Re: Calculator Dependence.
Dave L. Renfro
Gary Schnabl wrote:
http://groups.google.com/group/misc.education/msg/bed857ef919e3d8f
> I still remember those big trig and log tables in
> high school/college (and beyond) and am glad to not
> need them short of a power failure for my computer
> or missing my "solar" powered scientific calculator.
I've always thought it was interesting that texts
continued to include these tables throughout the 1980's
(and maybe even into the early 1990's), long after
no one was ever using them. Well, *almost* no one.
I sometimes used them to help students understand
certain concepts in a more concrete fashion. For
example, properties of logarithms such as
log(ab) = log(a) + log(b) and log(a^b) = b*log(a)
become more personalized when you spend a few
minutes using log tables to perform a numerical
computation such as the fifth root of 3500 times
the cube root of 260. Also, the rule that
10^log(a) = a gets reinforced when you're using
the antilog tables.
I once gave a talk at a T^3 (Teachers Teaching
with Technology) conference whose theme was a
little antithetical to the party line. One part
of my talk dealt with the topic "What has Technology
Subtracted?", and before giving a number of examples
(such as logarithm tables), I said:
Technology has made a number of previously taught
topics obsolete. Some of these obsolete topics were
useful for their reinforcement of facts that we
wanted students to know very well, some of these
topics were useful for their development of mathematical
thinking, and some of these topics do not appear
to have any significant merit beyond their necessity
before present technology.
I focused on these two areas:
A. Topics that Reinforce Certain Facts
There are several topics that once reinforced certain
mathematical facts, but no longer do so in the presence
of current technology. The impact of these omissions
has less effect on the teaching of gifted students
than on the teaching of average ability students,
since the former require less drill and practice.
Nonetheless, I often encounter very talented students
who display some striking weaknesses. I suspect that
these weaknesses are primarily caused by less drill
due to the omission of these topics.
I discussed "Numerical calculations with logarithm
tables", "Calculating square roots", "Arithmetic
operations with rational numbers", "Estimation,
consistency, and exact values", and then posed
the question: "Do we want students to be as proficient
with these skills as they once were? For those
skills in which the answer is "YES", what should
we include in the curriculum to give our students
the necessary drill?"
B. Topics that Develop Mathematical Thinking Skills
The presence of graphing and symbolic calculators has
turned some problems whose solution once involved higher
order thinking skills into drill problems.
I discussed "Curve sketching" and "Algebraic and numerical
computations". In the latter, I discussed the re-enforcement
of algebraic facts in simplifying numerical computations,
such as rewriting 5*7*11*2 as (5*2)*(7*11) and (3.8 x 10^-7)/5
being evaluated via (380 x 10^-9)/5 = (10)(38)/5 x 10^-9
= (2)(38) x 10^-9 = [(2)(30) + (2)(8)] x 10^-9
= 76 x 10^-9 = 7.6 x 10^-8). Then I posed the question:
"How do we compensate for the absence of topics
that once required higher order thinking skills,
but no longer do so?"
> But again, I was schooled in an era when we really
> learned arithmetic and reading text-based material,
> when phonetics education was universal, when being
> taught axiomatic geometry and calculus courses in
> high school/college were the norm, and when we were
> actually taught our non dumbed-down courses by
> competent teachers who weren't constantly bitching
> about not being overpaid enough (and, fortunately
> for those teachers, were not universally dissed).
Without getting into the other things you said,
I disagree with the part about calculus. In the
last 10 or 20 years many, many more U.S. high school
students (as a percentage of the population, even)
have taken calculus than in the decades before this.
My high school didn't even offer calculus (well into
the early 1980's, I think), but they do now. I think
you'll find the number of high school students taking
AP-calculus is far higher in recent years than it
was 30 or more years ago, before calculators were
in general use. Now, the question of whether this is
a good thing or not is another issue, one that is
often brought up by college faculty who have to
deal with students that took calculus in high school
and then wind up not_even_placing_into_precalculus.
By the way, I thought Cary Kittrell's idea (in this
thread) of a calculator that forces students to
determine the power-of-ten order of a computation was
a great idea. I think it has significant potential
(financial and educational), and she should
consider approaching a calculator company about it.
Dave L. Renfro
Dave L. Renfro
> One part of my talk dealt with the topic "What has
> Technology Subtracted?", and before giving a number
> of examples (such as logarithm tables), I said:
>
> Technology has made a number of previously taught
> topics obsolete. Some of these obsolete topics were
> useful for their reinforcement of facts that we
> wanted students to know very well, some of these
> topics were useful for their development of mathematical
> thinking, and some of these topics do not appear
> to have any significant merit beyond their necessity
> before present technology.
>
> I focused on these two areas:
>
> A. Topics that Reinforce Certain Facts
[snip]
> B. Topics that Develop Mathematical Thinking Skills
For completeness, I should add that I also discussed
C. Topics that can be left out
and then I went through the same three areas in the second
part of my talk, "What has Technology Added?". An example
of topic C (generalized to things that will probably
be irrelevant several decades from now as technology
becomes "smarter" and more user-friendly) is all the
calculator-dependent graphing menu knowledge a student
needs to be familiar with ("trace", turn off STAT-PLOTS
when graphing y = f(x), various ZOOM options, etc.).
Dave L. Renfro
Gary Schnabl
>
> Without getting into the other things you said,
> I disagree with the part about calculus. In the
> last 10 or 20 years many, many more U.S. high school
> students (as a percentage of the population, even)
> have taken calculus than in the decades before this.
> My high school didn't even offer calculus (well into
> the early 1980's, I think), but they do now. I think
> you'll find the number of high school students taking
> AP-calculus is far higher in recent years than it
> was 30 or more years ago, before calculators were
> in general use. Now, the question of whether this is
> a good thing or not is another issue, one that is
> often brought up by college faculty who have to
> deal with students that took calculus in high school
> and then wind up not_even_placing_into_precalculus.
>
AP calculus need not be axiomatically based and usually isn't. The bloom
is off the AP craze so much so that better colleges like Harvard now do
not accept any AP scores lower than 5. Still, many colleges today accept
AP scores of 3, 4, or 5. There's a huge difference between a 3 and a 5.
An honors curriculum usually teaches more depth and retention than the
"mile-wide, nanometer-deep" AP curricula.
Those HS students taking the lower quality AP calculus (especially when
not axiomatic based) do not receive the mathematical education you seem
to think they do. And because of AP, they do not take the higher quality
courses they would have otherwise taken. Therefore, they're worse off
academically. But what the hell--getting a good education doesn't matter
to many of today's crowd...
Lehigh doesn't seem to have high regard for HS AP courses: "Placement
and Advanced Placement Credit & Calculus Readiness Test"
"There is a big difference between calculus study at Lehigh and calculus
at most high schools. A solid high-school precalculus course is
necessary background for calculus at Lehigh. Students need a strong
foundation in functions (forms, graphs, roots) and trigonometry to
really thrive in calculus. Most students who take calculus in high
school are accustomed to using a graphing calculator. Calculators are
not permitted in Lehigh calculus classes. Lehigh has very high
standards, and calculus students are expected to learn calculus without
relying on calculators. Many students find a summer course in calculus
at a local community college to be helpful in bridging from high school
mathematics to Lehigh calculus."
"We recommend that students consider beginning Lehigh calculus a
semester below their advanced placement. If you have credit for Math 21
and are eligible for Math 22, consider taking Math 21; if you have
credit for Math 21 and Math 22 and are eligible for Math 23, consider
taking Math 22. You will relinquish some or all of your AP credit, but
experience has shown that many AP courses do not provide adequate
preparation for calculus at Lehigh."
BTW, Lehigh accepts AP grades 4 or 5.