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.