Re: Meaningless problems in algebra texts

[John Clement <clement@HAL-PC.ORG>]

The original question was about the problem that the book had not enough
formalism to help the students.  Most books have exactly the opposite
problem.  The "best" solution is to take an inquiry approach, which
apparently the first book has attempted.  Unfortunately it has confused
"free discovery" with inquiry.  Students can be given definitions after they
have been given guided exploration.  In other words it should follow a
learning cycle approach exploration, term definition, application which was
found by Karplus, Renner, and Lawson (still alive and still writing papers
on the subject a AZ State) to be the most effective approach.

But much more is needed.  Multiple representations have to also be used to
provide multiple connections in the student's mind.  They should have
pictorial, verbal (written), graphical, and equations simultaneously.
Usually these are present in some small amounts in the texts, but students
are typically only asked to produce or manipulate equations.  Students also
have to produce these representations in addition to seeing them.  Math
teachers concentrate almost exclusively on manipulations and formalism.  I
have seniors who came through geometry unable to use a protractor or a
ruler.

I disagree violently with the idea of the "magic X".  This sets up the
student for failure later by making X and Y the only variables.  Variables
should be introduced first as words such as money or time... then later
shorthand single letters could be used.  The widespread almost exclusive
usage of X, Y sets up students to not use other variables when they get to
chem. and physics.  Just inserting some equations involving t or V or
whatever doesn't help later.  The fact that these variables are introduced,
but students are never asked to attach any meaning to them, just makes them
random gobbledygook.  Learning some elementary computer programming can be
helpful here if it is taught in such a way as to promote metacognition.

The real problem has been alluded to in this message.  Math is introduced as
a formal system to students who are in general below the formal operational
level.  As a result they gain no understanding of what they are doing it is
just meaningless manipulation.  All of the math techniques should be
introduced in the context of some type of physical examples.  This could be
financial, physics, biology, chemisty, history, even English.  According to
Shayer and Adey students who are at the concrete operational can not
understand 3 variable equations such as F=ma, P=F/A ...  Since the majority
of freshmen are probably concrete operational, algebra at that time is
impossible to understand.

I realize that this does not answer the question, but now I will give a
reference.  www.project2061.org has been reviewing texts in science and math
for both accuracy and pedagogy.  You might find a good alternative from
their list.  Of course this problem is not unique to math.  If you look at
the Hubisz report you will find no middle school science books by major
publishers being acceptable.  Just yesterday I noted that Prentice Hall
physics science text said that chemical bonds store energy.  This is not the
most outrageous example, but it is certainly one thing that must be changed
in a large number of texts including college texts.

John M. Clement
Houston, TX

> I am resending this message because it may not have been transmitted the
> first time.
>  There is one solution that can halp to make more sense out of the
>  typical algebra texts that are full of mostly "meaningless problems".
> Why
>  not teach algebra together with physics in the junior or senior year of
>  high school?
>  It certainly makes sense from a logical view.  I suppose that the
>  major problem in doing so is that the mathematics teachers are generally
>
>  unable to teach physics and there are too few physics teachers
> available'
>
>  Herb