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Notebook[{
Cell[CellGroupData[{
Cell[TextData[{
"Calculus&",
StyleBox["Mathematica", "SBO"],
": \nInstructors' Perspectives on Continuing Controversies"
}], "Title",
CellTags->"0.1"],
Cell[TextData[{
"Concentrating on the computer\[Hyphen]based course, Calculus&",
StyleBox["Mathematica", "SBO"],
", the author considers the merits and addresses the concerns raised by \
critics of technology\[Hyphen]based instruction."
}], "Subtitle",
CellTags->{"S0.0.1", "1.1"}],
Cell["by Judy Holdener", "Subsubtitle",
CellTags->{"S0.0.1", "1.2"}],
Cell[TextData[{
"In the mid\[Hyphen]eighties the National Science Foundation launched a \
major initiative to improve the quality of calculus instruction at the \
national level. This initiative resulted in the development and \
implementation of numerous innovative calculus projects. Many of these \
projects incorporate technology, and now, over 10 years later, math \
departments world\[Hyphen]wide are using graphing calculators and/or \
computers in their calculus courses. According to a survey by the MAA [",
StyleBox["Tucker and Leitzel 1995, pp.29\[Hyphen]30", "TI"],
"], 80% of responding institutions make substantial use of calculators, and \
over half use computers in some way. Essentially all reform efforts appear to \
use technology. \nAmidst the increasing use and enthusiasm for computers in \
the classroom, however, not all mathematicians proclaim the joys of \
technology in education. As illustrated in a \"mini\[Hyphen]forum\" of four \
articles in the ",
StyleBox["American Mathematical Monthly ", "TI"],
"discussing calculus reform [",
StyleBox[
"Askey 1997; Kaput 1997; Knisely 1997; Ostebee and Zorn 1997", "TI"],
"]",
", mathematicians do not agree on the merits of technology in undergraduate \
math education. One article discusses the \"modern perspective of calculus\" \
stating: "
}], "Text",
CellTags->{"S0.0.2", "1.3"}],
Cell[TextData[{
StyleBox[
"\"Technology should be a tool we welcome with ex\[Hyphen] citement\
\[LongDash]no more rigging problems so the algebra comes out right", "TI",
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(TraditionalForm\`\(\[Ellipsis]\ \)\)], "InlineFormula",
FontColor->RGBColor[1, 0, 0],
CellTags->"S0.0.1"],
StyleBox[
". Our debate should be about how we should use technology, not if we \
should use technology.\" ", "TI",
FontColor->RGBColor[1, 0, 0]],
"[",
StyleBox["Knisley 1997, p. 726", "TI",
FontColor->RGBColor[1, 0, 0]],
"]",
StyleBox[" ", "TI",
FontColor->RGBColor[1, 0, 0]],
StyleBox[" ",
FontColor->RGBColor[1, 0, 0]]
}], "Abstract"],
Cell["\<\
Another author expresses his concern that technology will do more harm than \
good: \
\>", "Text"],
Cell["\<\
\"Technology has a place in mathematics instruction, but it needs \
to be used carefully. Until much more is learned about the drawbacks, it \
should not be pushed too much.\" [Askey 1997, p. 742] \
\>", "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
"\nIn this article we address some of the concerns regarding technology\
\[Hyphen]based education. We concentrate on the ",
StyleBox["Mathematica\[Hyphen]", "TI"],
"based course, Calculus&",
StyleBox["Mathematica", "TI"],
", developed by Uhl, Porta, and Davis [",
StyleBox["Davis, Porta and Uhl 1994", "TI"],
"]. We will focus primarily on learning issues. Our discussion is based on \
the compilation of opinions and findings of many experienced (past, as well \
as current) instructors of Calculus&",
StyleBox["Mathematica", "TI"],
". More specifically, instructors were asked, in light of their experience \
with Calculus&",
StyleBox["Mathematica", "TI"],
", to address issues such as proof\[Hyphen]abuse and students' lack of hand \
skills. For results of comparative studies done on Calculus&",
StyleBox["Mathematica", "TI"],
", the reader is referred to [",
StyleBox["Alarcon and Stoudt 1997", "TI"],
"], [",
StyleBox["Crocker 1991", "TI"],
"], [",
StyleBox["Holdener 1997", "TI"],
"], [",
StyleBox["Lefton and Steinbart 1995", "TI"],
"], and [",
StyleBox["Porzio 1995", "TI"],
"]",
". "
}], "Text"],
Cell[CellGroupData[{
Cell[TextData[{
"CALCULUS&",
StyleBox["MATHEMATICA", "TI"],
": BACKGROUND INFORMATION"
}], "Section",
CellTags->{"S0.0.2", "2.1"}],
Cell[TextData[{
"Calculus&",
StyleBox["Mathematica", "TI"],
" is a computer\[Hyphen]based course which uses ",
StyleBox["Mathematica", "TI"],
"'s symbolic, graphic, numeric, and word\[Hyphen]processing capabilities to \
illustrate the ideas of calculus. The \"textbook\" consists of a series of \
interactive ",
StyleBox["Mathematica", "TI"],
" notebooks, and students learn calculus by working through tutorial \
problems, experimenting with \"built\[Hyphen]in\" ",
StyleBox["Mathematica", "TI"],
" routines, and explaining graphical and computational output. Each lesson \
follows the same format with four components: "
}], "Text",
CellTags->{"S0.0.2", "2.2"}],
Cell[TextData[{
"\[FilledSmallSquare] The ",
StyleBox["Basics", "TI"],
" section introduces the fundamental ideas in the lesson by presenting \
problems along with their electronic solutions. Presentations are often \
visual in nature, and students are encouraged to experiment with the problems \
by changing functions and numbers and rerunning the ",
StyleBox["Mathematica ", "TI"],
"code. "
}], "Notes",
CellTags->{"S0.0.2", "2.3"}],
Cell[TextData[{
"\[FilledSmallSquare] The ",
StyleBox["Tutorial", "TI"],
" section, like the ",
StyleBox["Basics", "TI"],
" section, presents problems and solutions in parallel. ",
StyleBox["Tutorial", "TI"],
" problems concentrate on techniques and applications related to the basic \
ideas. "
}], "Notes",
CellTags->{"S0.0.2", "2.4"}],
Cell[TextData[{
"\[FilledSmallSquare] The ",
StyleBox["Give it a Try", "TI"],
" section is the heart of the course providing a list of problems to be \
solved by the student. Like the ",
StyleBox["Tutorial", "TI"],
" section, problems are often application\[Hyphen]based, incorporating \
models from economics, physics, chemistry and biology. Information needed to \
solve the problems are found in the ",
StyleBox["Basics", "TI"],
" and ",
StyleBox["Tutorial", "TI"],
" sections. "
}], "Notes",
CellTags->{"S0.0.2", "2.5"}],
Cell[TextData[{
"\[FilledSmallSquare] The ",
StyleBox["Literacy Sheets", "TI"],
" present problems to be worked out with paper and pencil. This section is \
typically completed after students have seen the concepts with the computer. \
"
}], "Notes",
CellTags->{"S0.0.2", "2.6"}],
Cell[TextData[{
"Although format may vary with the instructor, most Calculus&",
StyleBox["Mathematica", "TI"],
" classes are conducted in a computer lab. Typically, every third or fourth \
session meets in a classroom for a \"traditional\" chalkboard discussion. "
}], "Text",
CellTags->{"S0.0.3", "2.7"}]
}, Open ]],
Cell[CellGroupData[{
Cell["Common Concerns", "Section",
CellTags->{"S0.0.3", "3.1"}],
Cell[TextData[{
"Opponents of technology\[Hyphen]based instruction cite a wide range of \
concerns, but they are generally worried about lowering standards. In the \
context of Calculus&",
StyleBox["Mathe\[Hyphen] matica", "TI"],
", lowered standards are often attributed to one or more of the following \
three categories: gadgetry over intellect, \"proof abuse\", and the students' \
lack of necessary hand skills. We will address these concerns based on the \
reponses of the Calculus&",
StyleBox["Mathematica", "TI"],
" instructors. "
}], "Text",
CellTags->{"S0.0.3", "3.2"}],
Cell[CellGroupData[{
Cell["Gadgetry over Intellect", "Subsection",
CellTags->{"S0.0.3", "3.3"}],
Cell["\<\
Instructors were asked to respond to the following characterization \
of this concern: \
\>", "Text",
CellTags->{"S0.0.3", "3.4"}],
Cell[TextData[{
StyleBox[
"\"Computers reinforce the fascination with gadgetry, as opposed to \
intellect", "TI"],
Cell[BoxData[
\(TraditionalForm\`\(\[Ellipsis]\ \)\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
". Most software is based on immediate gratification and does not encourage \
sustained mental effort. While physically playing an active role, the pupil \
is intellectually passive, and has little opportunity to be creative; that \
is, the pupil is programmed to follow a path already laid out in detail by \
others.\" ", "TI"],
"[",
StyleBox["Koblitz 1996, p. 13", "TI"],
"]",
StyleBox[" ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Response", "TI"]], "Subsection",
CellTags->{"S0.0.3", "3.5"}],
Cell[TextData[{
"A majority agreed that \"most software\" had the downfalls Koblitz states \
but took exception in the case of Calculus&",
StyleBox["Mathematica", "TI"],
". "
}], "Text",
CellTags->{"S0.0.3", "3.6"}],
Cell[TextData[StyleBox[
"\"I think I agree with the letter of the statement, but probably not the \
spirit. The item I want to dissect is most software. Most software is junk. \
Yawn. But Koblitz does provide nice criteria for good software: it requires \
sustained intellectual activity and creativity. I am not familiar with all \
the available software, but Calculua&Mathematica and Project CALC make \
excellent use of technology. I haven't seen a supplement to a traditional \
text that makes decent use of technology. In fact, let's use these criteria \
to evaluate the traditional calculus curriculum taught using a traditional \
lecture. I don't think it fares particularly well. The students aren't even \
physically active.\" \n\n(David Olson, Michigan Tech.)", "TI"]], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["\n", "TI"],
"Several instructors emphasized the teacher's role in the successful \
implementation of technology. "
}], "Text"],
Cell[TextData[{
"\n",
StyleBox[
"\"I think this depends on the way that a particular teacher chooses to use \
the technology. If you make an assignment which requires little more than \
clicking, cutting, pasting, and printing, I agree with Koblitz. But I don't \
blame the computer; I blame the teacher. The computer should be used as a \
tool to help better understand mathematics (and perhaps better understand the \
computer). It is not a panacea. As always, the amount of student intellectual \
activity is determined by the teacher's expectations. If you ask the student \
to do an intellectually demanding project or assignment and then provide \
guidance with technological as well as mathematical obstacles, then I \
disagree with Koblitz. Such assignments are hard to craft and can take \
several iterations. This is the challenge of good teaching, and I think the \
Calculus&Mathematica materials have a lot of good starting points for \
precisely these kind of assignments. With proper guidance, I don't believe \
that using the computer will stifle student creativity. In fact, its novelty \
can provide a jolt of energy which could even stimulate creativity if \
properly harnessed. But, of course, it depends on the teacher.\" \n\n(Lew \
Lefton, University of New Orleans)", "TI"]
}], "Abstract",
FontColor->RGBColor[1, 0, 0]]
}, Open ]],
Cell[CellGroupData[{
Cell["\"Proof\[Hyphen]Abuse\"", "Subsection",
CellTags->{"S0.0.3", "3.7"}],
Cell["\<\
Proof\[Hyphen]abuse occurs when heuristic arguments replace precise \
proofs [Wu 1997]. In his paper titled \"The Mathematics Education Reform: \
What is it and why should you care?\", Wu writes: \
\>", "Text",
CellTags->{"S0.0.3", "3.8"}],
Cell[TextData[{
StyleBox[
"\"The cavalier manner in which the reform texts treat logical argument is \
nothing short of breath\[Hyphen]taking. Heuristic arguments are randomly \
offered or withheld and, in case of the former, whether these are correct \
proofs or not is never made clear. Rather than being the underpinning of \
mathematics, logical deduction is now regarded as at best irrelevant.\" ",
"TI"],
"[",
StyleBox["Wu 1997, pp.3\[Hyphen]4", "TI"],
"]",
StyleBox[" ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
"\nWu continues the debate in the light of technology. In reference to \
Calculus&",
StyleBox["Mathematica", "TI"],
" he writes, "
}], "Text"],
Cell[TextData[{
StyleBox[
"\"Problems of a different kind arise when heuristic arguments based on \
appeals to technology are made with increasing frequency: the computer or \
calculator begins to assume the role of arbiter of pure reason.\" ", "TI"],
"[",
StyleBox["Wu 1996, p. 1533", "TI"],
"]",
StyleBox[" ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
"\nWu then cites several examples to illustrate his point. One of his \
examples comes from Calculus&",
StyleBox["Mathematica'", "TI"],
"s coverage of approximations. Wu writes: "
}], "Text"],
Cell[TextData[{
StyleBox[
"\"On p. 31 of Approximations of Calculus&Mathematica, the discussion of \
finding the power series expansion of ", "TI"],
Cell[BoxData[
\(TraditionalForm\`\(1/\) \((1 - x)\)\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
" goes like this: One uses the computer to print out the first 50 terms, \
and of course the print\[Hyphen]out reads:", "TI"],
Cell[BoxData[
\(TraditionalForm\`\[Sum]\_\(n = 0\)\%50 x\^n\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
". The comment that follows is (p.32): Ain't no doubt about it. The \
expansion of ", "TI"],
Cell[BoxData[
\(TraditionalForm\`f\ \ \([x]\) = \(1/\) \((1 - x)\)\)],
"InlineFormula",
CellTags->"S0.0.3"],
StyleBox[" in powers of ", "TI"],
Cell[BoxData[
\(TraditionalForm\`x\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[" is ", "TI"],
Cell[BoxData[
\(TraditionalForm
\`1 + x + x\^2 + x\^3 + \[CenterEllipsis] + x\^k +
\[CenterEllipsis]\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
". By using computer print\[Hyphen]outs to replace the elementary \
derivation of the geometric series, Calculus&Mathematica and other like\
\[Hyphen]minded texts send out the unmistakable message to K\[Hyphen]12 that \
learning about the geometric series is no longer something of consequence.\" \
", "TI"],
"[",
StyleBox["Wu 1997, pp. 3\[Hyphen]4", "TI"],
"]",
StyleBox[" ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Response", "TI"]], "Subsection",
CellTags->{"S0.0.3", "3.9"}],
Cell[TextData[{
"Calculus&",
StyleBox["Mathematica", "TI"],
" veterans responded similarly to this criticism. Agreeing that Calculus&",
StyleBox["Mathematica", "TI"],
" didn't always provide complete proofs to mathematical statements used in \
the course, they indicated that the omission of complete proofs is sometimes \
pedagogically sound. Although a few instructors expressed concern regarding \
the preparation of math majors, most shared the belief that students at the \
introductory calculus level often gain more from heuristic arguments than \
from formal proofs. "
}], "Text",
CellTags->{"S0.0.3", "3.10"}],
Cell[TextData[StyleBox[
"\"For everything there is a season, and a time to every purpose under \
heaven: a time for proofs, a time for heuristics, a time for noticing \
patterns, a time for finding exceptions, a time to test hypotheses on the \
computer, and a time to doubt the output. \n\nFirst semester calculus \
students come to us with little or no understanding of proof. Many do not \
even know that a few examples do not constitute a proof, let alone understand \
more subtle errors. We can teach them what it means to prove things, but we \
have to acknowledge that this will be a difficult and time\[Hyphen]consuming \
task, which will reduce our opportunity to teach other things. We cannot \
teach all of mathematics in one course or sequence; we have to decide what is \
worth our time, and let the rest go. Depending on the goals of the course, it \
may be more important to give students an intuitive understanding of the \
derivative and the integral of \"nice\" functions than to develop precise \
proofs of general cases or to examine pathological counter\[Hyphen]examples. \
Detailed proofs may appeal to mathematicians, but if the student is not \
functioning at the necessary level, time spent on proof is wasted. If it is \
truly important that the students understand a particular proof or group of \
proofs, the instruct or will have to start at the level where the students \
are, and take them to the level where they can grasp the proof. This may mean \
spending weeks on basic ideas of proof. \n\nI once taught a one\[Hyphen]week \
unit on these ideas, emphasizing, among other things, that examples cannot \
make a proof unless all possible cases are considered. We considered \
mathematical examples along with practical examples like: \"It rained last \
Monday. It rained last Friday. Does this prove that it rained every day last \
week? If it did rain every day last week, how could you prove that? How about \
last year? Could you prove that it always rains on Monday? If so, how? If \
not, why not?\" At the end of the unit I gave a quiz which included several \
\"proofs,\" all but one of which were invalid. I asked the students to \
identify the one valid proof, and find the errors in the invalid ones. A \
substantial portion of the class was not able to find the error in the \
\"proof\" which consisted of a short list of examples. If these are the \
students to whom Wu wants me to present many detailed proofs, then I think he \
is deluding himself about the value of this experience for the students.\" \n\
\n(Lisa Murphy, University of Illinois)", "TI"]], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["\n", "TI"],
"Regarding Wu's criticism of Calculus&",
StyleBox["Mathemat\[Hyphen] ica", "TI"],
"'s coverage of approximation, Sidney Graham writes, "
}], "Text"],
Cell[TextData[{
StyleBox[
"\"The objection to this is logically correct but pedagogically unsound. \
The only decent learning tool that most students bring to the study of \
infinite series is a rudimentary ability to recognize patterns. Wu wants to \
tell students that their intuition about patterns is wrong and thereby \
further undermine what little confidence they have learning this concept. \
This is unconscionable, unless he is also prepared to teach the students what \
they need (e.g. mathematical induction) to make their pattern guesses \
rigorous. Personally, I think at this level, it's better to help the students \
improve their pattern recognition, and save the rigor for some other time.\" \
\n\n(Sidney Graham, Michigan Tech.) ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]]
}, Open ]],
Cell[CellGroupData[{
Cell["Lack of Necessary Hand Skills", "Subsection",
CellTags->{"S0.0.3", "3.11"}],
Cell[TextData[{
"Perhaps the greatest area of concern with increasing use of technology is \
that students will leave calculus courses lacking the hand skills necessary \
for success in later coursework. Andrews explores the \"concerns about the \
tendency toward inbalance in the current reforms of mathematics instruction\" \
[",
StyleBox["Andrews 1997", "TI"],
"]",
". He poses five questions to the mathematics community, including: \"Are \
standards being maintained?\" and \"Has the use of calculators impaired basic \
skills?\" "
}], "Text",
CellTags->{"S0.0.3", "3.12"}]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Response", "TI"]], "Subsection",
CellTags->{"S0.0.3", "3.13"}],
Cell[TextData[{
"Student attainment of necessary hand skills was also the most cited \
concern among Calculus&",
StyleBox["Mathematica", "TI"],
" instructors. Most respondents indicated that they were careful to ensure \
that their students learned the basics. \nSome used gateway exams, others \
collected written assignments regularly, and virtually everyone tested basic \
skills using written examinations. Here are some responses: "
}], "Text",
CellTags->{"S0.0.4", "3.14"}],
Cell[TextData[{
StyleBox[
"\"It is up to me to craft the class so that this (lack of necessary hand \
skills) does not happen", "TI"],
Cell[BoxData[
\(TraditionalForm\`\(\[Ellipsis]\ \)\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
".I assign hand work and correct it, work on it in lecture, and put \
students up at the board, etc.\" \n\n(Elizabeth Covington, Casper College)\n\n\
\"I felt at the time that the students' hand skills suffered a little; that \
is why I believe that we need to take the approach that computation must be \
used wisely. Students can appreciate the idea that some things are not worth \
turning on the machine for, like solving simple equations, taking \
non\[Hyphen]chain\[Hyphen]rule intensive derivatives and integrals, etc. BUT, \
having taught calculus both ways, I also know that just because students can \
do chain rule problems quickly and correctly doesn't mean they can \"abstract\
\" the chain rule for arbitrary functions, ", "TI"],
Cell[BoxData[
\(TraditionalForm\`f(g(x))\)], "InlineFormula",
CellTags->"S0.0.3"],
StyleBox[
", as needed to understand and do proofs in differential equations. In \
fact, the students having intuitive understanding of the ideas but slower \
computational skills handle the abstraction better!\" \n\n(Holly Hirst, \
Appalacian State University)\n\n\"This is a question of some concern. We do \
not, in fact, know what the role of skill at hand computation is in the \
natural history of the mature mathematical mind. We do ensure that our \
students can find derivatives by hand by use of \"gateway tests.\" Otherwise, \
we will have to wait and see if the Calculus&Mathematica approach leaves \
things out of balance. If so, then readjustments will be necessary. We do not \
currently believe that our students leave our (Calculus&Mathema tica) courses \
at any significant disadvantage. \n\n(Lou Talman, Metro State College, \
Denver)", "TI"]
}], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["\n", "TI"],
"Many instructors felt that the gain in conceptual understanding and the \
exposure to interesting applications offered a preferable trade\[Hyphen]off \
to the concerns regarding hand skills. "
}], "Text"],
Cell[TextData[StyleBox[
"\"I believe students are better prepared with Calculus&Mathematica. Their \
perceived disadvantage pales compared with the fact that they understand what \
is going on. They will not remember calculus as one more course needed for \
graduation, but as an interesting scientific experience. The necessary hand \
skills are practiced and tested in the gateway exams. \n\n(Juan Manfredi, \
University of Pittsburgh)", "TI"]], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["\n", "TI"],
"On the other hand, at least one instructor felt that software packages \
such as ",
StyleBox["Mathematica", "TI"],
", can ",
StyleBox["improve", "TI"],
" students' skills in symbolic manipulation. For example, David Olson noted \
that students often misinterpret function notation in compositions such as ",
Cell[BoxData[
\(TraditionalForm\`\(c o s\) \((\(l n\) \((x)\))\)\)],
"InlineFormula",
CellTags->"S0.0.3"],
", and apply the product rule instead of the chain rule when \
differentiating. "
}], "Text"],
Cell[TextData[{
StyleBox[
"\"Perhaps students can learn to understand function notation by working \
with a computer program. For example, Mathematica uses \"", "TI"],
StyleBox["[", "MR"],
" ",
StyleBox["]", "MR"],
StyleBox["\" to enclose the argument of a function and \"", "TI"],
StyleBox["( )", "MR"],
StyleBox[
"\" for algebraic grouping. In order to use the program, students must \
learn to make the distinction between function notation and algebraic \
grouping. ", "TI"],
" "
}], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell["\<\
He goes on to say \
\>", "Text"],
Cell[TextData[StyleBox[
"\"Calculus&Mathematica might do more for a student's symbol skills than one \
would conclude based on superficial considerations. If students learn some \
key conceptual ideas, they might develop better symbol skills than the \
students who spent their entire course working on symbol skills.\" \n\n(David \
Olson, Michigan Tech.)", "TI"]], "Abstract",
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["\n", "TI"],
"Finally, many people indicated in their responses that more thought should \
be given to the issue of symbolic manipulation. For example, how much \
symbolic manipulation is necessary at the calculus level? How much is \
necessary to ensure preparation for future coursework?",
Cell[BoxData[
\(TraditionalForm\`\(\[Ellipsis]\ \)\)], "InlineFormula",
CellTags->"S0.0.3"],
"and future employment? ",
StyleBox["Which", "TI"],
" hand skills should be included in a calculus course? What should be left \
out? "
}], "Text"],
Cell[TextData[StyleBox[
"\"Before one can address the issue of necessary hand skills, it is necessary \
to define what they are; they will most definitely differ from person to \
person. C&M (Calculus&Mathematica) students may be at a disadvantage in later \
courses if the emphasis of these courses is proficiency at hand skills, \
although I believe they will do fine on everything but the more complex \
symbolic manipulations that do require lots of practice in order to know the \
right \"trick\" to use. I have no problem with my C&M students being at a \
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Olson, Don Porzio, Allan Struthers, Lou Talman, Lee Wayand. \
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Perspective. ",
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in K\[Hyphen]13 Math Education (Kindergarten through Calculus), ",
StyleBox["The Mathematical Intelligencer. ", "TI"],
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Course. ",
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" (to appear) \n\nPorzio, D. 1995. Effects of Differing Technological \
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Representations and Their Understanding of Calculus. ERIC database. \n\n\
Tucker, A. and leitzel j., Editors. 1995. ",
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End of Mathematica Notebook file.
***********************************************************************)