MatheMatics teaching-ReseaRch JouRnal online 19 Vol 3, n1

MatheMatics teaching-ReseaRch JouRnal online

Vol 3, n1December 2008

Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Re-search Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All oth-er uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.

www.hostos.cuny.edu/departments/math/mtrj

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using clinical inteRView with low-peRfoRMingstuDents in MatheMatics

eRic fuchs & Violetta Menil





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Using Clinical Interviews with

Low-Performing Students in Mathematics

Eric Fuchs, Bronx Community College

Violeta C. Menil, Hostos Community College of The City University of New York (CUNY)

ABSTRACT

The purpose of this study was to describe the usefulness of clinical interviews

when working with low-performing high school students in mathematics. The subjects of

the study were public high school students participating in the New York City

Mathematics and Science Partnership program during the summer of 2007. Our work

was aimed at improving the teaching of high school mathematics through an innovative

approach called mathematics interviews, popularly called clinical interviews. The

methodology was geared to a target population consisting of the lowest third performing

Math A students.

Through the clinical interviews conducted with the members of the lowest

performing students, we were able to identify and correct misconceptions, identify areas

of weaknesses and gain a better understanding of the students’ mathematical thinking.

We gained an insight on how a student’s mind functions vis a vis mathematical thinking.

Based on this qualitative research, we concluded that clinical interviews could be

a useful tool for helping teachers gain insight into the students’ thought processes,

pinpoint misconceptions, develop a dialogue in response to the students’ problems and

raise the students’ self-confidence. By helping teachers become more responsive to their

students’ needs, clinical interviews ultimately help students become better learners.

This research addresses the usefulness of clinical interviews in conjunction with

other pedagogical tools to improve the teaching and learning of mathematics in urban

high schools and middle schools. This work also paves the way for enhancing the

methodology used with remedial mathematics students in community colleges.





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INTRODUCTION

Many students start having mathematics difficulties in middle school, especially if the schools they attend are urban schools in high poverty areas. According to Ball, these students will probably have a teacher who is “entirely unprepared in mathematics and has had no opportunity to study mathematics for teaching” (Hiebert and Ball, 2005). The condition worsens in the high school years. When these students enter community colleges without a mathematical base on which future knowledge can be scaffolded—and when they also have to support a family or hold down a job—their mathematics difficulties become an impediment to their going forward with their education.

During the several years we have been teaching mathematics in community colleges in the Bronx, we have worked with students in the Associate Degree program who struggle in arithmetic and algebra. Not entirely to our surprise, we found that the same difficulties are prevalent among high school students in the Mathematics and Science Partnership in New York City (MSPinNYC or simply MSP), a five-year project

funded by the National Science Foundation. In MSP, college faculty, high school teachers and selected tutors provide a six-

week intensive summer training on a college campus to several hundred students from Manhattan and the Bronx who failed the Regents exam in mathematics, chemistry or living environment. The students’ day was divided equally among classroom instruction, labs and tutoring sessions. They also participated in board of directors’ meetings where they discussed educational or social issues. After class, the teachers and tutors met for half an hour in an activity called “kid talk” discussing their observations about what worked and what needed improvement. At the end of the summer, the students took the Regents exam and most earned a passing grade.

The mathematics knowledge MSP students need to pass the Math A Regents exam is similar to the knowledge students in community colleges need to pass remedial mathematics classes. The topics with which students struggle—fractions, decimals and proportions, signed numbers, distributivity, estimating, simple word problems—have been taught in high school, middle school and even elementary school. Since the ethnic and socioeconomic composition of our college students is similar to that of our MSP students, our work with MSP students lays the groundwork for expanding the research to the teaching of mathematics in community colleges.

More than three years ago while working in the summer MSP program, two faculty researchers, Czarnocha of Hostos and Prabhu of BCC, postulated that our students did not think metacognitively in mathematics but simply performed operations algorithmically and got an answer as a result. They relied on memorization to solve certain types of problems instead of trying to understand the problem. With too many facts to “remember,” the students got confused. Lacking an understanding of elementary arithmetic, they could not make the transition to algebra and trigonometry. Czarnocha and Prabhu theorized that clinical interviews might provide a better gauge of students’ knowledge than the results of weekly mock exams.





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To better understand the MSP students’ mathematical thinking, Czarnocha, Prabhu and Fuchs experimented with one-on-one clinical interviews in the summer of 2005. The interviewers confirmed the above postulates about students’ mathematical thinking.

In the summer of 2007 we conducted this qualitative research with the lowest performing MSP Math A students at Lehman College. Our goal was to gain a deeper insight into our students’ minds. Why did they solve a problem in a certain way? How did they arrive at their results? What types of mistakes did they make, and what were their misconceptions? (Ambrose, Nicol, Crespo, Jackobs, Moyer & Haydar, 2003). The theoretical underpinning of the research consists of socio cultural theory, the sociology of emotions, elements of educational psychology and error detection and correction in mathematics. The Research Questions

Our research was driven by three questions:

• To what extent can the clinical interviews help teachers understand their students’ mathematical thinking?

• If clinical interviews help teachers understand their students’ mathematical thinking, how can they be used as a pedagogical tool?

• To what extent are clinical interviews useful with low-performing students in mathematics?

The impetus for the research came from the low retention and graduation rates of

our students enrolled in Associate Degree programs at Bronx Community College (BCC) and Hostos Community College of The City University of New York. Based on the data of CUNY Office of Institutional Research and Assessment (OIRA), less than 22% of a typical freshmen cohort was awarded the Associate Degree at the end of 6 years (OIRA, RTGI_0001).

Research Purposes and Assumptions

A primary goal of this research was to use clinical interviews to identify and possibly correct misconceptions and areas of weaknesses in the mathematical thinking of the bottom third of MSP Math A students. These students were our intervention group, referred to in this paper as the target population (TP). A second goal was to determine if clinical interviews and follow-up activities could help increase the percentage of students from the target population who passed the Regents examination.

The following assumptions guided this research study: • It is possible to identify early in the program the bottom third of Math A students

who are “doomed to fail” the Regents exam. • Clinical interviews can help uncover student misconceptions, areas of weaknesses

and faulty mathematical thinking. • Addressing student misconceptions and correcting these weaknesses are

instrumental in raising the passing rates of Math A students.





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THEORETICAL FRAMEWORK A framework for understanding teaching and learning in a democracy is provided by the intersection of three major components (Bransford, Darling-Hammond & LePage, 2005): knowledge of subject matter and curriculum goals, knowledge of teaching and knowledge of learners.

The first two components are based on Shulman’s article on the professionalization of teaching (Shulman, 1987.) Essentially, Shulman claims that teaching is a learned profession. In 1986, he introduced a new concept, the pedagogical concept knowledge, which includes

The most regularly taught topics…and also includes an understanding of what makes

specific topics easy or difficult; the conception and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently

taught topics or lessons (Shulman, 1986, pp 9-10)

Sockett criticizes Shulman’s theory on three accounts: (a) context as opposed to content, (b) the description of the moral framework for teaching and (c) the relation between reason and action in teaching (Socket, 1987). Sockett, who thinks teaching should be viewed through the lenses of socio-cultural perspective and the sociology of emotions, argues that the act of teaching cannot be separated from the knowledge of the student population. The moral framework considers a teacher’s excellence, grace, style, enthusiasm, commitment, integrity, care, passion and sense of fairness. Regarding reason and action, since teaching is praxis, teaching should be evaluated based on teaching outcomes, not on teacher’s ability to explain. The Emotional Aspect of Teaching

What makes a good teacher? What makes a poor teacher? All of us remember a good teacher we once had: someone who made us want to go to school or influenced our career choice.

Hargreaves, whose study is grounded in the sociology of emotions, points out that (a) teaching is an emotional practice, (b) teaching and learning involve emotional understanding, (c) teaching is a form of emotional labor and (d) teachers’ emotions are inseparable from their moral purpose (Hargreaves, 1998).

Based on Collins’s sociology of emotions, Tobin explains that good teaching is made out of successful interactions that are charged with positive emotions; bad teaching is made out of unsuccessful interactions that are charged with negative emotions (Tobin, 2006). During successful interactions, there is synchrony between the teacher and the students. A chain of synchronous practices is charged with positive emotional energy and leads to entrainment—for example, instead of copying from the board the students anticipate what the teacher will write next. In these moments there is solidarity in the classroom among all participants. The teaching happens without conscious awareness and through practices that are anticipatory, timely and appropriate, i.e. fluent (Tobin, 2006).

Hargreaves argues that the emotional dimension of teaching is largely ignored or underplayed by the policy makers (Hargreaves, 1998). Like painters, good teachers are born, not made!





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Cultural Responsive Teaching

Given the diverse background of the students in CUNY community colleges, it is vitally important for educators to be sensitive to the students’ culture. Research shows that “teachers’ attitude and expectations, as well as their knowledge of how to incorporate the cultures, experiences, and needs of their students into their teaching, significantly influence what students learn and the quality of their learning opportunities (Banks, Cochran-Smith, Moll, Richert, Zeichner, LePage, Darling-Hammond & Duffy with McDonald, 2005).

Villegas and Lucas state that because of the diversity of the student population, a good teacher cannot have one script for all. In a multicultural society, responsible educators continuously tailor instruction to individual children in specific cultural contexts. Doing so demands that teachers know their students well and have the skills to transform this knowledge into appropriate classroom practice (Villegas and Lucas, 2002).

The need for culturally responsive teaching is acute in community colleges like BCC and Hostos Community College, which are characterized by significant ethnic diversity and a large community of Black and Hispanic students. Villegas and Lucas, who espouse a constructivist approach in learning of science, point to research proving the effectiveness of working collaboratively in small groups of mixed ability.

Mathematics Difficulties

At the Bronx and Hostos community colleges, our students’ mathematics difficulties emanate from many years of inadequate mathematics education. Even those students who attend classes regularly and work hard on their homework make basic errors when working with fractions, percentages, decimals, proportions, distributivity, basic equations or order of operations. In this paper we refer to these errors as misconceptions (or conceptual errors).

In a typical urban classroom, an overworked classroom teacher has neither the time nor the patience to try to understand WHY their students make conceptual errors. Diagnosing these errors is time consuming and requires skills that many teachers do not possess. Consequently, for most teachers an error is an error. This is even truer in community colleges where the instructors are more detached from their students, where a “lesson” is called a “lecture”, and where homework is recommended but not mandatory and in most cases goes unchecked and uncorrected.

In mathematics education we see no difference between our community college remedial mathematics students and the MSP students. They not only have the same attitudes toward learning mathematics but are held back by low literacy skills, their inability to do mental mathematics or work with fractions, difficulties with word problems or problems requiring mathematical logic, the desire to be shown the algorithmic way of “how to” solve a problem rather than bothering to understand “why” and a lack of a “feel” for numbers.





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Type of Mathematical Errors

For simplicity’s sake, we distinguish among three types of errors: careless errors, calculation errors and conceptual errors. Careless Errors

Careless errors result when students rush. An example would be writing the number 8 instead of 6, reading the wrong number, or reversing digits. These errors are made randomly. Students can avoid them by “slowing down” and checking their work. A consistent pattern in these errors, such as a student consistently reversing digits, may indicate the presence of dyslexia or other learning disability. Calculation Errors

These errors result when students have a good grasp of concepts but are tired or are working under pressure. When adding numbers, students frequently make calculation errors by omitting the carry in one column while correctly adding the rest of the columns. Like careless errors, calculation errors can be minimized by having students concentrate on the calculation. Students can detect these errors by rechecking their work. Systematic, neatly organized work helps in reducing calculation errors. Conceptual Errors

In mathematics, conceptual errors (misconceptions) result from a fundamental misunderstanding of a concept. A student makes a conceptual error in geometry when confusing perimeter with area or not understanding that the opposite sides of a rectangle are equal.

In algebra, students make conceptual errors when distributing only to the first term in the parentheses. Example for many students 2(3x – 4) is equal to 6x – 4. Students usually confuse between

– 4 – 2, an addition of two negative numbers resulting in – 6 and (– 4)(–2), a multiplication of two negative numbers resulting in +8

Based on our experience we attribute conceptual errors to the following:

• Dependence on memorization and algorithms • Overdependence on calculator • Lack of understanding

Error Detection and Correction

Conceptual errors in mathematics are insidious. Students acquire them along the way and keep reinforcing them so that they become “fossilized”. Even when resorting to a calculator, the students fail to compare the calculator’s answer to the answer they would have obtained erroneously had they done the calculation by hand. An undetected or uncorrected conceptual error becomes part of the student’s mathematical construction.

Without outside intervention, most students cannot correct their mathematical conceptual errors. In his Winning at Math guide, Nolting states “...it is not the fault of the





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students if they have not been taught how to study math. Even students taking general study skills courses are often not taught how to study and learn it.” (Nolting, 2002). In our mathematics college teaching, we tell our students that “mistakes are their friend.” Students who understand and analyze their errors can capitalize on that knowledge and thus achieve a better understanding of the subject. (Borasi, 1994).

Scholarly Use of Clinical Interviews in Mathematics

Clinical interviews are becoming an important tool in numeracy projects because of their value in helping teachers understand children’s thinking while they are working on a problem Heirdsfeld (2002). Rather than waiting for mistakes to become fossilized, teachers in lower grades can use clinical interviews to help them detect children’s

misconceptions. Heidsfield corroborates Hunting’s statement that clinical interviews “allow students to teach teachers” (Heirdsfield, 2002).

A teacher can learn a lot about a student’s thinking, even if the student is an A student. The 8th grade student interviewed by Wheatley was presented with seven different problems involving arithmetic computations and applications with proportions, fractions, and geometry (Walbert, 2001). Using videotaping and interview transcripts, Wheatley found that while the student learned common procedures and how to apply them, she failed to make connections between concepts and unknown situations

Hunting (1997) contrasts the similarities and differences of clinical interviews used as part of mathematics research vs. interviews used in mathematics classroom for assessment purposes. The interviewers should possess interviewing skills, sound pedagogical content knowledge of mathematics, know what types of questions to ask and how to answer student’s questions and be capable of interpreting and making connections of students’ answers (Hunting, 2002). METHODOLOGY

Identifying the Target Population

Because no data on the students’ prior performance was available to us at the beginning of the research, we identified the target population based on the average of the first two mock Regents tests given on July 12 and July 19, 2007. Of the 51 Math A students at Lehman College, the 16 lowest scoring students (bottom third) became candidates for the target population. Their average score was 44.0, compared to 54.5 of the rest of the students. We then assigned the codes TP1 through TP16 to the 16 students. Our selection was corroborated by the classroom teachers. Clinical Interviews

The target population of 16 students was codified as TP1 through TP16 in a data collection book. We then interviewed each student and did so several times during the entire summer program.

In the first interview we built a rapport with the students by introducing ourselves and explaining the purpose of the research and its methodology. The students described their career aspirations, college plans and attitude toward math. Some students told us





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how much they hated mathematics while others said, “I don’t hate or dislike math, I just don’t understand it.” We were surprised to learn that several students were contemplating a career in engineering, technology or computer science. After the first interview, the students expressed their eagerness to meet with us again. They grasped that the goal was to improve their mathematical ability which in turn would increase their chance of passing the Regents exam.

During the clinical interviews, we asked the students to explain why and how they solved different multiple-choice questions in the mock Regents exam that they had taken. We did not assume that a correct answer in a multiple-choice question equated to correct mathematical thinking or knowledge in solving the question. We asked for evidence, such as an explanation of the way they solved the problem; sometimes we asked them to solve a similar problem. We also studied the students’ work and notes in their exam books.

After identifying students’ misconceptions, we revisited the concept or the procedure. We probed for understanding by asking the students to find mistakes in a wrong solution.

Interaction with Classroom Teachers and Tutors

After the clinical interviews, we alerted the teachers and tutors to the misconceptions we detected and to weaknesses characteristic of the target population. We conducted professional development sessions with the tutors during which we discussed our findings and suggested ways of addressing the misconceptions. FINDINGS FROM THE CLINICAL INTERVIEWS

Detected Misconceptions

Below are some common student misconceptions we identified though the clinical interviews. The item analysis helped us identify problems students struggled with. Order of Operations Although the students had spent the first two or three days of class on order of operations, many were still confused especially when it was time for them to work independently. It was hard to wean them from certain mnemonics, mainly PEMDAS (Please Excuse My

Dear Aunt Sally), when deciding what operations to do first. Many erroneously believed they should always do addition before subtraction, since the letter A comes before the letter S in that mnemonic.

As a case in point, many students answered that 4 – 2 + 1 = 1 Some students’ thinking was so ingrained with PEMDAS that they failed to relate the expression 4 – 2 + 1 to a practical problem like, “You have $4, spent $2 and then found $1; how much money did you end up with?” They failed to see the connection between mathematics exercises and mathematics as a representation of real life.

Likewise, because the letter M comes before the letter D in the mnemonic PEMDAS, some students answered 8 ÷ 2 ! 4 = 1. Since the letter R (for root) is not part

of the mnemonic PEMDAS, students had difficulty calculating 25 ! 4. They

multiplied 25 by 4, took the square root and got 100 = 10. Some students followed the

same procedure for 4 25 and obtained 10 as a result.





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Reliance on a mnemonic without understanding the underlying principle increased the likelihood of making an error. The students tried memorizing the “rules,” albeit incompletely or incorrectly, and could not see the relationship between mathematics and real life. Operations with signed numbers We covered order of operations and operations with signed numbers during the first week as a prelude to solving linear equations. By the time we conducted our first interviews, the students were well into solving linear equations. Yet in response to – 4 – 2 = ? Many students gave the answer 8. They “remembered” that two negatives give a positive and confused – 4 – 2 with (– 4)(– 2).

Lacking a feeling for negative numbers, the students had trouble translating the expression, “You spent $2 at breakfast and $4 at lunch” into a mathematical expression such as (– 2) + (– 4) or – 2 – 4. Some of the students were still confused even when asked to relate to a simple example, such as an elevator going up and down.

In elementary and part of middle school, students work exclusively with positive numbers; the negative numbers are introduced only as part of algebra. By that time the target students had difficulty with most mathematics concepts. Introducing negative numbers simply added to their confusion.

While teaching remedial mathematics classes in community colleges, we found that order of operations and operation with signed numbers posed a huge hurdle for many of our students. MSP students, like the rest of high school students, solve many mathematics questions with the TI 83 calculator. It should be noted that college students in remedial mathematics classes do not have access to a calculator. Division by 0 Many students in the target population correctly answered 0 to the operation 0 ÷ 6; however they provided the same answer for the divisions 6 ÷ 0, or 0 ÷ 0.

One student explained, “When I divide six apples to two people, each one will get three apples; but, if there are zero takers, I can keep the six apples.” Obviously, the student confused the quotient with the remainder. Operations with fractions Most students in the target population said they hated fractions. They regarded working with operations and fractions as a set of rules to be memorized and applied.

Since 4

1

2

1! =

8

1, a few students concluded that

2

1+ 4

1 =6

2. They assumed that

since in multiplying fractions one multiplies numerators and denominators, thus when adding fractions, one should add numerators and denominators

When asked how much is one half a dollar and one quarter of a dollar, students answered “75 cents”. The relationship between the three quarters of a dollar and $.75 was not clear. Generally, most students could relate to money but not to fractions. They knew neither how to divide fractions nor how to work with equivalent fractions. Working with proportions





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We identified several difficulties students had when trying to solve a word problem requiring the use of proportion:

• They lacked a sense of magnitude when working with fractions and fraction equivalents.

• They struggled translating the words of a problem into a simple proportion. For example, “If 5 workers assembled 9 computers a day, how many workers will

be required to assemble 18 computes in a day?” • They lacked a sense of proportionality.

For example, “Suppose you want to enlarge a picture measuring 3” by 5”. The enlarged picture is 9” wide; how long is it?”

Distributivity property Some students we interviewed equated 3(2x + 3) with 6x + 3. Since they did not understand that 3(2x + 3) is equivalent to (2x + 3) + (2x +3) + (2x +3), these students remembered to multiply but distributed the 3 only to the first term in the parenthesis. Solving linear equations This is one topic on which students relied on memorization of an algorithm par excellence. Their lack of understanding was obvious in many cases: For example in one student’s solving equations

2x = 4 yielded x = 2 as a solution, but 2x = 10 yielded x = 8 as a solution.

The student was trying to eliminate the coefficient of x. Instead of dividing both sides of the equation by 2, the student subtracted 2 from both sides. The student thought that “taking away” 2 on both sides, x will be left alone on the left side of the equation. In working the first example the student got the right answer for the wrong reason! Evaluating algebraic expressions Students had difficulty solving linear equations because they had no idea of what these expressions mean.

For example, one student evaluated the expression 2x – 3 when x = 1 as 18. The student substituted x with 1 to obtain 21 – 3 = 18. The student did not understand the difference between 2(1) and 21.

Other students who got the correct answer to the previous question had difficulty finding the value of 2x – 3 when x = – 1. Perimeter and Area Many students confused perimeter with area. The question was to find the perimeter of a rectangle in which the length was equal to 5 and the width equal to 3. Most students chose 15 as an answer for the perimeter. Other students thought that “not enough information given” since the picture had measurements of only one length and only one width of the rectangle. Units of measurement None of the students interviewed knew how many square inches are in a square foot.





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Logic problems The logic problems on the Regents exam are hard for all students. The concepts of inverse, converse and contrapositive were confusing not only to the target population students but also to the other students. Some students relied on a mnemonic memorization of how to find the inverse, converse or contrapositive (logical equivalent) of the statement, “If I have money, I go to the movies.” They remembered that to find the inverse you convert the positive to a negative and vice versa.

When answering this kind of question on the Regents exam, some students chose the equivalent of “I go to the movies if I have money” as being the converse of “If I have money, I go to the movies.” They apparently thought the converse of “If p then q” is “then q if p.” The “if” and “then” conjunction was carried with the subordinate statement it preceded.

Over reliance on mnemonics may trap students who have not mastered a mathematical concept.

Solving Linear Equations Below are examples of how a student tackled the questions below:

6(x – 2) = 36 – 10x 6x – 12 = 36 – 10x

4x – 24

4

4x

4

24!

x = –6 The student showed knowledge of distributive property of multiplication over

addition, but failed to recognize the equality symbol which makes this expression an equation. As a result the student did not know how to combine 6x and – 10x due to the absence of the equality symbol. Also, to this student the sum of – 12 and 36 resulted to –24, probably since the first number was negative. The equality symbol was totally ignored. Another student’s attempt to solve the same equation is shown below:

6(x – 2) = 36 – 10x 6x – 12 = 36 – 10x

6x – 48 = 10x –10x –10x

________________ 4x – 48

4

4x

4

48! = 12

x = 12 This student showed knowledge of distributive property of multiplication over

addition; however the student had a fuzzy idea of solving linear equation. The student was confused on how to work with an additive inverse. After adding –10x to both sides of the equation, the student did not know how to proceed. Evaluating Algebraic Expression





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Below is the work of a student attempting to evaluate the expression ab – b2 when a = 3 and b = –1 ab – b

2 = 3(-1) – (–1)

2

= –3 + 1 ab – b

2 = –2 The student confused between – (–1)

2 and (–1)2 .

Another student attempted to find the value of x – 3y2 when x = – 4 and y = 3 x – 3y

2 = –4 – 3(3)

2

= –4 – 92

= –4 – 81 = –85 The student equated 3(3)2 with (3! 3)2. ATTAINING RESEARCH OBJECTIVES

Based on this research we concluded that clinical interviews could be a useful tool for helping teachers gain insight into the students’ thought processes, pinpoint misconceptions, develop a dialogue in response to the students’ problems and raise the students’ self-confidence. By helping teachers become more responsive to their students’ needs, clinical interviews ultimately help students become better learners. An important contribution of this research was showing how clinical interviews helped bolster the weakest students’ performance in mathematics in the statewide August 2007 Regents exams. In fact a total of 25% of the target population passed the Regents exam with a score of 65 or higher. At the same time, 56% of the target population obtained a score of 55 or higher, which is sufficient for high school graduation. . RESEARCH CONCLUSIONS

Following our research, we concluded the following:

• Clinical interviews were useful for uncovering students’ misconceptions and weaknesses.

• Used properly, the clinical interviews could constitute an important pedagogical tool for the teachers.

• To leverage on the interviews, the results have to be communicated by the interviewers to the classroom teachers and tutors and acted upon in error correction.

• Our target population struggled with word problems, compounded by the fact that they paid less attention to mathematics terminology; tended to jump to conclusions faster and were less inclined to have a plan to solve the problem.

• The target population students relied on memorizing algorithms (mnemonically or procedurally) to solve certain types of problems. They relied on the TI83 calculator even for simple operations.





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• As a result of the intervention, the percentage of the lowest performing students passing the Regents or getting grades for high school graduation increased significantly compared to the previous years.

CLINICAL INTERVIEWS vs OTHER PEDAGOGICAL TOOLS

We concluded that clinical interviews could be a powerful pedagogical tool in mathematics education. The value of the interviews is helping teachers become better teachers. Equipped with understanding of students’ mathematical thinking, classroom teachers could design strategies and exercises to correct misconceptions, strengthen students’ knowledge, promote integration of mathematical concepts, and encourage metacognition.

From a socio-cultural perspective, clinical interviews help in lowering the barriers between the interviewer and the student. When the interviewer is the classroom teacher the interviews contribute to fostering a climate of mutual trust, conducive to future learning.

While the value of well-designed, well-conducted clinical interview cannot be overstated, clinical interviews have inherent limitations. Training and developing an “army” of capable, knowledgeable interviewers is costly and labor-intensive. As such it is hard to fathom the clinical interviews as a panacea for the mathematics difficulties encountered by our middle school, high school and community college students.

The power of clinical interviews would be greatly enhanced when used in conjunction with other pedagogical tools:

• Examination of individual students’ work presentation and calculation to gain insight into students’ weaknesses

• Collaboration between teachers and tutors: exchanging information about individual students’ strengths, weaknesses and misconceptions, and discussion of joint strategies to promote error correction and enhance learning

• Use of item analysis in conjunction with multiple choice tests to get an understanding of topics in which the teaching was not effective.

• Expert design of multiple-choice tests used with item analysis for error detection • Use of personal response systems (PRS or clickers) for error detection

FUTURE RESEARCH We recommend that the research should be expanded in two areas:

• Use clinical interviews with the highest performing students. The interviews may be more helpful to students who are better prepared in mathematics.

• Use clinical interviews with remedial mathematics students in community colleges. In conjunction with other pedagogical tools, the clinical interview may play an important role in increasing student retention and graduation rates.





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MatheMatics teaching-ReseaRch JouRnal online 19 Vol 3, n1