Another paper I had them write was about quadratic equations and quadratic
functions. They had to solve some quadratic equations analytically as well
as graphically, discuss the characterics of the graph of a quadratic
function, what the graph looks like, the effect of a, b, and c on the
parabola, how to recognize the formula for a quadratic equation, how to
distinguish both analytically and geometrically a quadratic function from
other polynomial functions, how the parabola is related to the solution of
the "sister" quadratic equation, etc. How can you tell from solving a
quadratic equation if the related parabola has 1, 2, or no x-intercepts?
They also had to include two real-life examples of parabolas that we had
not discussed in class. I made a grid for grading on this one also that
they got with their instructions for the paper. I had them do some
attached drill/practice problems--identifying quadratic functions, linear
functions, etc, solving equations (graphically and analytically),
identifying nature of roots, etc. The again did not have "pretty number"
coefficients and did not necessarily show up in a 10x10 window.
I gave these papers instead of major tests and gave more frequent quizzes
that were skills-based , short (generally 2 questions) and tricky.
Students who missed any part of any of a question had to come to me for a
replacement question that was also tricky but with a different trick.
The quizzes at first scared the students as most of them got 0 points the
first time. However, the "re-takes" were very effective. It got students
to analyze what they missed and clear up misconceptions. That was the
first time that I realized that most students do not make "silly mistakes."
They usually think the way they did the problem was correct. For example,
I had a student who correctly used distributive law, cleared fractions and
several other steps to solve an equation and correctly came down to x =
-3/(-4) and simplified the answer to -3/4. I was tempted to back down on
my all or nothing policy for points. However, in going through the quiz
with the student, I said you just made a mistake in the last step. She
said, "but there is a minus sign, the answer is negative isn't it?" Not
giving the points gave me the opportunity to clear up a misconception.
The papers were very effective in students learning and retaining what they
knew. For a measuring device, I gave a traditional final exam that I had
given before. I also gave the same final in another class I taught more
traditionally. In the two classes, I had about the same drop rate. In my
traditional class, 19 out of 32 students took the final and in the writing
approach class, 20 out of 32 students took the final. I had about the same
average score in the two classes. 70.3 in one class and 70.4 in the other.
The standard deviation was much higher in the traditional class. I had
two students in the traditional class who made 10 and 17. The low score in
the writing approach class was 58.
Another interesting statistics is that 19 of the 20 people who took the
final exam in the writing-approach class were women. Changing the teaching
method did have an impact on what group of students succeeded. It seems
that if we want to find ways to teach mathematics to groups of students who
have traditionally been unsuccessful in learning mathematics, maybe we need
to figure out ways to teach other than how we learned since we know large
groups of students fail from the methods we, the most educated, learned.
Writing can be a very effective way to teach mathematics. It can be a lot
of work on the instructor. Making out the grids is helpful. Having
students "grading" each other's drafts is a time saver for the instructor
and a learning experience for the students.
Martha
>
> I'd suggest simulations and global modeling as a place where
> writing and algebra intersect. You have those flow charts of
> relationships like in the back of a SimCity book, showing
> how a tick up in taxes nets more for police but raises
> citizen dissatisfaction and perhaps lower population density
> in higher taxed areas and... SimAnt or SimEarth are other
> places to start (even if you don't play the game, the ideas
> are accessible).
>
> Talk about terawatts from the sun as an energy source and
> Earth as a spherical "motherboard" with ecosystemic circuitry
> interpenetrated by human economic circuitry (econosphere
> embedded in the ecosphere) and suggest functions and relations
> tie the trends together in a kind of Gaia thing which may or
> may not be of sufficient integrity to gravitate towards a
> sustainable situation for Earth's billions (Club of Rome
> suggested it was hopeless but we've had a lot of rethinking
> since then).
>
> You can toss in functions of arbitrary complexity, making
> wild guesses as to what's exponential to what. Just make
> sure your variables represent real quantities (population,
> food supply, energy income...) and you'll have set the
> stage for a lot of co-variable relationships (direct,
> inverse). Stick these functions in boxes (only a few
> variables per function) and get some arrows going from
> one to another (e.g. increase health -> increase longevity
> -> family security -> lower birth rate per family -> lower
> growth rate).
>
> Use the web to add reality to your assumptions or to
> discover that your global model needs rethinking.
>
> Just a suggestion. Could be fun.
>
> Kirby
>
>
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