TITLE: CALCULUS REMEDIATION USING DATABASES AUTHOR: Russell Jay Hendel AFFILIATION: Health Care Finance Administration, Phil PA ADDRESS: POB 34662 Phil PA 19101 ACKNOWLEDGEMENT: (Written while at University of Louisville) PHONE: 215 596 6830 FAX: 215 596 5084 EMAIL: RHendel @ Phi1 . SSW . DHHS . GOV REFERENCE NUMBER: C39 1:INTRODUCTION:The goal of this presentation is to illustrate how the concepts of database theory can aid in calculus remediation. An outline of this paper is the following: Using some TYPICAL CALCULUS II PROBLEMS (section 2) we precisely define WHICH TYPES OF REMEDIAL STUDENTS WE CAN HELP (section 3). We then review TYPICAL THOUGHT PROCESSES OF GOOD STUDENTS(4). We show how these thought processes of good students are identical with DATABASE QUERY METHODS(5).These 5 sections present our BASIC IDEA. The remaining sections of the paper present the TECHNIQUE DATABASE(6), USE OF ADVANCED DATABASE THEORY(7) in remediation, and CONCRETE SUGGESTIONS FOR CALCULUS TEACHERS(Section 8). 2: TYPICAL CALCULUS II PROBLEMS EXAMPLE 2.A: Integrate the function x(1-x^2). HINTS: POSSIBLE INTEGRATION TECHNIQUES (i) EXPANSION (ii) GENERAL SUBSTITUTION (iii) TRIG SUB EXAMPLE 2.B: Integrate the function x / {x^2 - 4}. HINTS: POSSIBLE INTEGRATION TECHNIQUES (i) TRIG SUB (ii) PARTIAL FRACTIONS (iii) GENERAL SUB 3:TYPES OF REMEDIAL STUDENTS:We define three types of students: SATISFACTORY STUDENTS,TOTAL REMEDIAL, and INTERMEDIATE REMEDIAL. An INTERMEDIATE REMEDIAL STUDENT could not solve either of the two typical calculus problems presented in section 2; However, the INTERMEDIATE REMEDIAL STUDENT could completely solve these problems if they were provided with the HINTS mentioned there. In other words the INTERMEDIATE REMEDIAL STUDENT doesn't know how to START a problem but can COMPLETE it if given a hint. Still another formulation is that the INTERMEDIATE REMEDIAL STUDENT is characterized by his(her) asking, "WHERE DO I BEGIN." By contrast, the SATISFACTORY STUDENT can solve a problem EVEN if NOT given a hint while the TOTAL REMEDIAL STUDENT cannot solve a problem EVEN IF GIVEN a hint. Our methods can only help the INTERMEDIATE REMEDIAL STUDENT. 4:THE SATISFACTORY STUDENT:To help the INTERMEDIATE REMEDIAL STUDENT we first explore how SATISFACTORY STUDENTS would solve the problems presented in section 2 and then attempt to transfer these skills to the INTERMEDIATE REMEDIAL STUDENT. I think most instructors would agree that the typical calculus problems presented in section 2 would be solved using the following two step process: "STEP-A: FIND OUT WHICH INTEGRAL TECHNIQUES APPLY TO THE INTEGRANDS PRESENTED IN SECTION 2 AND THEN (STEP-B) APPLY THOSE TECHNIQUES TO THE INTEGRANDS AND INTEGRATE." STEPS A and B suggest the following strategy to help INTERMEDIATE REMEDIAL STUDENTS: Help them find the right integration technique so that they can then complete the problem themselves. 5:DATABASE QUERY METHODS: Our fundamental thesis is as follows: "STEP A above-THE SEARCH FOR INTEGRAL TECHNIQUES THAT APPLY TO A GIVEN INTEGRAND-IS IDENTICAL TO THE PROCESS OF A DATABASE QUERY" This IDENTITY of SATISFACTORY STUDENT thought processes with DATABASE QUERY METHODS can be succinctly illustrated by the following one-to-one correspondence showing how a SATISFACTORY STUDENT would solve EXAMPLE 2.B. ============================================================== SATISFACTORY STUDENT DATABASE QUERY -------------------- -------------- Which SELECT * integration techniques FROM Techniques apply to integrands WHERE Integrand-Form like the rational function = "Rational Function" in EXAMPLE 2.B =======================TABLE 5.1============================= The QUERY column of table 5.1 can be understood with the TECHNIQUE database presented in table 5.2 whose full meaning will be explored in sections 6 and 7. For those unfamiliar with database theory we give a brief introduction using EXAMPLE 2.B. The DATABASE is basically a TABLE consisting of ROWS and LABELLED COLUMNS. The DATABASE QUERY presented in table 5.1 can be interpreted as follows: SELECT *(i.e. ALL COLUMN ENTRIES) FROM THE TECHNIQUE DATABASE but only from THOSE ROWS where the INTEGRAND-FORM COLUMN has the VALUE "Rational Function." The RESULT of this query will be those rows in the database table which have "Rational Function" in the INTEGRAND FORM COLUMN. In other words the DATABASE QUERY "returns" a "part" of the total DATABASE TABLE. It is then immediately seen that the ROWS that the DATABASE QUERY returned all deal with the technique of PARTIAL FRACTIONS. Thus the DATABASE QUERY would "SUGGEST" using the method of PARTIAL FRACTIONS for solving EXAMPLE 2.B. The STEPS for performing the PARTIAL FRACTION technique is also presented in the table. ===============THE TECHNIQUE DATABASE========================== SYMBOL METHOD INTEGRAND CASES STEP-1 STEP-2... ------ ------ FORM----- ----- ------- ------- ------ ------ --------- ----- ------- ------- F-1-n Partial Rational Degree 1 Integrand Find the Fractions Function No multiplicity Equals constant multiples of multiples linear of linear reciprocals inverses F-1-m Partial Rational Degree 1 ..... Fractions Function Multiplicity ..... Present T-o Trig Rational Powers of ...... Int Functions sin and cos of sin & are both odd cos S-t-1 Trig Rational The quadratic Set x = Sub Functions function can sec A of x and be reduced to a the form x^2-a^2 ... Quadratic =========================TABLE 5.2============================= SECTION 6:THE TECHNIQUE DATABASE:A fragment of the 6-column TECHNIQUE database is presented in TABLE 5.2. The structure of the database is transparent enough. We review column meanings and give more complete descriptions of column contents. =============COLUMNS 1(SYMBOL) and 2(METHOD)===================== T S F R P G B L Trig Trig Partial Radicand Parts General Basic Linearity, Int Sub Fractions Sub List Expansion, Factor ====================TABLE 6.1============================== INTEGRAND FORM: The most challenging part of constructing the DATABASE is to give proper, concise rigorous definitions of the INTEGRAND FORMS associated with each integration technique. We use the succinct notation of STEIN & BARCELLOS: Let R(,) denote a rational function of its arguments. Then the main METHODS of tables 5.2 and 6.1 apply when the INTEGRAND FORM has the functional forms presented in table 6.2. ================COLUMNS 1(SYMBOL) and 3(INTEGRAND FORM)========= T S F R P G B L R(sin,cos) R(x,Quadratic) R(x) R(x,n-th root) x^n*f(x) ff' x^n. of ax+b e^x*trig f/f' 1/x Inv trig ... e^x =================TABLE 6.2======================================= CASES: "PARTS" problems, for example, typically fall into one of three classes: (i) a polynomial power times a trig or exponential function, (ii) an exponential function times a trigonometric function or an (iii) inverse function (of trigonometrics or an exponential).Similarly PARTIAL FRACTION techniques have 4 cases depending on the degree and multiplicity of denominator factors. Two comments are in order: First we have used the SYMBOL column as the PRIMARY KEY (that is, UNIQUE IDENTIFIER) in table 5.2. This PRIMARY KEY allows a short easy way to name each row, in the TECHNIQUES DATABASE, uniquely. Second, because of the CASES, the SYMBOL column must be modified by placing ATTRIBUTE SYMBOLS after the MAIN SYMBOLS T,S,F,R,P,G,B or L. Thus F denotes the PARTIAL FRACTION method and F-1-m denotes the PARTIAL FRACTION METHOD with the CASE of degree "1" and "Multiplicity" in the denominator. To ease readability we however have deleted these extra attribute symbols from the main SYMBOL in tables 6.1 and 6.2. STEP-1,STEP-2...: The remaining columns give the STEPS for each METHOD. To ease readability we have entered steps for only one METHOD. It should be clear how to enter it for the other METHODS. One point is worth emphasizing here: There is no one TECHNIQUES DATABASE. Rather each instructor and student will use the column labels and develop their OWN VERSION. As a simple example of how versions would differ, the number and contents of the STEPS columns will depend on the individual instructor or student. 7:USE OF ADVANCED DATABASE THEORY: The TECHNIQUES DATABASE of table 5.2 when fully developed would be quite big and difficult for a student (or instructor) to memorize for a course or test. In other words, the database would be intractable. A solution to this problem of intractability would be to breakup or DECOMPOSE the database into smaller more tractable databases. The challenge in such a decomposition is not to LOSE any information from the original database. Database theorists call such a DECOMPOSITION, if it exists, LOSSLESS. We give details. Table 6.1 contains COLUMNS 1 and 2 of table 5.2 and is itself a table, the METHODS database. Similarly Table 6.2 contains COLUMNS 1 and 3 and forms an INTEGRAND-FORM database. We could also form a CASE database consisting of columns 1 and 4 and a STEP database consisting of columns 1,5,6 and 7. Note how these four databases all have column 1--the SYMBOL column--the PRIMARY KEY. If we now JOIN the four databases, that is, if we form all 7-column rows made up by selecting rows from each of the four databases with the same primary key value then it should be clear that we get back the original TECHNIQUES database. Consequently we say the four databases form a LOSSLESS JOIN DECOMPOSITION of the original TECHNIQUES database. Note that the 4 databases used to make this join are all more tractable then the original database. The above LOSSLESS JOIN DECOMPOSITION allows us to articulate many pedagogical insights. For example: (i) Most class and textbook time is spent teaching the STEP database; (ii) "A" students (vs "B" students) probably spend more time studying the CASE database; (iii) Since all textbooks must teach the STEP database a "good" textbook is characterized by its also emphasizing the "recognition" methods contained in the INTEGRAND-FORM database. 8.CONCRETE SUGGESTIONS FOR CALCULUS TEACHERS: We recommend the following to instructors wishing to use our methods: 8.A: FOCUS REMEDIATION ON YOUR INTERMEDIATE REMEDIAL STUDENTS: These can be identified by diagnostic quizzes after about two weeks. The quizzes should contain "integrate" and "integrate using the following method" questions. The INTERMEDIATE REMEDIAL students can be recognized by their ability to answer the "integrate using" questions but not the "integrate" questions. 8.B: USE A DOUBLE ASSESSMENT VEHICLE: The methods of this presentation requires that students first master EACH PARTICULAR METHOD (by knowing the contents of the STEPS database). Only then is it meaningful to learn how to correctly select appropriate techniques for an arbitrary problem. The best way to implement this two step learnings is to have two assessment vehicles, say quizzes and tests. On the quizzes only "integrate using" problems are given while tests present general problems. This assessment method encourages a two step learning process. 8.C: TEACH/TEST THE FOUR DATABASES: In addition to the traditional "integrate" problems (whose solution requires the STEP database) it would also be useful to have "Which technique should be used here" problems (whose solution requires the INTEGRAND database) as well as "which case is this" problems (whose solution requires the CASE database). 8.D: TEACH DATABASE THEORY: Say one or two lectures should be devoted to DATABASE THEORY. More specifically students should obtain minimal practice in going to TABLES selecting ROWS and COLUMNS based on criteria and forming a NEW TABLE. It is however not necessary to teach them database theory language.