teaching-research communities
19
algebra-english as a second language
Bronislaw Czarnocha
Chapter 5.1 Algebra/English as A Second Language (ESL) teaching Experiment
Introduction
A teaching experiment in correlating the instruction of courses in Elementary Algebra and Intermediate ESL is described whose results suggest a measurable transfer of thought organization from algebraic thinking into written natural English. It is shown that a proper context to situate this new effect is (1) the Zone of Proximal Development (ZPD) of L. Vygotsky and (2) a new concept of the “Relative ZPD” characterizing the relationship between the ZPD of arithmetic/algebra and ZPD of native/foreign language. The chapter is the amplification of the Czarnocha and Prabhu (2002) paper with the same title.
The relationship between teaching mathematics and English as a Second Language (ESL) has been the topic of many papers and presentations (Anderson, 1982; (Birken, 1989; Connolly, 1989; Luria and Yudovich, 1971). Yet the literature and research on the subject suffer from several shortcomings. First, the majority of the research deals with the role of language in learning mathematics, leaving the reciprocal relationship, that addresses the influence of learning mathematics on the development of language, almost totally unexplored. Furthermore, while several benefits of writing as an instructional tool in teaching mathematics, such as better understanding of conceptual relationships (Birken, 1989) or the facilitation of “personal ownership” of knowledge (Connolly, 1989; Mett, 1998) have been proposed, there has been, until recently, little evidence to explicitly demonstrate these benefits (Powell and Lopez, 1989). Finally, there is a relative absence of theoretical considerations that could provide a context in which to properly situate the reciprocal relationship between the development of mathematical understanding and mastery of language.
ESL related literature presents us with more or less the same situation and focuses on the role of the mathematics instructor as “a teacher of the language needed to learn mathematical concepts and skills” (Cummins, 1980). The methodology of classroom practice based on this principle was formulated in (Dale and Cuevas, 1987). An important theoretical distinction in the area of second language acquisition has been introduced by Cummins in (Cummins, 1980), who asserted that the process of language acquisition has at least two distinct levels: (1) the Basic Interpersonal Language Competency (BILC) level of everyday use, and (2) the Cognitive Academic Language Proficiency (CALP) level.
This presentation addresses the shortcomings listed above. A brief discussion of certain ideas of Vygotsky in (Vygotsky, 1986) outlines a context in which the relationship between mathematics and language can be situated. This is followed by a new and interesting result obtained during a teaching experiment at CUNY’s Hostos Community College, in which an Elementary Algebra course was pedagogically linked with an ESL course. The findings of this experiment suggest a potentially powerful influence of mathematical reasoning on the development of descriptive writing.
Theoretical Background
The existing literature contains sporadic hints about the relationship between mathematical understanding and the acquisition of language. Recognizing the similarities between writing skills and problem-solving skills, as pointed out by Kenyon in (Kenyon, 1989) this relationship can be appreciated by the necessity of mastery of a common set of problem-solving strategies. This point of view, that doesn’t take into account the peculiarities of each of the disciplines, is strongly supported by Anderson’s Adaptive Control of Thought theory (Anderson, 1982).
A point of view that gives justice to the richness of relationships between thought and language can be found in some early works of Vygotsky (Vygotsky, 1986). Following Vygotsky, thought and language exist in a “reciprocal relationship of development” (Kozulin, 1986). Vygotsky writes, “Communication presupposes generalization … and generalization … becomes possible in the course of communication” (Vygotsky, 1986). In other words, in order to communicate, we need to think; and in order to think, we need to communicate. Such a view opens, in a very natural way, the possibility that thought, in our case, more specifically, mathematical thought, has the potential to shape natural language. One of the ways through which this process can take place is across the Zone of Proximal Development (ZPD) (Vygotsky, 1986).
The ZPD arises in Vygotsky’s theory through his distinction between spontaneous and scientific concepts. It represents the depth to which an individual student can develop, with expert help, his or her spontaneous concepts concerning a particular task or problem, as opposed to his ability to do it alone.
Valsiner had noted that the development of the ZPD can be fostered even further if the environment is structured in a way that leads the student to use elements that are new and yet unfamiliar, but are reachable from his or her ZPD (Valsiner, 1993). One of the essential characteristics of the upper level of the ZPD, as compared to the level of the corresponding spontaneous concepts, is its higher degree of systemic structure. In the experiment explored here, the abstract character of elementary algebra had created exactly that type of ZPD with respect to the “spontaneous” level of natural English.
Experimental Realization
To confirm Vygotsky’s highly dialectical view one would need to clearly detect the presence of two different directions of developmental progression: the acquisition of the English language under the influence of mathematical thinking, and the acquisition of mathematical understanding under the influence of a sufficient grasp of the English language. While the main topic of the current discussion is the first of the two directions, we note that the importance of the second has been confirmed, for the first time, in a recent experiment by Wahlberg (Wahlberg, 1998). Measuring the level of students’ understanding of calculus when assisted by a systemic incorporation of essay writing, she observed a substantially higher increase in the experimental group as compared to the control group.
Elementary Algebra/Intermediate ESL teaching experiment
The general goal of the ESL sequence at Hostos is to develop what Cummins calls the Cognitive Academic Language Proficiency (Cummins, 1980), and, what Vygotsky calls the language of “scientific concepts”. Our experiment had two goals: to see how far algebra can help in that process, and to investigate the cognitive correlation in the acquisition of both. More precisely, the questions of the teaching experiment stated above were translated into the following goals:
– to formulate a series of tested instructional strategies for teaching English and improving critical thinking skills within both the mathematics and ESL courses;
– to analyse the degree to which the syllabi of the content courses need to be modified to incorporate language instruction;
– outline major problems encountered during the interdisciplinary collaboration;
– understand the learning process of English acquisition through the teaching of math in the context of our student population;
– to identify a series of hypotheses concerning the cognitive relationship between learning English and mathematics by students whose primary language is not English
Methodology
A group of seventeen students was enrolled in an intermediate ESL class and in a remedial Elementary Algebra class taught in English. In the previous semester, these students passed the second lowest level ESL course as well as the first remedial mathematics course (Basic Arithmetic). The Algebra class was the only class they were taking in English, and, thus, constituted their only exposure to academic English. Although the classes were separate, the communication between the instructors was frequent and substantive, involving weekly meetings, exchange of materials, and mutual class visits. The methodology of the experiment was based on two assumptions. First, since we were interested in the influence of the algebraic language upon the natural one, we needed to verbalize the symbolic algebraic language to the highest possible degree. That meant we needed to make the symbolic notation of algebra explicit in speech and/or writing – to verbalize the procedural steps and the content of algebraic thinking. Second, these elements, having been made explicit in their algebraic context, needed to be transferred into the context of the ESL class, both on the semantic and the grammatical level. As a result, student discussions in the Algebra class often, by design, involved a level of academic discourse somewhat above the students’ capacity at the given time. We hypothesized that it is this increased level of student effort to communicate the comparatively abstract mathematical ideas in English is at the root of their eventual linguistic improvement. At the same time, the ESL class deliberately involved discussions of the linguistic peculiarities of algebraic language, such as the role of word order and sentence structure with the aim of improving their mathematical reasoning skills. Below are examples of specific instructional strategies in both classes. A special attention was paid to the careful observation of cognitive difficulties experienced by the students during the actual process of learning. This was a way to reflect upon, improve, and increase our understanding of how teaching and learning takes place. In particular, we were interested in understanding the details of the developmental learning process in the context of collaborative instruction of mathematics and English. One of our main goals in using this methodology was to create a profile of a teacher-researcher paradigm at Hostos Community College through which:
– The teacher becomes engaged in critical reflection and experimentation of his/her own instructional practices,
– The instructor’s teaching practices are critically evaluated in relation to the available theoretical research, and
– The theoretical research is critically applied in the instructional context for further questioning and development.
We believed that this new profile would help us gather data and more successfully understand the challenges concerning the acquisition of English through content areas. Furthermore, it would allow for the formation of a supportive intellectual structure to address and to solve other present and future pedagogical issues such as shortening the remediation time of students enrolled in Mathematics and English courses.
New algebra instructional strategies
Verbalization of algebraic procedures
Example – Solving linear equations:
Solve 2x + 5 – 5=9-5 for x.
Solution / Steps (to be explicitly written by students)/ First, I add -5 to both sides of the equation in order to eliminate the +5 on the left side.
Second, I cancel the opposite numbers and add the like terms.
Third, I divide both sides by 2 in order to have X alone.
The answer is X = 2
Explication of algebraic symbolism through writing paragraphs
Example:
(a) Write a paragraph explaining the difference between 3×5 and 5×3. What does it mean to you that 5×3 = 3×5?
(b) What is the difference in the meaning of the equality symbol in the following two expressions?
3×16 = 48 and x + 5 = 12
Analysis of algebraic rules and principles
Example:
(a) Compare the rule for the addition of signed numbers with different signs with the rule for the multiplication of signed numbers with different signs.
(b) Write a paragraph addressed to a fellow student, who missed a couple of classes, explaining how to solve the problem below. Clearly verbalize to him/her the order of steps in the procedure, warning against any possible errors and reminding him/her of the rules, which justify your steps in the solution.
Simplify
Readings and linguistically adjusted word problems
One of the most important problems that became evident was the absence of sufficient readings about mathematics suitable for our students. To address this, a series of special word problems were developed. They were made out of paragraphs from stories read in the ESL class. Additional explanatory paragraphs and questions about mathematics itself were developed to assist student learning. For instance:
– What does it mean to you that 3×5 = 5×3?
– How do you differentiate the operation of subtraction from addition?
– What is the difference in the meaning of the equality sign in and?
These paragraphs of explanations were motivated by the advice from the mathematics educators L. Steffe and Smock (1975), in their work entitled Model for Learning and Teaching Mathematics. These authors proposed that a “…carefully arranged interplay between spoken words which symbolize a mathematical concept and the set of actions performed in the process of constructing a tangible representation of the concept should be mandated ...A mathematical vocabulary should be developed… to explicate and provide embodiments for the concept.”
Example 1 – A literary word problem from a paragraph from The Pearl by John Steinbeck:
Kino awakened in the near dark. The stars still shone and the day had drawn only a pale wash of light in the lower sky to the east. The roosters had been crowing for some time, and the early pigs were already beginning their ceaseless turning of twigs and bits of wood to see whether anything to eat had been overlooked. Outside the brush house in the tuna clump, a covey of little birds chattered and flurried with their wings.
Kino’s eyes opened, and he looked first at the lightening rectangle that was the door and then he looked at the hanging box where Coyotito slept. His eyes wandered again to the rectangle of the door, to its familiar elongated shape. They doors were much shorter in width than in the height. Kino knew their dimensions by heart because it was him who made the doors when he and his wife, Juana, moved in here. The height was exactly three times the width, which made it a tall and narrow entrance. Sometimes, though rarely, Juana would cover the entrance with her long blue shawl whose lengths of four sides added to 32 units. The shawl fit exactly the opening of the door. He turned his head to Juana who lay beside him on the mat, the blue head shawl over her nose and over her breasts and around the small of her back. What were the dimensions of the shawl?
Example 2 – An explanatory paragraph utilizing a house cleaning analogy to demonstrate commutativity:
The product of your work at home is the cleanliness of your house. If you do it carefully and with thought, you know that some actions must be taken before other actions. For example, you have to dust the surfaces before you clean the floor. Otherwise, you will have to clean your floor twice. This means that the action dust the surfaces and the action clean the floor are not commutative. The order in which you do them matters for the efficiency of your work. On the other hand, if you have two bedrooms, both coming out into the hall, it doesn’t matter which you clean first. The action clean the first bedroom commutes with the action clean the second room. Cleaning the two bedrooms is like multiplication. It doesn’t matter which number (or bedroom) comes first.