Inclusive and Supportive Education Congress
International
Special Education Conference
Inclusion: Celebrating Diversity?
1st - 4th August 2005. Glasgow, Scotland
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Inclusive and Supportive Education Congress 1st - 4th August 2005. Glasgow, Scotland |
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Dr. Miriam Ben-Yehuda
Beit-Berl Collage, Kfar-Sabba, Israel
Miriamby@zahav.net.il
In this article we present a new diagnostic tool designed to investigate the thinking and learning mechanisms that affect student success or failure in mathematical studies. Using the tool (Arithmetical Discourse Profile, ADP), teachers and diagnosticians can formulate a detailed profile of each student’s mathematical discourse, describe their particular strengths and weaknesses in their course toward an arithmetical solution and build a suitable learning program for each student accordingly. We adopted in our research the communicational approach that presumes there to be a close relationship among discourse, thinking and learning, inasmuch as when people speak, they are telling us what they think. Each discourse is characterized by repetitive patterns of decisions. Therefore, in order to describe a mathematical discourse, we need to describe its repetitive characteristics .
The development of a tool for detailed, targeted, individual analysis was motivated by numerous meetings with children and young people who could not manage even the simplest arithmetical calculations. This experience highlighted the current lack of appropriate tools for analyzing the process students undergo when they solve an arithmetical problem as well as the absence of a clear terminology to properly describe this process. Our own sense of helplessness was exacerbated by the low student achievements in mathematics manifested in reports from Israel and on international examinations (Geist, 2000; NCTM Standards, 1989, 2000). According to teacher estimates, 20 to 30% of students, and even more in certain areas, find mathematics a major hurdle. Some students in this group (6-10%) are categorized as “learning disabled”, and others, whose difficulties may be equally troubling, are categorized as having “learning difficulties”. While the rates continue to climb yearly, the definitions of learning problems in mathematics remain ambiguous, and the methods of their diagnosis poor (Ginsburg 1997). Studies have shown that the instruments currently used for testing and diagnosing student performance are insufficient. It has become clear that there is a need for an intercultural cross-situational study aimed at understanding sociocultural-based learning and thinking processes and at explaining their effect on learning Math (Stigler, Galimore & Hiebert, 2000; Hoard et al., 1999; Ginsburg 1997).
The widespread confusion in the field has had a major impact on the accepted diagnostic methods in Israel (Report of the Committee for Evaluating Potential in Students with Learning Disabilities, Margalit,1997) and this is especially true for mathematics. The lack of testing tools in Mathematics urged us to develop a new and accurate diagnostic tool that would meet the needs of teachers and diagnosticians.
For this purpose, we adopted an original approach based on the use of the mathematical discourse wherein student participation is analyzed in real time during problem solving.
The main aim of the study was developing an instrument to identify the factors responsible for the high rate of student failure in mathematics. An instrument which can help testers and teachers formulate a detailed profile of each student’s arithmetical discourse, and describe the reasons for his (or her) success or failure. The Arithmetical Discourse Profile (ADP) is divided into five main criteria and various secondary components that underscore students’ ways of thinking and the expression of their strengths and weaknesses. We assumed that each student's discourse is characterized by a unique composite of features among which we would be able to identify those that could yield good results even in low achievers.
Students were divided into three groups according to their current achievements in mathematics (high, moderate, low), and their ADPs were compared in order to identify similarities and differences between the groups and within each group.
In this research we adopted the communicational approach to studying mathematical thinking and learning. This approach regards thinking as a special case of communicative action, since we a person thinking is essentially communicating with himself. This form of communication does not have to be audible or verbal (Sfard, 2000, 2001a,b). In this article, we will regard communication as an activity in which one tries to make an interlocutor, possibly oneself, react in a certain way. There is a close relationship among discourse, thinking and learning, and the concept of discourse is indivisible from the concept of thought. When people think, they are in effect communicating with themselves, and when they speak, they are telling us what they think. A change in discourse reflects learning and thereby a change in thinking (Sfard, 2001b). A child understands a topic when he is capable of engaging in a discourse on it with others in compliance with the rules of that discourse (Harre & Gillet, 1994; Sfard, 2000, 2001a,b).
The communicational mechanism underlying the thinking-learning-discourse pathway is essentially dialogical, regardless of whether it occurs in words or images, orally or in writing. When we converse with a partner, we are informing him about something. Accordingly, when we think, we are informing ourselves. We argue, we ask questions, and we anticipate a response from ourselves to the questions we raised. As such, all discourses are characterized by repetitive patterns of decisions, and each interlocutor has certain rhetoric features typical to him that are manifested in the discourse he conducts. Therefore, to describe a student, we need to describe the repetitive characteristics of his, in this case, mathematical discourse (Cazden, 1988; Sfard 2000; Yackel & Cobb, 1996).
Mathematical Discourse has been proposed by many researchers as a useful method to help students, especially students who perform poorly in math, learn mathematics in the classroom. When students “speak math”, they build their thought processes through words, and when they work in a team, they must clarify their ideas so that others would understand (Cazden, 1988; Yackel & Cobb, 1996; Schwartz, Neuman & Biezuner, 2000). In addition to its teaching value, mathematical discourse has been found to be an essential adjunct for the evaluation and diagnosis of students. Interviewing students can be regarded as discoursive evaluation, especially when the student is asked to solve problems out loud (Meichenbaum, 1985; Meichenbaum & Goodman, 1971 ).
The specific research methods used here were derived from the communicational theory. The function of the researcher is to listen, document, and analyze students’ discourses during problem solving. The discourse transcription is the record of the student’s thinking, and the discourse analysis is an analysis of the student’s ways of thinking. A change in the discourse indicates a change in thinking, i.e. learning. The goal of the researcher is to identify and to characterize the repetitive patterns and underlying mechanisms in order to obtain adetailed, in-depth discourse profile.Contrary to the traditional discourse researcher who evaluates and interprets the interlocutor’s schemata and the concepts he forms in his head, the new communicational approach requires the researcher's presence and participation in the discourse, and that he test the obvious - the statements themselves - and interpret them.
Our study sample included 16 nonrandomized students. Fourteen of them were selected from two 7 th grade classes of a single school in central Israel. They included children with high, moderate, or low achievements in mathematics. Background characteristics varied widely as well. The group categorization was done by a research team of mathematics teachers and school counselors using the following criteria: general scholastic performance as evaluated by the student’s present teacher; general scholastic performance in elementary school, as specified in the annual report card data; grades of the two most recent final exams in mathematics; the results of the 7 th grade math proficiency test, taken at the beginning of the school year to determine the distribution of students into study groups by performance; and individual data, such as previous didactic and psychological diagnoses and remedial classroom help.
The two remaining students were 17-year-old girls with a long history of problems in mathematics. These cases were included to exemplify the use of our new diagnostic tool and the method of building a ADP. They served as the basis for our earlier study on the unique problem-solving characteristics of students with difficulties in math (Ben-Yehuda, Lavy, Lynchevski & Sfard, 2005).
All students were interviewed in order to collect personal and other background data and to examine their reaction to different statements related to learning mathematics and to caricatures depicting classroom situations (for example, students looking perplexed during a math exam or a teacher glaring angrily at a student).
Thereafter, they were presented with two similar examination papers. Both papers included a set of numeric problems and a set of word problems depicting everyday situations. Each student was then asked to solve the mathematical problems in the first examination paper and to explain his solution out loud. The second examination paper was given to two students simultaneously, and they were asked to solve the problems independently and to explain their solutions orally to each other.
All mathematical tasks presented were on the 7 th grade level as established according to the results of the proficiency test taken by the students at the beginning of the year (and comprised of the following subjects: concept of the number and decimal system, solving problems with whole numbers and fractions, decimals and percentages).
Three accepted means of student evaluation were employed: Recorded interviews, observation of the student while solving mathematical problems and observation of the student in a group, during a lesson.
The ADP constitutes a set of discourse components that is unique to the specific student. A student's ADP is formed by a detailed analysis of his discourse segments collected in the two research settings (single or paired participants) with respect to 5 major criteria.
(1) Communications Management: The manner in which a student conducted his discourse and conveyed his ideas was measured by the following criteria: Externalization (accessibility to the student's self-communication), fluency (continuity of discoursive flow in the conversation), connectivity among and the participant’s expressions and within each expression. The segments of each discourse were analyzed using an interaction flowchart (introduced by Sfard and Kieran 2001a), which describes the student-student interactions.
(2) Mediational Modes: To analyze the means whereby the communication was mediated, we focused on the manner in which students processed the written symbols they scanned visually. Every expression in the discourse was reviewed for this factor. Our experience has shown symbols can be handled through objectified or syntactic mediation. In objectified mediation, the student perceives the symbol as representative of an object; he relates to its features and performs manipulations on it accordingly. In syntactic mediation, the student translates one numerical expression into another by using a series of statements that are constructed following the recognized rules of syntax. The discourse is considered flexible when there is mediational diversity and the student uses a variety of mediational modes in his discourse.
(3) Discoursive self-control: The effort made by the student to control the discourse was analyzed by counting the number of times the student erred, stopped and retraced his steps in search of mistakes.
(4) Routine Task Interpretation: Here we focused on those routine ways of dealing with tasks that give arithmetic discourse its distinctive characteristic. To describe the discoursive routine, we needed to relate to the discourse on two levels: the operative enacted rules, that regulate the student's calculations, wherein we can directly observe the student’s actions, and the endorsed rules, that influence the student's decisions to a great extent (for example, whether the student can reach proficiency in math or is "destined" to fail).
(5) Self-Imaging: We tested the manner in which the students presented themselves while engaging in discoursive activity: self-assessment and self description.
(A). The major finding of this study is that it is possible to define a unique composite of discoursive traits for every student. That is, every student reaches the solution to mathematical problems with the help of different combinations of features, and there are no two individuals, regardless of the group they belong to (high, moderate, low), whose discoursive characteristics are identical.
(B).Despite the differences within each group, we can still identify several discoursive features shared by most group members.
Several elements were found to repeat themselves in the profiles of all members of any individual group.
(1). The discourse of all the high achievers in math was managed with fluency and connectivity. The students recognized their teachers as the source of the meta-rules for their discourse. They valued themselves as high/moderate achievers and saw a connection between success and commitment.
(2). The discourse of the low-achiever group members was poorly managed: it lacked fluency, connectivity, continuity and integrity. The students lacked the skills to follow the mediational mode they chose, didn’t reach discoursive self-control - made many mistakes, and did not correct their mistakes. They also considered themselves to have low abilities in mathematics.
(3). The students in the moderate achiever group made many errors and were not proficient in using the mediation mode they chose. They preferred colloquial meta-rules and could not explain the manner in which they reached a solution. Their sense of ability was low.
(4).Comparison of the ADPs of the different groups revealed several similarities and differences:
(A). The main conclusion derived from the present study is that in mathematics, as in all quests, goals can be reached via several different routes. Many paths lead to the right solution, and there are no two individuals – whether they reach the right solution or not – whose discoursive characteristics are identical. All that is needed for one’s success is to find this person’s special strengths and bypass his particular weaknesses. For each of our participants, we identified a unique composite of discoursive features that could lead him to the solution. This finding indicates that it is possible to help every student follow a personal path to success, and that even for students with difficulties in math, we can find discourse components to suit their skills.
(B). The ADP is a multifaceted tool that offers a novel means to observe, in real time, the thinking processes and mechanisms used by students to reach mathematical solutions. Instead of a limited description of student performance of the 4 arithmetic functions, the ADP simultaneously checks the whole range of factors that affect the student's problem solving process, yielding a holistic, comprehensive picture. The ADP provided us with detailed knowledge of the unique discoursive features of each student, so that we were able to identify the factors responsible for his success or failure in math.
(C).The ADP is not intended for the differential diagnosis or categorization of students, but as a practical instrument to describe the manner in which the particular student operates.
(D). There is no one profile or one component that determines whether a student will succeed or not. It is the relationship among the discourse features that is key. For example, the discourse of Talia, one of the two 11 th grade girls in the sample, was conducted according to the meta-rules she acquired in school but it was not effective. She perceived the meta-rules to be obligatory and strict. Therefore, she did not relate to numbers as objects, used inflexible syntactic means of mediation and did not control her discourse.
(E).The ADP can serve as a basis for targeted intervention. Even for low achievers, a composite of discoursive features can be identified and used to develop an intervention program.
(F). The tendency to explain student failure on the basis of a uniform grading system disregards their personal differences and leads to the clumping of a wide range of cases under a single label or numerical category.
(G). Comparing the ADP with traditional measures such as proficiency tests and report card grades, we found that our new tool helps to uncover differences even among students with similar marks. Students within the same group (which was put together according to our study criteria mainly by grade) had different profiles, with different sets of discourse characteristics and different strong and weak discourse points.
Some students, even within the high achiever group, had discourse components which led them to failure. In some cases, for example, although the discourse flowed well, was connected and integrated and involved the use of varied mediators, it was not well controlled, so the student failed to check and correct his errors.
(H). Most of the students in our sample were not skilled in conducting an interpersonal discourse. Most did not succeed to solve the problems in the paired testing conditions. As was claimed by Sfard and Kieran (2001a) - discourse needs to be learned. We cannot assume that students know how to conduct a discourse between themselves in an effective manner without first learning how to do so.
(K). During the course of data collection and analysis, we found further support for the conjecture that the traditional categorical diagnostic approach leads to over diagnosis of children with learning disabilities. The use of labels such as “learning disablity”, “dysgraphia”, ”dyslexia” or “dyscalculia” can sometimes be deleterious, as many students soon learn that they can be manipulated for ulterior motives. For instance, several students in our sample refused to do written or other types of tests because they were officially certified as “learning disabled”. Furthermore, the very existence of the label is a kind of “self-fulfilling prophecy”. The result is a vicious cycle of rejection that causes feelings of incapability, lack of application, and failure (Jones, Wilson & Ghojwani, 1997). Therefore, we believe we should instead focus on the process of the student’s participation in the task itself, and not stop at evaluating the result alone. This can lead to improved operative applications and targeted individual intervention.
The aim of our painstakingly detailed analyses was to produce results that would effectively guide us in helping the student in his struggle against difficulties. The awareness of the complexity of arithmetic tasks, good acquaintance with students’ relative strengths and weaknesses as they are manifested in his mathematical discourse, and sensitivity to the issues of identity and positioning that can affect student performance – all these are necessary conditions for effective intervention.
Ben-Yehuda, M., Lavy, I., Lynchevski, L., & Sfard, A. (2002). Doing wrong with words or What bars students’ access to arithmetical discourses. To appear in The Journal for research in Mathematics Education.
Cazden, C.B. (1988). Classroom discourse: The language of teaching and learning. (2 nd ed). Portsmouth, NH: Heinemann.
Geist , E.A.(2000). Lessons from the TIMSS videotape study. Teaching children mathematics. 7 (3), 180-186.
Ginsburg, H.P. (1997). Mathematics learning disabilities: A view from developmental psychology. Journal of learning disabilities, 30(1), 20-33.
Harre, R. & Gillet, G. (1995) . The discoursive mind . thousands Oaks. Sage Publication.
Hoard, M.K., Geary, D. C., Hamson, C. O. (1999). Numerical and arithmetical cognition: Performance of low- and average-IQ children. Mathematical Cognition5(1), 65-91.
Jones, D. E., Wilson, R. & Bhojwani S., (1997). Mathmatcs instruction for secondary students with learning disabilities. Journal Of Learning Disabilities, 30(3), 171-163.
Meichenbaum,D. & Goodman (1971). Training impulsive children to talk to themselves: A means of developing self control. Journal Of Abnormal Psychology. 77,115-126.
Meichenbaum,D. (1985). Teaching thinking :A cognetive behavioral perspective,In Sigal , Chipman & Glaser, (Eds). Thinking and Learning. New York: Plenum.
National Council of Teachers of Mathematics .(1989). Curricular and Evaluation standards for school Mathematics. Reston, Va.: The Concil.
National Council of Teachers of Mathematics (2000). Professional Standards for Teaching Mathematics INC.,Reston , Verginia.
Sfard , A. (2000). Steering (dis)course between metaphor and rigor: Using focal analysis to investigate the emergence of mathematical objects. Journal of Research in Mathematics Education, 31(3) 296-327 .
Sfard , A. ( 2001a). Learning Mathematics as Developing a Discourse. In R. Speiser, C. Maher, C. Walter (Eds), Proceedings of 21 st Conference of PME-NA (pp. 23-44). Columbus, Ohio: Clearing House for Science, mathematics, and Environmental Education.
Sfard , A. (2001b). There is More to Discourse than Meets the Ears: Learning from mathematical communication things that we have not known before. Educational Studies in Mathematics, 6(1/3),
Sfard , A. & Kieran, C. (2001). Cognition as communication: Rethinking learning-by-talking through multi-faceted analysis of students’ mathematical interactions. Mind, Culture, and Activity, 8(1), 42-76.
Stigler , J W., Galimore, R., & Hiebert, J. (2000). Using video surveys to compare classroom and teaching across cultures: examples and lessons from the TIMSS video studies. Educational Psychologist, Spring 2000,35 (2) 87-101.
Schwartz, B.B., Neuman, Y.,& Biezuner,S.,(2000). When tow wrongs argue together, one may make it right! Cognition and Instruction., no 18 .495-461.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation and autonomy in mathematics. Journal For Research In Mathematics Education, 27, 458-477.
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