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«The Exclusion of the Students’ Dynamic Misconceptions in College Algebra: a Paradigm of Diagnosis and Treatment Mohammad A. Yazdani, Ph.D. † ...»

The Exclusion of the Students’ Dynamic Misconceptions in

College Algebra: a Paradigm of Diagnosis and Treatment

Mohammad A. Yazdani, Ph.D. †


We refer to dynamic misconceptions in college algebra as systematic errors,

which are the consequences of misapprehensions and misconstructions of algebraic

concepts and skills. In summer of 2004 an experimental study was designed and

implemented to identify students’ misconceptions in college algebra. In addition, a

method was devised to eliminate the misconceptions. We employed a selected response assessment instrument to evaluate the students’ knowledge in college algebra.

Furthermore, we utilized an uncomplicated approach to measure the participants’ degree of certainty of their selections. An item analysis of each answer in conjunction with the degree of certainty for that response directed us to determine the student’s miscomprehension and/or misinterpretations of the concepts. As a result we identified the participant’s level of knowledge (appropriate, improper, or misconception) with reference to the measured information and/or skills. We utilized a combination of instructional strategies such as individual tutoring, peer tutoring, cooperative learning groups, and related computer programs to assist the students to discern their misconceptions. As a final point, we re-taught the misunderstood concepts and skills.

Background The literature suggests that assessment of students’ advancement in mathematics is interwoven with the learning and teaching of this discipline. Stiggins (1997) writes, “Assessment and instruction can be one and the same if and when we want them to be”.

Glatthorn, et al. (1998) state, “the achievement cycle begins with curriculum and moves from there to assessment and then into instruction”. De Lange (1999) expresses, “New views of assessment call for tasks that are embedded in the curriculum, the notion being that assessment should be an integral part of learning process rather than an interruption

of it”. The objectives of assessment in mathematics are the followings:

• To accumulate data regarding students’ learning and achievement in mathematics.

• To analyze this information for feasible revising of the instructional design, delivery, and possible re-teaching of the concepts which students did not attain the expected proficiency in.

• To provide feedback to students, parents or guardians, and school administrators.

The widely used instruments to assess students’ knowledge and achievement in a subject matter are standardized test in the form of Selected Response Assessment (multiple choice). These instruments are constructed using a psychometric model. The aim of this model is to be as objective as possible. Nevertheless, there always remains a doubt en route for the possibility of guessing the correct answer or unintentionally choosing the proper response by the student. Taking into consideration the 20% probability of guessing or coincidently selecting the correct response in a multiple choice Journal of Mathematical Sciences & Mathematics Education 32 question with five possible answers, the scores obtaining from such instruments would not endow us with a precise assessment of students’ achievement in mathematics.

Consequently, we are unable to present an accurate feed back to students, parents or school administrators. Moreover, it would not supply us with adequate information to assist students to triumph over their deficiencies by revisiting the contents. The precise information is essential to analyze and demonstrate student’s mastery of required concepts and skills discussed in such instruments. To re-teach a mathematical skills effectively we should investigate if the deficiency is the end result of students’ lack of knowledge or it is the consequence of their misapprehensions of the mathematical concepts and skills. In the present document we refer to such misapprehensions and misinterpretations as misconceptions. Since misconceptions in all probability are components of an imperfect cognitive structure shaped because of erroneous knowledge instructed to the students by educators and/or students’ life experience, it is indispensable to determine the sources of students’ misconceptions. Mestre, J. (1989) writes, “Misconceptions are a problem for two reasons. First they interfere with the learning when students use them to interpret new experiences. Second, students are emotionally and intellectually attached to their misconceptions, because they have actively constructed them. Hence, students give up their misconceptions, which can have such a harmful effect on learning, only with great reluctance”. Hasan, et al. (1999) claim, “Misconceptions are strongly held cognitive structures that are different from the accepted understanding in a field and that are presumed to interfere with the acquisition of new knowledge”. The treatment to prevail over lack of knowledge requires reteaching, reconstructing, and reinforcing the concepts and skills which students are deficient in. Meanwhile, the treatment to overcome the misconception requires the elimination of the misconception followed by re-teaching, reconstructing and reinforcing the correct concept and skills.

As mathematics educators it was of our interest to search for a reliable and an effective strategy to assist us to interpret the information obtained from the Selected Response Assessment instruments. Furthermore, we were concerned to not only discover a strategy to eradicate the possibility of guessing the correct response by students, yet assist us in identifying the students’ deficiencies as well.

The Method

To achieve this goal we employed a method to measure the students’ degree of certainty (DC) regarding to the correctness of their responses. We requested the students select a response for each question, as well as, indicate, on a scale 1-4, how assured they were of their choices. We asked the students to enter a numerical code, 1 through 4, in a space provided beneath each question to indicate their degree of certainty for that particular question. Table I illustrates the characteristics for any given entry code.

–  –  –

Journal of Mathematical Sciences & Mathematics Education 33 There are 8 possible combinations of different answers collectively with various degrees of certainty for each distinct question in this scheme. Table II demonstrates the possibilities of a student’s response in conjunction with the degree of certainty (DC) for a particular question. The table also illustrates the diagnosis of the student’s deficiency as well as the appropriate treatment.

–  –  –

⎝1 ⎠ Where ADC is the group’s average degree of certainty for any given response, DC is the degree of certainty of any single student for that particular response, and n is the number of the students forming the group. Table III demonstrates different possibilities of ADC in conjunction with the response for any particular question. The table also indicates the diagnosis and suggested treatments for each possibility.

–  –  –

Journal of Mathematical Sciences & Mathematics Education 34 The Experimental Study Two classes of College Algebra students participated in our experimental study (54 students). The instrument used in the study was selected from the college algebra textbook publisher’s recommended assessment. The instrument was consistent with the content information that was presented to the students. Therefore, the instrument was valid because it exactly measured what it was supposed to measure. The test consisted of Polynomials and Rational Functions, Exponential Functions and Their Applications, and logarithmic Functions and Their Applications. The instrument contained 25 multiple choice questions with maximum possible score of 25 points. We assigned the numerical value of “1” for each correct response and “0” for every wrong answer. The followings

are samples of the instrument as well as the direction used in the present study:

–  –  –

We assessed the students’ responses; in addition, we evaluated their answers in conjunction with their degrees of certainty. Table IV displays the item analysis for one individual student. Table V exhibits the item analysis for a group of students.

–  –  –

The data, suggested that the majority of our students had misconceptions regarding the mastery of the subject matters discussed in items # 6, 18, 19, 22, and 23. In the present investigation the misconceptions were gave an account for the items involving the Exponential and Logarithmic Functions and Their Applications. The item analysis indicated that many of the subjects misunderstood the rules of logarithms and failed to employ the correct rules in problem solving. Our task was to remove these misapprehensions and misinterpretations. As mathematics educators and classroom teachers we have experienced the students’ resistance to discern their misconceptions.

Furthermore, re-teaching the concepts and skills requires us to differentiate between the students’ misconception and their improper knowledge. We modified the design of our Journal of Mathematical Sciences & Mathematics Education 37

instructional content and delivery. Moreover, we employed such learning strategies as:

small cooperative group, peer tutoring, individual tutoring, and utilized related instructional software to further encourage the students to allow their misconceptions to surface. Finally we presented the students with the correct concepts and skills.

A remarkable characteristic of the present study was its simplicity and straightforwardness. This attribute encouraged the students to unveil their misconceptions as a part of their assessment. Hence, they assisted us to discover and extract the misconceptions. We strongly recommend that mathematics educators of grades 9 – 16 utilize this strategy to examine the students’ genuine mathematical knowledge and to identify their misconceptions.

† Mohammed A. Yazdani, Ph.D., University of West Georgia, Carrollton, GA, USA

–  –  –

De Lange, J. (1999). Framework for Classroom Assessment in Mathematics. National Center for Improving Student Learning and Achievement in Mathematics and Science. Madison, Wisconsin. Freudental Institute.

Dugopolski, M. (2003). College Algebra. Boston, Massachusetts, Pearson Education, Inc.

Glatthorn, A., et. al (1998). Performance Assessment and Standard-Based Curricula: The Achievement Cycle. Larchmont. New York, Eye On Education, Inc.

Hasan, S., et. al (1999). Misconceptions and the Certainty of Response Index. Teaching Physics. September 1999. Physics Education 34 294 – 299 Mestre, J. (1989). Hispanic and Anglo Students’ Misconceptions in Mathematics. ERIC Clearinghouse on Rural Education and Small Schools Charleston WV.1989-03-00 Stiggins, R. (1997). Student-Centered Classroom Assessment. Upper Saddle River, NewJersey: Prentice-Hall, Inc.

Journal of Mathematical Sciences & Mathematics Education 38

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