Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsExploring the newman modified error hierachical model as a diagnostic approach in the teaching and learning of quadratic equation at the senior secondary school level in Zaria, Nigeria
Volume 3(2) · Pages 402–407
Research areasMathematics Education and PedagogyMathematics Education and Teaching TechniquesCognitive and developmental aspects of mathematical skills
Abstract
Author-writtenThe study explored the Newman’s modified error hierachical model as a diagnostic approach in the teaching and learning of quadratic equation at the senior secondary school level in Zaria, Nigeria. A total of 120 Senior Secondary 2 mathematics students randomly selected from two private schools in Zaria with a mean age of 17 constituted the sample size for the study. The Mathematics achievement tests (MAT) and Quadratic diagnostic test (QDT) were used as the instruments of this study that included three components: the use of factorization, completing the square and formulae. Diagnostic interview in the form of semi-structured questionnaire were also used to identify at which level students’ errors occur in solving problems and the perception of students towards quadratic equation was that success can be achieved in problem solving with a good grasp of the subject. The type of error is based on Newman Error Hierarchy Model that includes reading type error, comprehension, transformation, process skill, and encoding error. Data was analyzed using mean, percentage, frequency, Duncan’s (Multiple-Range) and Wilcoxon test at P
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Mathematical concepts are complex and interconnected, requiring students to use various elementary operations and translation skills. View evidence ↗
- Students with limited analytical and logical reasoning skills often struggle with mathematical problem solving. View evidence ↗
- The Newman Error Analysis Model provides a hierarchical framework to categorize student errors based on different levels of the problem-solving process. View evidence ↗
What they did
- The sample consisted of 120 randomly selected SS2 students (mean age 17) from two private secondary schools in Zaria, Nigeria. View evidence ↗
- A custom-designed Quadratic Equation Diagnostic Test (QEDT) was used to identify the types of errors students committed. View evidence ↗
- Following the written tests, students were interviewed using a modified Newman Error Hierarchical Model procedure to pinpoint where errors occurred. View evidence ↗
- Data analysis was performed using mean, percentage, frequency, Duncan’s Multiple-Range test, and the Wilcoxon test, with significance set at P < 0.05. View evidence ↗
What they found
- When solving by factorization, the most frequent errors were process skill (321) and transformation (275), while no reading errors were recorded. View evidence ↗
- For the completing the square method, process skill errors (380) and transformation errors (261) were the most significant. View evidence ↗
- When using the formula method, process skill errors (380) and transformation errors (265) were the most prevalent. View evidence ↗
- Errors related to encoding and general carelessness were found to be infrequent among the students. View evidence ↗
- The errors observed in solving quadratic equations were attributed to students' weaknesses in basic algebra and arithmetic operations. View evidence ↗
What it means
- The authors conclude that regardless of the method used, transformation and process skill errors are the most common mistakes students make. View evidence ↗
- The authors recommend that teachers encourage a stepwise, logical approach to problem solving to reduce difficulties with quadratic equations. View evidence ↗
- The authors state that the results are limited to secondary schools in Zaria, Nigeria, and may not be generalizable to other institutions. View evidence ↗
Visual summary
2 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
1: Comprehension vs Transformation vs Process Skill vs Encoding
Replotted from Table 3 (page 3) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
2: Comprehension vs Transformation vs Process Skill vs Encoding
Replotted from Table 3 (page 3) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 1
Table 2
Table 3
Table 4
Table 5
Table 6Introduction
Mathematics concepts are vast, interrelated and possess interconnected elements. The interrelationship of mathematical concepts can be identified in the use of elementary operations of division, ratios, percentage, addition, subtraction as well as translation of word problems (Charles-Ogan and George, 2015) [4] . Students of low logical reasoning and analytical prowess would, therefore, find certain concepts difficult especially when it has to do with problem solving. Problem-solving is an activity that can generate Higher Order Thinking Skills (HOTS) among students. However, only some of the students are capable of solving problems and some are having difficulties. The difficulties have caused students to make various kinds of errors. Hence, this study conducted is to identify and analyze students' errors in solving problems that involve HOTS in the topic of Fraction (Abdullah et al. 2015) [1] . Problem-solving is an activity that involves various actions in the mind of thought including accessing and using knowledge and experience (Lester & Kehle, 2003) [10] . Thus, teaching strategies that involve the use of non-routine problems in the classroom give students the opportunity to develop higher order thinking skills in the process of understanding, exploration and application of mathematical concepts (Polya, 1973) [16] . Diagnostic interview was also used to identify at which level students' errors occur in solving problems (Huang and Cheng, 2010) [8] . The type of error is based on Newman Error Hierarchy Model that includes reading type error, comprehension, transformation, process skill, and encoding error.
Guided by the phases described by Polya and Newman, students can solve problems more easily and systematically even if they are given problems with various levels of difficulties. However, not all students are able to solve the problems as they are having difficulties in the specific phases. Based on the studies conducted by Effandi and Siti Mistima (2010) [7] and Wijaya et al. (2014) [22] students have difficulties in the transformation and process skills phases while solving problems of Quadratic Equations. Students' difficulties in problem-solving process have caused them to make various errors. Widiharto (2008) [23] observed that mathematical concepts and skills that are not fully mastered by students have led to difficulties and errors in solving mathematical problems. A study conducted by Susanti et al. (2014) [20] found out that students found it difficult to solve problems that involve the use of HOTS and among the difficulties faced by them are a) reading and interpreting data, b) determining and delegating data and c) making conclusions and arguments. Therefore, teachers need to realize the difficulties and errors encountered by students and take appropriate approaches to improve their teaching practices (Ashlock, 2005) [3] . In conclusion, teachers are responsible to identify in advance the difficulties in learning mathematics, which led to errors before teaching other mathematical topics. According to Effandi and Siti Mistima (2010) [7] , the Newman's Error Analysis Model [13] has the hierarchy that categorizes types of error based on the levels of problemsolving by the students. The above statement is in line with Ellerton and Clements (1996) [5] , who stated that Newman used the "hierarchy" as he gave the reason that students who fail at any level of problem-solving prevented them from getting the required solution. In the process of problemsolving, there are two factors that make students unable to produce the correct answer, namely a) problems in language fluency and conceptual understanding (reading and comprehension) and b) problems in mathematical processing (transformation, process skills and encoding) (Prakitipong & Nakamura, 2006) [17] . The body of academic literature in mathematics education research has shown that students have misconceptions and make errors (Effandi and Siti Mistima, 2010) [7] . This clearly shows that students need to understand the meaning of the questions before going through the mathematical processing in order to produce the correct answers. In addition, the use of the Newman's model in error analysis has overtime attracted considerable attention (Huang and Cheng, 2010) [8] . The first type of error in the Newman's Error Analysis Model is reading, which is the ability of students to read the mathematical problems given and to identify the sentences and mathematical symbols used. The second type of error is comprehension, which is the ability of students to understand the mathematical problems given. The next error is transformation, which sees the ability of students to choose the appropriate mathematical solution methods. The following error is process skills, where students can perform mathematics process correctly or not, and lastly the error of encoding, which is the ability of students to express the final answer. In this exploratory study, the focus is on the utilization of the Newman's model as a diagnostic approach to analyze the error students face in solving problem involving quadratic equations at the senior secondary school level in Zaria, Nigeria.
Purpose of the study
The main purpose of the study was to analyze students' error in learning of quadratic equation which focused on the manipulation of quadratic equation using factorization, completing the square and formulae. Specifically, the study sought to: 1. Determine the extent to which individual students commit error in the learning of quadratic equation 2. Find out the possible categorization of these errors in the learning of quadratic equation.
Research question
Three research questions were posed for this study: i. To what extent will individual students' error influence the learning of quadratic equation? ii. What are the possible categorizations of these errors in the learning of quadratic equation? iii. What are the student's answers that help teachers explore the students' thinking and reflection about learning of quadratic equation?
Hypotheses
The following null hypotheses were tested at P<0.05 Ho There is no significant difference in the type of error committed by students in solving problems involving quadratic equation. HA there is no significant difference in the perception of students in solving problems involving quadratic equation.
Methodology
The population for the study consists of all senior secondary schools in Zaria, Nigeria. Zaria and it's environ is made up of over twenty senior secondary schools with a population of over seventeen thousand (17,000) students. The Government funded schools unlike the privately own ones are noncoeducational. Apart from having qualified teachers, the students offer mathematics as a major subject. The private schools were therefore considered appropriate for the study and it is from this population that the study sample was drawn.
Sample
The study was conducted using 120 senior secondary (SS2) students randomly drawn from two non-governments funded (private) secondary schools in Zaria metropolis in Nigeria.
The mean age of the students was 17years.
Instrumentation
The study used a set of quadratic equation diagnostic test (QEDT) questions designed by the investigator to identify type of error committed by students. Prior to the commencement of the study, the questions were subjected to face and Pearson's product moment content validated test respectively. The reliability coefficient as computed from Pearson's product moment correlation for the Diagnostic test (QEDT) was r= 0.89. The value suggest that the test question were reliable and as such would test what it was designed for.
In addition, content and face validity of the instrument was carried by the investigator who is a specialist in mathematics and other specialists in cognate field especially senior colleagues in the department of mathematics, Federal College of Education, Zaria as compared to the overall objective of the curriculum.
Prior to the investigation, the respondents were subject to general mathematics achievement test. The primary objective of the MAT items was to categorize the students into high achievers (A) for those that scored 65% and above and achievers (B) for those that scored 50-64%. This is shown in Table 1.
| Profile | High Achievers | Achievers Total Percentage (%) | — | — |
|---|---|---|---|---|
| (A) | (B) | |||
| Male | 32 | 30 | 62 | 51.7 |
| Female | 30 | 28 | 58 | 48.3 |
| Total | 62 | 58 | 120 | 100 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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The investigator also interviewed the respondent based on the modified procedure of N= 120. Ratings were scored on a 5-point likert scale, where 5= strongly agree, 4= agree, 3= neutral, 2= disagree and 1= strongly disagree. *Significant deviations of student responses to questions from the neutral score of 3 were analyzed using a one-sample Wilcoxon sign-rank test P value of <0.05 were considered to be of statistical significance.
Discussion
The most frequent errors made by student in solving problems in quadratic equation include comprehension error, transformation error and process skill error. Error in carelessness was minimal. Most comprehension errors occur when students do not understand how to approach a given problem from the concept. For instance, in solving quadratic equation by the use of factorization the predominant error were those involving comprehension, transformation and process skill. Students' often misunderstand the demand of the question. This may be due to the lack of emphasis by teachers in teaching the simplification of concepts as they appear. It may also be due to rote learning on the part of the learner. In fact, students usually don't immediately know what to do when a novel problem is encountered for the first time. Therefore, they might attempt to recall a similar problem encountered in the past and try to use that to solve the current one (Huang and Cheng, 2010) [8] . This means that students' problem solving is based on their past experience in a problem solving episode (Robertson, 2001) [18] . Anderson (1993) [2] argued that all skill learning occurs through analogical problem solving, in which examples have an important rule. Therefore, teachers must ensure that the teaching of mathematical concepts must be balanced with the arithmetic skills. The findings from present study is consistent with previous reports (Delice, 2002; Intanku 2003; Weber, 2005; Effandi and Siti Mistima, 2010) [6,9,21,7] . The error type in transformation occurred during computation process especially during multiplication. This takes place due to computation problem especially among achievers. Most students make error at the process skill level especially in the manipulation of quadratic equation using formula. For example in item 3 and 12 that were virtually the same the students were required to find the roots of the equation x 2 + 7x + 10=0, it was found that most students who got the item by the use of factorization method could not replicate same answer using the formula as a result of appropriate placing and manipulation. The findings of the study support the research of Norasiah (2002) who noted that most average students face difficulty in performing algebraic operations especially when square roots are involved. In a related study involving the use of completing the square method, learners were unable to divide by the coefficient of x 2 if the equation was greater than 1 and less than zero,multiplying by half the coefficient of x and also additive inverses of any constant given (Makgakga,..) [12] . some of the reasons which contributed to these difficulties were learners' laziness, lack of participation, teachers commitment and conduct towards learners' among others. The author is of the view that with periodical interaction with learners' and encouragement, some of these diagnosed errors could be used as a basis to understand learners' challenges in the aspect of mathematics. This will no doubt go a long way in making up for appropriate problem solving strategies that will minimize errors in subsequent interactions. The error in process skill seems more pronounced in the use of formula than the factorization method. This error may be due to the fact that students failed to understand and describe what is required by the questions. This results in failure to solve the problems. Results of this study concurred with the findings of Norasiah (2002) [14] in which problematic students failed to translate mathematical problems into mathematical form and also having problem in understanding the special terms in mathematics.
Results of this study are in line with the research findings of Liew and Wan Muhammad (1991) [11] who observed that emphasis on algorithmics skills without explanation on the concept or principle are the factors that cause difficulties in mathematics and by extension poor achievement in quadratic equation.
Conclusion and Recommendations
The findings suggest that in the exploration of the Newman's model, most errors that are committed by students irrespective of the method used are transformation errors and process skills errors. It further shows that generally, students irrespective of different cognitive ability are susceptible to error in solving problems in quadratic equation. This implies that academic performance is gender bias. In summarizing our findings, we conclude that these results apply to only Secondary School in Zaria, Nigeria. One cannot say whether these relationships apply to other institutions of secondary education. The results certainly suggest avenues for further research and it would be desirable to examine these relationships using data from more than one institution. Nonetheless, the novelty of the findings from this study will undoubtedly enable mathematics teachers to diligently plan their teaching in an effective and meaningful way. Apart from also identifying the roots of the problems in students' learning, the findings will further add to the library of data in this line of research for diversified group of respondents. In view of the findings above, it is therefore recommended that:
1. Teachers as facilitator of the teaching-learning process should encourage the learners to concentrate on one point at a time and proceed stepwise in a logical manner to reduce attendant difficulty faced in quadratic equation. 2. Teacher should make quadratic equation lessons exciting by encouraging group work with frequent activity-based demonstrations so as to demystify the difficulty encountered in problems involving quadratic equation. 3. Learners should be given enough opportunities to do regular problem exercises as this will go a long way in assisting them and increasing their reasoning skills. 4. This study should be repeated with other similar mathematically based concepts so as to compare findings.
Appendix
Appendix
Figures and tables from the paper
| Skill | Item number |
|---|---|
| Solving quadratic equation by factorization | 1-6 |
| Solving quadratic equation by completing the square | 7-12 |
| Solving quadratic equation by formula | 13-18 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| Item | Reading | Comprehension | Transformation | Process Skill | Encoding | Carelessness |
|---|---|---|---|---|---|---|
| 1 | 0 | 18 | 40 | 46 | 10 | 6 |
| 2 | 0 | 14 | 46 | 51 | 9 | - |
| 3 | 0 | 16 | 45 | 47 | 8 | 4 |
| 4 | 0 | 9 | 46 | 60 | 5 | - |
| 5 | 0 | 10 | 50 | 56 | 4 | - |
| 6 | 0 | 8 | 48 | 62 | 2 | - |
| Total | 0 | 67* | 275** | 321** | 38* | 6 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| Item | Reading | Comprehension | Transformation | Process Skill | Encoding | Carelessness |
|---|---|---|---|---|---|---|
| 7 | 0 | 10 | 40 | 60 | 6 | 4 |
| 8 | 0 | 6 | 48 | 61 | 3 | 2 |
| 9 | 0 | 6 | 45 | 65 | 4 | - |
| 10 | 0 | 7 | 40 | 68 | 5 | - |
| 11 | 0 | 8 | 44 | 62 | 6 | - |
| 12 | 0 | 7 | 44 | 64 | 5 | - |
| Total | 0 | 44* | 261** | 380** | 29* | 6 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| Item | Reading | Comprehension | Transformation | Process Skill | Encoding | Carelessness |
|---|---|---|---|---|---|---|
| 13 | 0 | 5 | 45 | 60 | 6 | 4 |
| 14 | 0 | 6 | 48 | 61 | 3 | 2 |
| 15 | 0 | 5 | 45 | 65 | 5 | - |
| 16 | 0 | 6 | 40 | 68 | 5 | - |
| 17 | 0 | 7 | 43 | 62 | 6 | 2 |
| 18 | 0 | 7 | 44 | 64 | 5 | - |
| Total | 0 | 36 | 265** | 380** | 30* | 8 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| Item | Agree % | Neutral % | Disagree % | Total |
|---|---|---|---|---|
| Interesting but extremely challenging Very important topic in mathematics Could very difficult due to it | 75 | 15 | 10 | 100 |
| 70 | 24 | 6 | 100 | |
| Abstractness Has a lot of misconception Teacher makes it more simpler Success in solving problem is due to: | 65 | 21 | 14 | 100 |
| 70 | 15 | 15 | 100 | |
| 69 | 14 | 15 | 100 | |
| -translating worded problem to equation | 80 | 12 | 8 | 100 |
| - writing appropriate equation | 80 | 10 | 10 | 100 |
| - using appropriate formula/method | 72 | 18 | 10 | 100 |
| Conceptual understanding help me to Solve problems easily Mathematical ability help in problem solving | 85 | 8 | 7 | 100 |
| 70 | 20 | 10 | 100 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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