marked the quizzes by highlighting the mistakes; no score was given. The students were allowed time in class or at home to make corrections with justifications.

The same posttest was administered to both groups at the conclusion of the equation and inequality chapter, and a delayed posttest was administered 6 weeks later. The delayed posttest also asked the students in the treatment group to respond to an open-ended request to “Please provide some feedback on your experience”. The test scores were analyzed for significant differences using independent samples t tests. The responses to the open-ended request were coded and analyzed for similarities and differences, and then, used to determine the students’ perceptions of the benefits or disadvantages of using error analysis in their learning.

At the conclusion of gathering data from the assessments, the researcher interviewed the teacher to determine the differences which the teacher observed in the preparation of the lessons and students’ participation in the lessons [ 6 ]. The interview with the teacher contained a variety of open-ended questions. These are the questions asked during the interview: (a) what is your opinion of using error analysis in your classroom at the conclusion of the study versus before the study began? (b) describe a typical classroom discussion in both the control group class and the treatment group class, (c) talk about the amount of time you spent grading, preparing, and teaching both groups, and (d) describe the benefits or disadvantages of using error analysis on a daily basis compared to not using error analysis in the classroom. The responses from the teacher were entered into a computer, coded, and analyzed for thematic content [ 6 , 27 ]. The themes that emerged from coding the teacher’s responses were used to determine the kind of benefits or disadvantages observed when error analysis was included in students’ lessons and assignments versus a traditional approach of learning through correct examples only from the teacher’s point of view.

Findings and discussion

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.

The students participating in both the control and treatment groups from the two-seventh-grade mathematics classes at the IB school in a suburban charter school in Northern Utah initially learned the concepts taught in the equations and inequality unit statistically just as well with both methods of teaching. The control group had the information taught to them with the use of only correctly worked examples. If they had a question about an exercise which they did wrong, the teacher would show them how to do the exercise correctly and have a discussion on the steps required to obtain the correct solutions. On their assignments and quizzes, the control group was expected to complete the work by correctly solving the equations and inequalities in the exercise, get a score on their work, and move on to the next concept. On the other hand, the students participating in the treatment group were given erroneous examples within their assignments and asked to find the errors, explain what had been done wrong, and then correctly solve the exercise with justifications for the steps they chose to use. During lessons, the teacher put erroneous examples from the students’ work on the board and generated paired, small groups, or whole group discussion of what was wrong with the exercise and the different ways to do it correctly. On the quizzes, the teacher highlighted the errors and allowed the students to explain the errors and justify the correct solution.

Both the method of teaching using error analysis and the traditional method of presenting the exercise and having the students solve it proved to be just as successful on the immediate unit summative posttest. However, the delayed posttest given 6 weeks after the posttest showed that the retention of knowledge was significantly higher for the treatment group. It is important to note that the fact that the students in the treatment group were given more time to discuss the exercises in small groups and as a whole class could have influenced the retention of mathematical knowledge just as much or more than the treatment of using error analysis. Researchers have proven academic advantages of group work for students, in large part due to the perception of students having a secure support system, which cannot be obtained when working individually [ 10 , 12 , 13 ].

The findings of this study supported the statistical findings of other researchers [ 1 , 16 , 17 ], suggesting that error analysis may aid in providing a richer learning experience that leads to a deeper understanding of equations and inequalities for long-term knowledge. The findings of this study also investigated the teacher’s and students’ perceptions of using error analysis in their teaching and learning. The students and teacher used for this study were chosen to have the same teacher for both the control and treatment groups. Using the same teacher for both groups, the researcher was able to determine the teacher’s attitude toward the use of error analysis compared to the non-use of error analysis in her instruction. The teacher’s comments during the interview implied that she no longer had an unenthusiastic and skeptical attitude toward the use of error analysis on a daily basis in her classroom. She was “excited to implement the error analysis strategy into the rest of her classes for the rest of the school year”. She observed error analysis to be an effective way to deal with common misconceptions and offer opportunities for students to reflect on their learning from their errors. The process of error analysis assisted the teacher in supporting productive struggle in learning mathematics [ 18 ] and created opportunity for students to have deep discussions about alternative ways to solve exercises. Error analysis also aided in students’ discovery of their own errors and gave them possible ways to correct those errors. Learning through the use of error analysis was enjoyable for many of the participating students.

According to the NCTM [ 18 ], effective teaching of mathematics happens when a teacher implements exercises that will engage students in solving and discussing tasks that promote mathematical reasoning and problem solving. Providing erroneous examples allowed discussion, multiple entry points, and varied solution strategies. Both the teacher and the students participating in the treatment group came to the conclusion that error analysis is a beneficial strategy to use in the teaching and learning of mathematics. Regardless of the two negative student comments about error analysis not being helpful for them, this researcher recommends the use of error analysis in teaching and learning mathematics.

The implications of the treatment of teaching students mathematics through the use of error analysis are that students’ learning could be fostered and retention of content knowledge may be longer. When a teacher is able to have their students’ practice critiquing the reasoning of others and creating viable arguments [ 19 ] by analyzing errors in mathematics, the students not only are able to meet the Standard of Mathematical Practice, but are also creating a lifelong skill of analyzing the effectiveness of “plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is” ([ 19 ], p. 7).

Limitations and future research

This study had limitations. The sample size was small to use the same teacher for both groups. Another limitation was the length of the study only encompassed one unit. Using error analysis could have been a novelty and engaged the students more than it would when the novelty wore off. Still another limitation was the study that was conducted at an International Baccalaureate (IB) school in a suburban charter school in Northern Utah, which may limit the generalization of the findings and implications to other schools with different demographics.

This study did not have a separation of conceptual and procedural questions on the assessments. For a future study, the creation of an assessment that would be able to determine if error analysis was more helpful in teaching conceptual mathematics or procedural mathematics could be beneficial to teachers as they plan their lessons. Another suggestion for future research would be to gather more data using several teachers teaching both the treatment group and the control group.

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Rushton, S.J. Teaching and learning mathematics through error analysis. Fields Math Educ J 3 , 4 (2018). https://doi.org/10.1186/s40928-018-0009-y

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• Standards of mathematical practices

Teaching Students Error Analysis: A Pathway to Critical Thinking

Last year, I visited classrooms with April Shackleford, the principal of Lucama Elementary, recently recognized as a Blue Ribbon School. We walked into a 4th-grade math classroom, and the first thing I noticed were students working in pairs and discussing math problems. 

As we walked around the classroom, it became clear that students were working together, reviewing a test they had just taken. They discussed each problem, and when they encountered one that they both missed, they used an Error Analysis process to find the error point(s). If one student missed the problem, the other student became the teacher and went through the mathematical process step-by-step with the other student until they identified the error(s). If the students got stuck, they would raise their hands and ask for help from the teacher, who was circling the classroom and listening to the conversations. This process went on until the students went through the entire test.  

How to Adapt and Implement Error Analysis In Your Classroom

This visit sparked the idea for this article. I have seen Error Analysis used in classrooms from Kindergarten to High School and in many subjects. This test review process was just one example of how teachers can incorporate Error Analysis into a classroom. So, I set out to collect examples, share research, and provide you with ideas to adapt and implement in your classroom.

In this article, you will learn:

• the importance of integrating Error Analysis into your teaching methodology, highlighting its impact on student learning. 
• practical strategies and examples to effectively incorporate Error Analysis into your lessons, transforming how students perceive and learn from their mistakes.

Table of Contents

Error Mindset

• Importance of Error Analysis
• Benefits of Error Analysis and its Impact on Critical Thinking
• Using Error Analysis in English Language Arts
• Using Error Analysis in Math
• Using Error Analysis in Social Studies
• Using Error Analysis in Science

Cultivating a Classroom Culture Around Error Analysis

Benefits of error analysis for students, embracing error analysis for empowered learning.

Traditionally, classrooms have viewed mistakes as obstacles, hindering the journey toward academic achievement. Yet, envision a transformative shift in mindset, where these errors are reconceptualized as essential instruments for gaining insight and deepening understanding. Consider Thomas Edison’s words regarding his endeavors to invent the lightbulb: “I have not failed; I just found 10,000 ways that won’t work.” This statement epitomizes the core philosophy of Error Analysis. 

What Is Error Analysis? 

Error Analysis is an approach that invites students to engage deeply with their errors and the errors of others, to understand them not as failings but as opportunities for growth and learning.

In this revised perspective, errors are no longer the end of learning but the beginning. They open avenues for exploring why a certain approach didn’t work and what can be learned from it, enriching the student’s understanding and approach to problem-solving. This is the heart of the Error Mindset – viewing each mistake not as a roadblock but as a guidepost on the path to knowledge.

Importance of Error Analysis in Cultivating Higher Order Thinking

Many studies have provided evidence that when students learn and independently use Higher Order Thinking strategies, their achievement increases. This increased focus on Higher Order Thinking in standards, along with corresponding changes in how students are being assessed, has made the integration of Higher Order Thinking an expectation that must be emphasized in every classroom.  Error Analysis is one of the nine Higher Order Thinking Strategies students should be taught and expected to use independently.

Building on an Error Mindset: Error Analysis is a critical and versatile tool for learning relevance in mathematics, science, language arts, and more. This approach extends far beyond the mere identification of student mistakes by teachers; it is an empowering process that enhances clarity, deepens understanding, and cultivates strategies to mitigate similar errors in the future, thus promoting higher order thinking skills.

Benefits of Error Analysis and its Impact on Critical Thinking:

• Deepens Learning and Critical Analysis: Error Analysis drives students beyond basic understanding to a more profound engagement. It involves examining the ‘why’ and ‘how’ behind mistakes, an exercise in critical thinking. This process doesn’t just clarify concepts; it strengthens knowledge retention and encourages learners to actively challenge and rectify their misconceptions, enhancing their analytical skills.

• Fosters Growth Mindset and Creative Thinking: At the heart of Error Analysis is the belief that mistakes are not failures but opportunities for growth and learning. This philosophy nurtures a growth mindset among students, encouraging them to view challenges as chances to learn and improve. It’s a shift that cultivates resilience and a lifelong passion for learning, as each error becomes a stepping stone towards mastery. Additionally, this mindset invites creative solutions, pushing students to think outside the box in understanding and overcoming their errors.
• Enhances Metacognitive Abilities: Engaging in Error Analysis contributes significantly to developing metacognitive skills. As students reflect on their mistakes, they gain insights into their thinking processes and learning strategies. This heightened self-awareness is a cornerstone of higher order thinking, enabling students to regulate their learning and adapt strategies for improved outcomes.

The significance of Error Analysis in education cannot be overstated. It transforms how students perceive and interact with their learning process, equipping them with critical skills for academic success and beyond.

Having explored the Error Analysis conceptual framework and critical aspects, you might wonder how this translates into tangible classroom practices. To bridge this gap, let’s delve into a few concrete examples.

Error Analysis Examples Across the Curriculum: Subject-Specific Strategies

Below are practical examples illustrating how this approach can be tailored to different disciplines, demonstrating its versatility and impact in fostering deeper student understanding and engagement. These subject-specific strategies for integrating Error Analysis demonstrate how it can transform the classroom learning landscape.

Using Error Analysis in English Language Arts: 

Goal: Students will identify and correct misinterpretations of the central idea in given text excerpts.

Activity: Explain the importance of accurately determining the central idea of a text. Divide the class into pairs and provide each pair with text excerpts and the corresponding inaccurate summaries. Students read the excerpts and discuss in pairs why the provided summaries do not accurately reflect the central idea. Each pair writes a corrected summary, focusing on the actual central idea of the text. Utilize Pairs Squared to share their corrected summaries and discuss the common mistakes made in the initial interpretations.

Goal: To enhance students’ ability to provide accurate text evidence and develop critical thinking skills by identifying and correcting misinterpretations in literary analysis.

Activity: Select a short story, poem, or passage from a novel rich in thematic elements and literary devices. Prepare a worksheet with a series of statements about the text. Half of these statements should contain common misinterpretations or incorrect text evidence, while the other half are accurate. Students will go through the text, finding evidence to support their decisions about the accuracy of each statement (Think).  Ask students to mark up the text, highlighting or noting where it contradicts or supports the statements (Ink). Divide students into collaborative pairs (Pair). Students will share their findings with their partner, explaining why certain statements are misinterpretations and providing the correct text evidence (Share). To ensure that each student shares, select one student to share the odds and the other to share the evens. Close the activity by discussing how it changed their approach to reading and analyzing text and emphasizing the importance of supporting arguments with accurate textual evidence in literary analysis.

Using Error Analysis in Math :

Goal: To demonstrate the process of using Error Analysis.

Activity: Present the following word problem to students: Jake built a backyard pen for his new pet pig. The length of the pen is 12 feet, and the width is half that. What is the area of the pen? Lisa says 6 sq ft, Tom says 72 sq ft, and John says 84 sq ft. 

• Who is correct? 
• How do you know? 
• What errors did each of the other two make? 

Do a Think Aloud to work through the problem, circling important information and showing work with the area formula.

Goal: To allow students to identify and discuss the most common errors when determining an area.

Activity: Organize students into Pairs and ask them to identify who is a one and a two. In Numbered Head pairs, ask 1s to tell 2s how to solve for the area; 2s tell 1s a mistake to avoid when solving for the area. In Collaborative Pairs, students will solve two similar problems on mini whiteboards, but they will complete the problem incorrectly. The two students will exchange the whiteboards and then identify and correct the error their partner made. They will share with their partner to ensure they identified the correct error. You may continue this activity for additional practice.

Goal: Use Error Analysis so that students demonstrate an understanding of the mathematical process and the common mistakes.

Directions: Use the information you have learned from this lesson to explain how the error can be corrected or why the problem is correct. Then, use the Record Media Button, and you will submit your explanation. As part of your explanation, show me your work and self-evaluate using the rubric. 

Problem: Mark multiplied 2.56 x 10 and said the answer was 0.256. Is Mark’s answer correct? Explain your reasoning. Be sure to include the correct answer if Mark is incorrect.

Assessment Rubric

Using Error Analysis in Social Studies :

Goal : Demonstrate that errors in social studies often arise from oversimplifications or misunderstandings of complex events.

  Activity: Present students with a historically inaccurate statement (e.g., “The Civil War was solely about slavery”). Ask them to validate or refute it using primary sources. Through this, they might realize that while slavery was a central issue, there were other economic and political factors at play. Error analysis here helps students grasp the multifaceted nature of historical events.

Goal: To identify and correct misconceptions about the impact of specific innovations on North Carolina’s and the nation’s development.

Activity (Jigsaw): Prepare various articles or sources of key innovations (like the cotton gin, the Wright brothers’ airplane, and the telegraph) with descriptions that include intentional errors or misconceptions. Divide students into groups of 4 students. This will be their home group. Assign each student a different article. Each person will research their assigned innovation, focusing on its impact on North Carolina and the nation, and identify errors in the provided descriptions. Place students with the same article in an expert group where they discuss their findings, highlight the errors, and provide corrected information. The experts then return to their home group and share their findings with the students in their home group. Close the lesson by asking students to answer the following question: How did these innovations impact the development of North Carolina and the nation?

Using Error Analysis in Science :

Goal:  Demonstrate the patterns of weather (seasons) and factors that affect weather. 

Describe the task:   Provide students with graphic organizers that have events with cause and effect included. However, some causes or effects are false- the cause does not create the event or effects indicated. Students work in pairs to identify what is invalid and provide support for their conclusions. Students place an X in a bright color over the cause or effect they believe is wrong and write a correct cause or effect beside the incorrect information. Place a box around the new answer. Provide models for students on how to find and correct an error.  Go over the rubric with students.

Grading: Rubric for Error Analysis – “Seasons”

Having explored how Error Analysis can be effectively applied in various subject areas through targeted activities and quizzes, let’s now shift our focus to the broader impact of this approach on students. Teaching Error Analysis is about shifting the perspective on mistakes. Instead of viewing them as dead-ends, they become crossroads, leading to deeper understanding and refined skills. When students learn to analyze their errors, they take an active role in their education, becoming detectives of their thought processes. In doing so, they’re better equipped to navigate challenges, not just in academics but in any endeavor they undertake.

Incorporating Error Analysis into the Classroom offers numerous benefits:

• Identify Learning Needs: It helps teachers better understand their students’ learning needs. By identifying the types of errors that students are making, teachers can gain insight into their students’ misconceptions and areas of weakness. This allows teachers to address the root causes of student errors by developing targeted instruction and interventions.
• Enhanced Metacognition: It helps students to learn from their mistakes. Error Analysis can help students to identify and correct their errors. By dissecting errors, students can recognize patterns in their misunderstandings and adjust their strategies. This can lead to their ability to self-correct when applying new knowledge.
• Deliberate Erroring: Students can identify misconceptions up front by asking students to make deliberate errors, which helps students and leads to better understanding.
• Better Academic Performance: It promotes critical thinking and problem-solving skills in all subjects. When students analyze their errors, they are forced to think critically about their thinking process and why they made the mistakes that they did. This can help them to develop stronger problem-solving skills.
• Increased Resilience: It creates a more positive learning environment. When students feel comfortable making mistakes and know they will be used to help them learn, they are more likely to be engaged and motivated in their learning.

As we conclude this exploration of Error Analysis in education, it’s clear that this approach transcends traditional teaching methods, offering a pathway to deeper, more resilient learning. By viewing mistakes not as failures but as opportunities for growth, we empower students to become active participants in their learning journey, fostering a classroom culture that values curiosity, critical thinking, and the courage to explore.

Incorporating Error Analysis into your classroom is a journey of discovery for both you and your students. As we have seen through various examples across subjects, the application of this method can be as diverse as it is impactful. Error Analysis provides tools for enhancing student learning.

As you embark on this journey, here are some final thoughts to ponder, actions to consider, and ways for educational leaders to guide this transformative practice:

Questions to Ponder:

• How can Error Analysis reshape students’ attitudes towards mistakes and challenges in learning?
• In what ways can incorporating deliberate errors enhance students’ critical thinking skills?
• How can Error Analysis be adapted to suit different subjects?
• What role does teacher feedback play in guiding students through Error Analysis?
• How can Error Analysis be integrated into assessment methods to provide a more comprehensive understanding of student learning?
• What are the potential challenges in implementing Error Analysis, and how can they be overcome?

Actions to Consider:

• Start small by introducing Error Analysis in one lesson or subject, and gradually expand its use.
• Create a classroom environment where mistakes are openly discussed and analyzed without judgment.
• Collaborate with colleagues to share strategies and experiences in implementing Error Analysis.
• Regularly solicit feedback from students on the effectiveness of Error Analysis in their learning.

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Error Analysis

Introduction, significant figures, the idea of error, classification of error, a. mean value, b. measuring error, maximum error, probable error, average deviation, standard deviation, c. standard deviation, propagation of errors.

Initial Thoughts

Perspectives & resources, what is data-based individualization.

• Page 1: Overview of Data-Based Individualization

How can school personnel use data to make instructional decisions?

• Page 2: Collecting and Evaluating Data
• Page 3: Progress Monitoring
• Page 4: Analyzing Progress Monitoring Data
• Page 5: Diagnostic Assessment
• Page 6: Error Analysis for Reading

Page 7: Error Analysis for Mathematics

• Page 8: Making Data-Based Instructional Decisions for Reading
• Page 9: Making Data-Based Instructional Decisions for Mathematics
• Page 10: References & Additional Resources
• Page 11: Credits

• Identify the steps a student can perform correctly (as opposed to marking answers as correct or incorrect, which might mask the steps the student can perform correctly)
• Identify patterns of errors
• Determine whether the error is a one-time miscalculation or whether it is a persistent error indicating an important misunderstanding of a math concept or operation

To conduct an error analysis for mathematics, the teacher can analyze the student’s errors on a worksheet, test, or progress monitoring measure. The teacher should score each problem, marking each incorrect digit in the student’s answer from RIGHT to LEFT for addition, subtraction, and multiplication problems. By scoring from right to left, the teacher can be sure to note incorrect digits in the place value columns. Division problems should be scored LEFT to RIGHT. Evaluating each numeral in the answer allows the teacher to gain more information.

Click here to view an example of a mathematics worksheet containing addition and subtraction problems in which the teacher has noted the student’s errors .

Description

Cole’s Math Worksheet

This document is an example of Cole’s mathematics classwork. It consists of 25 mathematics problems, some of which Cole’s teacher has marked as incorrect. The problems are as follows: Problem 1 is 105 plus 25. Cole has answered 130, showing how he regrouped to solve the problem. Problem 2 is 47 minus 39. Cole has answered 12, a response he reached by subtracting the 7 from the 9 without regard to their position in the problem. The teacher has marked out both numbers of the 12 with a blue pen. Problem 3 is 312 minus 0. Cole has correctly answered 312. Problem 4 is 98 minus 1, and Cole has recorded the correct answer, 97. Problem 5 is 26 minus 12. Cole has correctly answered 14. Problem 6 is 154 minus 80, but Cole’s response, 134, is incorrect. His teacher has marked out all three numbers in his response with a blue pen. Problem 7 is 406 minus 295. Cole has incorrectly answered 291, and his teacher has marked out the 2 and the 9 with a blue pen. Problem 8 is 126 minus 29. Cole has incorrectly responded 103. His teacher has marked out all three numbers in his answer with a blue pen. Problem 9 is 13 plus 7. Cole has correctly answered 20, showing how he regrouped to solve the problem. Problem 10 is 409 plus 80, and Cole has correctly answered 489. Problem 11 is 375 plus 301, and Cole has correctly answered 676. Problem 12 is 432 minus 189, but Cole has incorrectly answered 366, all three digits of which his teacher has marked out with blue pen. Problem 13 is 222 plus 100, and Cole has correctly answered 322. Problem 14 is 59 minus 57, and Cole has correctly answered 2. Problem 15 is 68 plus 32, and Cole has correctly answered, showing how he regrouped to solve the problem. Problem 16 is 429 minus 421, which Cole has correctly answered as 008. Problem 17 is 109 minus 92. Cole’s answer, 197, is incorrect. His teacher has marked out all three digits of his response with a blue pen. Problem 18 is 416 minus 80. Cole’s answer, 376, is incorrect. His teacher has marked out the 7 with a blue pen. Problem 19 is 493 minus 36. Cole’s response, 463, is incorrect. His teacher has marked through the 6 and the 3 with a blue pen. Problem 20 is 272 minus 27. Cole’s answer, 255, is incorrect. His teacher has marked through the 5 in the tens position with a blue pen. Problem 21 is 211 minus 85. Cole’s response, 174, is incorrect. His teacher has marked through all three digits with a blue pen. Problem 22 is 305 plus 101. Cole’s answer, 406, is correct. Problem 23 is 219 minus 79, but Cole’s response, 260, is incorrect. His teacher has crossed through the 2 and the 6 with a blue pen. Problem 24 is 48 minus 18, which Cole answers correctly with 30. Finally, Problem 25 is 207 plus 111. Cole’s answer, 318, is correct.

For Your Information

• Unlike for reading in which the PRF is a general indicator of a student’s reading skills, there is no general indicator of a student’s mathematics skills.
• In the absence of appropriate progress monitoring measures or work samples, the National Center on Intensive Intervention provides information about ways to assess a student’s mathematics skills: counting, basic facts, place value concepts, whole-number computation, and fractions as numbers.

Handout: Mathematics Assessment Supplement

After marking the errors, it is important to analyze them further to help identify what types of errors have been made. Several of the most common errors students make with mathematical computation can be found in the table below.

Common Types of Computational Errors

• The example is 7 plus 4, which the student has answered incorrectly with 13.
• The first example is 28 plus 9. The student’s answer, 19, is incorrect. The second example is 10 plus 9. The student’s answer, 91, is also incorrect.
• The first example is 23 plus 78. The student’s response, 91, is incorrect. The second example is 34 plus 57. The student incorrectly answers 811.
• The example is 102 minus 31. The student incorrectly answers 131.
• The first example is 234 minus 45, which the student incorrectly answers 279. The second example is 3 plus 2. The student’s answer, 6, is incorrect.
• The example is 321 plus 245. The student answers incorrectly with 124.
• The example is ¾ plus 1/3, which the student answers as 4/7.
• The equation in the example is ½ divided by 2 equals ½ times 2/1 equals 2/2 equals 1.
• The example is 6.45 plus 72.1, which the student has answered incorrectly as 137.5.
• The example is 7.2 times .3. The student’s answer, 21.6, is incorrect.

Now that you have reviewed several types of commonly occurring mathematical errors, let’s revisit Cole’s worksheet from above. Click here to view the worksheet .

Notice that the only problems that Cole answered incorrectly were subtraction problems. As we analyze these problems, it appears as though he always subtracts the lesser number from the greater number, regardless of whether the lesser number is on the top (minuend) or the bottom (subtrahend). The first three problems he answered incorrectly are described in the table below.

Problems Analyzed from Cole’s Worksheet

• The problem is 47 minus 39. Cole has answered 12.
• The problem is 154 minus 80. Cole has answered 134.
• The problem is 406 minus 295. Cole has answered 291.

Now that you have seen how Cole’s incorrect problems were analyzed, it is time for you to practice analyzing Student 2’s mathematics errors. Click here to begin .

For this activity, click on each problem you wish to mark incorrect. For each incorrect response, analyze the correct and incorrect digits and try to determine the student’s error pattern(s). Next, in the description box below the probe, fill in the type(s) of errors Student 2 is making. After doing this, check your answer by clicking on the feedback link.

View feedback

Student 2’s data seem to indicate that he fails to find the common denominator when adding fractions. He made this mistake on all of the problems he got incorrect, except for the problem 11. For this problem, it appears as though he made a careless error: He answered all other problems with a common denominator correctly.

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Resources for Addressing Mathematics Skills

A2: Data Analysis 3 – Error

DOWNLOAD A2 MODULE

The most updated lab writing instructional modules are available: engineeringlabwriting.org

Learning Objectives

This module is designed to assist engineering instructors in strengthening lab instruction materials so that students should be able to:

• Define systematic and random error.
• Calculate the systematic error (aka bias) in a sample and explain its source.
• Calculate the random error (aka uncertainty) in a sample and recommend ways to reduce it.
• Differentiate systematic and random error.
• Present error in both absolute (as a quantity) and relative (as a percentage) terms.

What is Error?

Error is a difference between an expected value and a measured value and is categorized as either systematic (aka repeatable) or random (not following a pattern). Systematic errors can be attributed to problems in calibration or test configuration that can often be addressed or explained. Random errors are often associated with the precision of instruments used in measurement and can be addressed only by improving precision and test standardization.

Why Does the Technical Audience Value Error Analysis?

Quantifying the error in reported values provides an indication of the precision of the result. This is conveyed commonly by correctly reporting significant figures. For example, a value of 5.3 mm indicates a precision of ± 0.1 mm (± 2%). However, a rigorous error analysis might show that the uncertainty is actually ± 0.7 mm (± 13%), which significantly impacts the confidence I might have in the result. If an error analysis is not provided, your audience will likely take your results at face value, or worse, question your work for lack of rigor.

How is an Error Analysis Performed?

An error analysis can be conducted on either univariate or bivariate data. For univariate data, the analysis is simple:

• Calculate the differences of a series of measured values from a single expected value.
• Calculate the average of these differences. This is the systematic error, or bias, and it is either greater than or less than the expected value. Its sign is important.
• Calculate the standard deviation of these differences. This quantifies the random error, or uncertainty, and it occurs on either side of the average measured value. Two standard deviations capture 95% of the likely error. Three standard deviations capture 99.7% of the likely error.

For bivariate data, the analysis is similar, but rather than comparing to a single expected value, you are comparing data to expected values estimated by a trendline.

This is all better explained with an example. See an example of this here: Error Analysis Example .

What Expectations Does the Technical Audience Have for an Error Analysis?

• Ensure accuracy of your procedure and results.
• Report both the bias (or systematic error) and the uncertainty (or random error) in both absolute (with units) and relative (as a percentage) terms.
• Describe a likely source or sources of the bias.
• Describe the reasons for the random error.
• Describe improvements to a testing procedure to reduce error.

 What are Some Common Mistakes Seen in Poorly Written Engineering Lab Reports?

• Error is represented incorrectly with values presented with excessive and unrealistic levels of precision (e.g. 5.2343 mm).
• Error analysis is not conducted; results stand alone without any discussion of bias or uncertainty.
• Kim, J., Kim, D., (2019) “How engineering students draw conclusions from lab reports and design project reports in junior-level engineering courses,” The Proceedings of 2019 ASEE Annual Conference and Exposition, Tampa, FL, June 2019. Available: https://peer.asee.org/how-engineering-students-draw-conclusions-from-lab-reports-and-design-project-reports-injunior-level-engineering-courses.pdf
• “Argument Papers”, Purdue University, Purdue Online Writing Lab, Argument Papers, Available: https://owl.purdue.edu/owl/general_writing/common_writing_assignments/argument_papers/conclusions.html
• “Student Writing Guide”, University of Minnesota Department of Mechanical Engineering, Available: http://www.me.umn.edu/education/undergraduate/writing/MESWG-Lab.1.5.pdf

Using Error Analysis to Inform Meaningful Instruction

Parent: What did you study in math today?

Child: Fractions.

Parent: What did you learn?

Child: About one-tenth of what I was supposed to!

My mathematics teaching career began during the 1970s in an individualized middle-school setting. Working with students one-on-one taught me the importance of using diagnostic testing and error analysis to develop pinpoint instruction and intervention. In that setting, I formed my belief that simply marking an incorrect answer to a math problem as being WRONG provides limited information to both teacher and student. The teacher needs to know the nature of the error in order to properly inform instructional activities. The student needs to know what he/she understands as well as know of any specific misconceptions he/she may have.

Years later when I taught developmental mathematics in a learning laboratory at the community college level, I discovered that many of the systematic errors made in basic math (whole number computation, fractions, early algebra) at this level were similar to those I had found during my early teaching experience. A community college student who computes ¼ + 3/4 and obtains 4/8 probably has had the same misconceptions about the meaning of a fraction throughout grade school and high school. This motivated me to examine the academic research on the subject to find ways to address this problem.

According to Siegler (2003), “… children who lack … understanding frequently generate flawed procedures that result in systematic patterns of errors. The errors are an opportunity in that their systematic quality points to the source of the problem and this indicates the specific misunderstanding that needs to be overcome” (p. 291). My strong desire to teach for conceptual understanding based both on what the student knows and on what the student does not know, led me to write a collection of diagnostic tests with targeted interventions.

Consider the subtraction problem below from a diagnostic test, with the four given possible answers. The correct answer is (C). On a well-written diagnostic test, the foils are all based on common error patterns. The choice (D), Not Here, should be included so that students are encouraged NOT to guess when they are unsure about an answer. On a diagnostic test, guessing into a correct or incorrect answer provides no useful information.

                   83

                 – 57

(A) 34             (B) 36             (C) 26             (D) Not Here

For foil (A), a teacher should ask him or herself, “How was the answer 34 obtained?” Inspection of the problem (or a discussion with the student) likely will reveal that the student evidently subtracted the smaller digit (3) from the large digit (7) in the ones position. For foil (B), it appears that the student only partially renamed the minuend. Instead of renaming 83 as 7 ten and 13 ones, the student “renamed” 83 as 8 tens and 13 ones.

Beattie and Algozzine (1982) note that when teachers use diagnostic tests to look for error patterns, “testing for teaching begins to evolve” (p. 47). There are several ways to provided targeted intervention for the above problem. The use of base-10 blocks or play money on a place-value mat is my preference. Students should display the minuend using the 8 tens and 3 ones.

An intervention for foil (A) would involve posing these key questions: “Do we have enough ones to take 7 ones from 3 ones?” (No.) “So what should we do?” (Break apart one of the tens, and combine the resulting 10 ones with the 3 ones. Now we have 13 ones.) “Now do we have enough ones to take 7 ones from 13 ones? (Yes.) It is important to note that we should never tell students, “You cannot subtract a larger number from a smaller number.” Although such language may be expedient, it generally leads to misconceptions in later grades when students use integers to subtract larger numbers from smaller numbers.

For foil (B), key questioning should focus on the renaming process. Students need to understand that when the renaming is done correctly, the renamed minuend is equal to the original minuend . Ask: “Does 8 tens and 13 tens = 83?” (No.) “So something is wrong here. What did you do to obtain the 10 ones that were combined with the 3 ones?” (I traded one of the tens for 10 ones) “Good. So since you traded away one of the tens, we now have only 7 tens left, not 8 tens. So what should be the renamed minuend?” (7 tens and 13 ones.)

For both foils, a question related to reasonableness could also be asked: “Is it reasonable that subtracting a number close 60 from 83 would produce a number that is greater than 30?” (No. The answer should be less than 30.)

Let’s revisit the error pattern ¼ + ¾ = 4/8. Some students make this error because they apply whole-number concepts to fractions. They may read the fraction ¾ as “three fours” and may be struggling with how the numerator and denominator are related. It is interesting to note that in Korean, Chinese, and Japanese (and other languages), a fraction, say, three fourths, is read “of four parts, three” — with the denominator being stated first. Based on research with first and second graders, Miura and Yamagishi (2002) concluded that “the Korean vocabu­lary of fractions appeared to influence conceptual understanding and resulted in the children having acquired a rudimentary understanding of fraction concepts prior to formal instruction” (p. 207).

Language can also be used to illustrate that the denominator tells the unit being counted. For example, a fraction such as ¾ can be thought of as a measurement, say, 3 miles. Explain that in 3 miles, the unit of measure is miles, and we are talking about 3 of those units. In ¾, the unit of measure is fourths, and we are talking about 3 of them. So, just as 1 mile + 3 miles = 4 miles , we can use the word name “fourths” to show that

1 fourth + 3 fourths = 4 fourths.

Yet another way to address the error is to ask questions related to reasonableness: “Which is greater, ¾ or ½?” (3/4.) “So, is it reasonable that adding ¼ to ¾ would produce a fraction equivalent to ½? (No. The answer will be greater than ½.)

Of course, some errors that occur in students’ work are “random”—generally due to carelessness or incorrect recall o facts. However, I have found that many more errors are due to misconceptions and the use of incorrect strategies—often strategies that were engrained through rote memorization rather than through conceptual understanding. In such cases, I have found that providing (additional) meaningless drill is not the answer. (Would simply assigning ten more problems akin to 83 – 57 or ¼ + 3/4 really help?) Rather, a targeted approach based on teaching for conceptual understanding that addresses the nature of the error seems to provide a more efficient and long-lasting solution.

In her case study, Bray (2011) concluded that teachers “would benefit from a greater awareness of common student errors and how these errors are related to key mathematics concepts” (p. 35). Bray believes that teachers need support in developing teaching practices that use student errors in the classroom as springboards for class discussion. I fully agree, and I encourage teachers to seek resources that provide powerful diagnostic and intervention tools based on identified error patterns. The use of such resources will clearly save teachers time—thus enabling them to diagnose the nature of student errors and provide meaningful instruction based on student conceptual understanding.

Bray, W. S. (2011).  A collective case study of the influence of teachers’ beliefs and knowledge on error-handling practices during class discussion of mathematics. Journal for Research in Mathematics Education, 42 (1), 2–38.

Beattie, J. & Algozzine, B. (1982). Testing for teaching. Arithmetic Teacher, 30 (1), 47–51.

Miura, I. T., & Yamagishi, J. M. (2002). The development of rational number sense. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions. Yearbook of the National Council of Teachers of Mathematics (pp. 206–212). Reston, VA: National Council of Teachers of Mathematics.

Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 289–303). Reston, VA: National Council of Teachers of Mathematics.

Written by David Spangler

David B. Spangler is the recipient of the 2014 Lee E. Yunker Mathematics Leadership Award, sponsored by the Illinois Council of Teachers of Mathematics. The award honors an Illinois teacher for providing outstanding resources to mathematics teachers.

David is the author of two Corwin Press titles: Strategies for Teaching Whole Number Computation and Strategies for Teaching Fractions . As a mathematics educator, his goal is to teach mathematics for meaning rather than in a way that promotes rote memorization. Both of his research-based books were written to help teachers achieve that goal.

David’s professional career of more than 40 years has been exclusively devoted to mathematics education. He has literally worked with thousands of students and teachers during his career David began as a middle-school mathematics teacher in an individualized setting. Later he taught at Triton Community College, where he gained direct experience interacting with struggling students in a developmental math laboratory.

Currently David teaches mathematics methods courses through National-Louis University and ActiveMath Workshops, a professional development company he co-founded in 1994 (activemath.com). He has written numerous articles for mathematics journals, such as the popular “Cartoon Corner” for Mathematics Teaching in the Middle School.

David lives with his wife, Bonnie, in Northbrook, Illinois. They have three grown children, Ben, Jamie, and Joey.

Test Scores Plateauing? Here’s What’s Missing in School Improvement Efforts

Instruction that builds lasting learning, latest comment.

F. Williams / December 28, 2015

Great article. I think it is equally important to teach our students to look at their work, identify errors, correct it and articulate a strategy that they need to remember to avoid making the error in the future. There are great activities for this on http://www.matherroranalysis.com .

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Errors in Numbers of Participants in Studies Included in Meta-Analysis

• 1 Department of Psychiatric Laboratory, Tianjin Medical University, Tianjin, China
• Original Investigation Association of Schizophrenia With the Risk of Breast Cancer Incidence: A Meta-analysis Chuanjun Zhuo, MD, PhD; Patrick Todd Triplett, MD JAMA Psychiatry

To the Editor On behalf of my coauthor, I write to report errors in our Original Investigation, “Association of Schizophrenia With the Risk of Breast Cancer Incidence: A Meta-Analysis,” 1 published online first on March 7, 2018, and in the April 2018 issue of JAMA Psychiatry .

After a reader contacted the journal with questions about some of the numbers of participants in the studies included in our meta-analysis, the journal asked us to review all data and we agree that corrections are needed.

Zhuo C. Errors in Numbers of Participants in Studies Included in Meta-Analysis. JAMA Psychiatry. Published online May 15, 2024. doi:10.1001/jamapsychiatry.2024.1030

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Covering the Other Manhattan Trial

Tracey Tully’s reporting domain is New Jersey. But for the next six weeks, she’ll journey across the Hudson River to report on the federal corruption trial of Senator Robert Menendez.

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By Terence McGinley

Times Insider explains who we are and what we do, and delivers behind-the-scenes insights into how our journalism comes together.

Follow live updates on the bribery trial of Senator Robert Menendez.

Former President Donald J. Trump’s trial is not the only case The New York Times is covering in Lower Manhattan this week.

Less than a quarter-mile from Manhattan criminal court, where Mr. Trump faces 34 felony counts, Senator Robert Menendez of New Jersey is being tried on a web of federal corruption charges. Prosecutors say Mr. Menendez steered weapons and government aid to Egypt, propped up a friend’s halal meat business and meddled in criminal investigations that targeted his allies. The government has accused Mr. Menendez and his wife of accepting bribes for these favors in the form of cash, gold bars and a Mercedes-Benz.

Tracey Tully was one of five Times reporters in Federal District Court on Monday for the first day of jury selection in the trial. Ms. Tully, who covers all things New Jersey, joined The Times in 2017 after stints at The New York Daily News and The Albany Times Union. Three years into her role editing news about the New York region, she switched to reporting, which was a full-circle moment: Ms. Tully began her journalism career more than 30 years ago as a reporter in Hudson County, N.J., where Mr. Menendez grew up and where he rose as a political figure.

In an interview conducted after Monday’s proceedings , Ms. Tully spoke to Times Insider — from a hallway in the courthouse — about the case against Mr. Menendez, and its reverberations in New Jersey. This conversation has been edited.

This is not the first time Menendez has faced corruption charges, right?

Seven years ago, in 2017, he was on trial related to allegations of accepting bribes from a doctor, who was a close friend of his. That ended in a hung jury, resulting in a mistrial. The scope of the allegations in 2017 was more limited than what he is facing in this trial.

Your reporting has described Menendez as a powerful person in the Democratic Party in New Jersey. How have the charges resonated there?

The charges were a statewide political cataclysm. During the first trial, in 2017, people really stood with him, or at least they withheld their criticism. This time around he was almost immediately abandoned by most Democratic leaders in New Jersey. The day of the indictment, Governor Phil Murphy, a Democrat, said Menendez should resign. There was a cascade of fellow Democrats calling for his resignation. It took a few days, but finally, Senator Cory Booker of New Jersey, a longtime friend of Menendez and a very close political ally, called for his resignation. That was remarkable because Booker testified as a character witness during Menendez’s first trial.

What the senator has said is, Please, let me have my day in court — I deserve an opportunity to be innocent until proven guilty. He has refused calls to resign.

And his current Senate term ends this year, right?

People smelled blood in the water, and it was a frenzy for his Senate seat among fellow Democrats and Republican hopefuls. The first lady of New Jersey, Tammy Murphy, began running as a Democrat for the seat. That set up a clash with Andy Kim, a congressman from South Jersey. There was a lawsuit over the design of election ballots, which are incredibly important to the political party structure in New Jersey. The federal indictment against Menendez fueled all of this. There’s been a domino effect.

Seven years ago, you joined The Times as an editor. What made you want to go back to being a reporter?

I missed it. I just got the itch again. Being an editor was great for 20 years. I worked part time while I raised three kids. It suited my schedule. Nick Corasaniti, who covered New Jersey for The Times, went to the Politics desk to cover the 2020 election. There was an opening, and I had the audacity to ask, Can I do this again?

You’re normally responsible for covering news across an entire state. How does it feel to be assigned to one courthouse to cover a trial that could last more than six weeks?

I’m a regional reporter and a generalist; anything that happens in New Jersey, I get to cover it. Which is a blessing and a curse. This will be narrowly focused. I’m going to be traveling from New Jersey to Lower Manhattan every day court is in session. So that’s different. But the assignment is also no different from what journalists do all the time. You dive into whatever the story of the moment is, and you cover that.

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Trump’s trial has become a staging ground  for Republicans, including House Speaker Mike Johnson  and Senator J.D. Vance of Ohio , to prove their fealty to the former president.

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Key Inquiries: Trump faces several investigations  at both the state and the federal levels, into matters related to his business and political careers.

Case Tracker:  Keep track of the developments in the criminal cases  involving the former president.

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Getting an Overview of the Core Terms in Margin Analysis

After completing this lesson, you will be able to:

• Get an Overview of the Core Terms in Margin Analysis

Overview of the Core Terms in Margin Analysis

https://learning.sap.com/learning-journeys/outline-cost-management-and-profitability-analysis-in-sap-s-4hana/outlining-profitability-analysis_b5b7efbb-55ea-4ff5-bc70-15d39d8a14eb

Introduction to Margin Analysis

The following video provides an overview of Margin Analysis.

Master Data

Master data in margin analysis include profitability characteristics and functional areas. Functional areas break down corporate expenditure into different functions, in line with the requirements of cost of sales accounting.

These functions can include:

• Production.
• Administration.
• Sales and Distribution.
• Research and Development.

For primary postings, the functional area is derived according to fixed rules and included in the journal entries. For secondary postings, the functional area and partner functional area are derived from the sender and receiver account assignments to reflect the flow of costs from sender to receiver.

Profitability Characteristics

Profitability characteristics represent the criteria used to analyze operating results and the sales and profit plan. Multiple profitability characteristics are combined to form profitability segments. The combination of characteristic values determines the profitability segment for which the gross margin structure can be displayed. A profitability segment corresponds to a market segment.

For example, the combination of the characteristic values North (Sales region), Electronics (Product group) and Wholesale (Customer group) determine a profitability segment for which the gross margin structure can be displayed.

True vs Attributed Account Assignments

Each activity relevant to Margin Analysis in the SAP system, such as billing, creates line items. G/L line items can carry true or attributed account assignments to profitability segments.

• Goods issue item or billing document item in a sell-from-stock scenario.
• Manual FI posting to profitability segment.
• Primary Costs or Revenue.
• Secondary Costs.
• Balance Sheet Accounts with a statistical cost element assigned.

The derivation of attributed profitability segments is based on the true account assignment object of the G/L line item. This object can be of the following types:

• Cost Center.
• Sales Order.
• Production Order (only for Engineer-to-Order process.)
• Maintenance Order.
• Service Document (service order or service contract.)

After the profitability characteristics are derived, the resulting data is mapped to the G/L line item according to specific mapping rules. An attributed profitability segment is derived to fulfill the requirement of filling as many characteristics in the item as possible to enable the maximum drilldown analysis capability.

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H.J.Res. 109: Providing for congressional disapproval under chapter 8 of title 5, United States Code, of the rule submitted by the Securities and Exchange Commission relating to “Staff Accounting Bulletin No. 121”.

This was a vote to agree to H.J.Res. 109 in the Senate.

Joint Resolution Passed. Simple Majority Required. Source: senate.gov .

The Yea votes represented 59% of the country’s population by apportioning each state’s population to its voting senators.

Seat position based on our ideology score .

What you can do

Vote details, study guide.

Sen. Cory Booker (D), the Senate Democratic Policy & Communications Committee Vice Chair, voted Yea against his party.

Sen. Charles “Chuck” Schumer (D), the Senate Majority Leader, voted Yea against his party.

Somtimes a party leader will vote on the winning side, even if it is against his or her position, to have the right to call for a new vote under a motion to reconsider . For more, see this explanation from The Washington Post.

We do not know the rationale behind any vote, however.

“Aye” and “Yea” mean the same thing, and so do “No” and “Nay”. Congress uses different words in different sorts of votes.

The U.S. Constitution says that bills should be decided on by the “yeas and nays” (Article I, Section 7). Congress takes this literally and uses “yea” and “nay” when voting on the final passage of bills.

All Senate votes use these words. But the House of Representatives uses “Aye” and “No” in other sorts of votes.

Statistically Notable Votes

Statistically notable votes are the votes that are most surprising, or least predictable, given how other members of each voter’s party voted and other factors.

How well do you understand this vote? Use this study guide to find out.

You can find answers to most of the questions below here on the vote page. For a guide to understanding the resolution this vote was about, see here .

What was the procedure for this vote?

• What was this vote on?

Not all votes are meant to pass legislation. In the Senate some votes are not about legislation at all, since the Senate must vote to confirm presidential nominations to certain federal positions.

This vote is related to a resolution. However, that doesn’t necessarily tell you what it is about. Congress makes many decisions in the process of passing legislation, such as on the procedures for debating the resolution, whether to change the resolution before voting on passage, and even whether to vote on passage at all.

You can learn more about the various motions used in Congress at EveryCRSReport.com . If you aren’t sure what the Senate was voting on, try seeing if it’s on this list .

• What is the next step after this vote?

Take a look at where this resolution is in the legislative process. What might come next? Keep in mind what this specific vote was on, and the context of the resolution. Will there be amendments? Will the other chamber of Congress vote on it, or let it die?

For this question it may help to briefly examine the resolution itself .

What is your analysis of this vote?

• What trends do you see in this vote?

Members of Congress side together for many reasons beside being in the same political party, especially so for less prominent legislation or legislation specific to a certain region. What might have determined how the roll call came out in this case? Does it look like Members of Congress voted based on party, geography, or some other reason?

• How did your senators vote?

There are two votes here that should be more important to you than all the others. These are the votes cast by your senators, which are meant to represent you and your community. Do you agree with how your senators voted? Why do you think they voted the way they did?

If you don’t already know who your Members of Congress are you can find them by entering your address here .

• How much of the United States population is represented by the yeas?

GovTrack displays the percentage of the United States population represented by the yeas on some Senate votes just under the vote totals. We do this to highlight how the people of the United States are represented in the Senate. Since each state has two senators, but state populations vary significantly, the individuals living in each state have different Senate representation. For example, California’s population of near 40 million is given the same number of senators as Wyoming’s population of about 600,000.

Do the senators who voted yea represent a majority of the people of the United States? Does it matter?

Each vote’s study guide is a little different — we automatically choose which questions to include based on the information we have available about the vote. Study guides are a new feature to GovTrack. You can help us improve them by filling out this survey or by sending your feedback to [email protected] .

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