Introduction
One of the professional tasks of a competent modern educator is continuous work on students’ mistakes, aimed at the qualitative improvement of their skills and awareness, which is realized through reducing the number of mistakes. Strictly speaking, wrong answers, errors in the course of a solution, or wrong choice of an algorithm or method, regardless of the discipline taught, are a natural part of the learning process, which means that such obstacles and barriers cannot but exist on the way to the result. It would be erroneous to postulate that it is the educator’s task to identify such errors and grade the student based on the number of errors.
On the contrary, a professional teacher who practices a holistic approach and is interested in the planned development of students will identify a child’s errors promptly, identify their causes and determine why the student makes them, and develop a comprehensive plan aimed at reducing their number in the future through reflection and more informed instruction. This report critically examines a case study of a Dalton student struggling with mastery of math. In more detail, the report is structured first to identify the problems Dalton faces in solving arithmetic examples, and then to discuss, critically, pedagogical strategies and their implications relevant to this scenario.
Brief Description of Case Study
The center of the present scenario is 12-year-old Dalton, a 7th-grade student with tangible achievement problems in mathematics. The problems have become particularly evident when studying multiplying decimals, despite Dalton’s basic math skills being rated quite strong. To analyze the errors, Dalton provided a completed homework assignment from a previous class on decimals. The first nine assignments are analogous examples with different meanings for multiplying two decimals; the tenth assignment has the same meaning but is written differently; the eleventh assignment is a text problem on a real-life example; and the twelfth and final assignment applies decimal multiplication skills to a geometry case. Thus, all of the homework is built around learning Dalton’s skills toward this math competency.
Error Analysis
When comparing Dalton’s answers to how the 12 problems should have been solved, it is evident that the student’s problem is a failure to understand where the decimal point goes when multiplying decimals. The first ten examples were solved almost correctly by the student, but they should have been found incorrect because, in every case, Dalton made the same mistake: shifting the decimal point one or two places to the right. It follows that, at a minimum, the student does not understand the basic rules of multiplying decimal fractions, which state that one must first multiply the numbers as if there were no decimal place in them and then place the decimal point at such a position to the right as many digits after the point were totaled in the numbers being multiplied.
For example, if Dalton multiplied 0.78 by 9.6, the student’s first steps were correct: he would get the number corresponding to the product of the supposedly whole numbers, 78 by 96, namely 7488. However, Dalton did not account for the correct choice of decimal point position. Both in the numbers 0.78 and 9.6 after the dot are a total of three digits (7, 8, and 6) — so in the final number 7488, Dalton should have separated exactly that many digits to the right, getting 7.488.
In the second rule of the algorithm, Dalton made a repetitive procedural error, according to Brown and Skow (n.d.). In procedural errors, the essence is that the student incorrectly singles out the very actions necessary to achieve the correct answer. There is no indication of what causes this error: it may be ignorance of the rules, a lack of understanding of their principle, or a lack of proper information.
When analyzing Dalton’s solutions, it can be assumed that the student has a minimal understanding of multiplying decimals. In all cases, the resulting numbers are correct; however, the correct placement of the decimal point is the problem. It can be assumed that the cause of this error is a lack of awareness of the second rule of the multiplication algorithm. This finding is consistent with the sound evidence of Joung and Kim (2022), who showed that problems with incorrect placement of the decimal point predominantly originate from a lack of proper awareness on the part of the student and a lack of checking the reasonableness of the answer given. However, more research is needed to provide a more comprehensive and substantive analysis, with the ultimate goal of improving Dalton’s mathematical literacy.
Strategies
Several strategies may be pedagogically effective for the more detailed detection and management of errors students make when solving problems. First and foremost, these can include discussing the student’s mathematical background and playing a card game, which Yang et al. (2021) show to help detect thinking errors. In short, the card game is a tool the teacher can use to foster a more engaged interaction with the student, and thus, this strategy should be seen as supportive of implementing the other two discussed below.
Brown and Skow (n.d.) report that interviewing the student can be a valuable practice for identifying errors in decisions. It is not necessary to ask a direct question to do so; according to the authors, a more effective strategy is to use leading questions: “How did you come to this decision?” “Why do you think this is the right thing to do?” This kind of interviewing, as a strategy, aims to build the educator’s understanding of the student’s thinking system and to identify potential knowledge gaps rather than issuing ready-made algorithms. This can be useful because this strategy provides a basis for working with the gaps in more detail.
The second strategy is to discuss his errors with the student. This strategy is based on a dialogue in which both the student and the teacher come to a mutual understanding of the way forward to improve performance. It is appropriate to have such a discussion after the interview, when the teacher has articulated an understanding of the errors in Dalton’s mathematical thinking.
During this strategy, the teacher briefly describes the errors to the student, and then they work together to identify opportunities to avoid them in the future (Brown & Skow, n.d.). Dialogue that identifies milestones of progress is a valuable strategy for conceptualizing the student’s educational process and building trust with the teacher. An additional benefit is the opportunity to increase Dalton’s engagement in learning. Such methods are expected to deepen the fundamental understanding of the problem Dalton faces and to create resources for further improvement.
References
Brown, J. & Skow, K. (n.d.). Mathematics: Identifying and addressing student errors. IRIS Center.
Joung, E., & Kim, Y. R. (2022). Identifying preservice teachers’ concept-based and procedure-based error patterns in multiplying and dividing decimals. International Journal of Education in Mathematics, Science and Technology, 10(3), 549-567.
Yang, Z., Yang, X., Wang, K., Zhang, Y., Pei, G., & Xu, B. (2021). The emergence of mathematical understanding: Connecting to the closest superordinate and convertible concepts. Frontiers in Psychology, 12, 1-13.