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Topic: upper middle schoolers that haven't yet mastered multiplication facts

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Subject:   RE: upper middle schoolers that haven't yet mastered multiplication facts
Author: Mathman
Date: Jan 27 2005
On Jan 26 2005, ihor wrote:
> Geri writes:

David replies:

Ihor chimes in:

Geri and David echo to
> interesting approaches to the same problem, but in my thinking they
> are vastly different. I was personally the beneficiary of the
> approach that Geri promotes and the victim of the kind that David
> espouses. And yet I dont mean to suggest that David's suggestions
> cant be effective.  With the proper support students can benefit
> from the more disciplined approach that is neccessary to succeed. I
> didnt master my multiplication facts until I was a teacher (really,
> that 7 x 8 was always a killer for me. I still sometimes think its
> 54). But it didnt keep me from getting a math degree. What finally
> got me to shed some perspiration towards learning hard things was
> some inspiration. We know that perspiration resulting from practice
> is important. But how that gets motivated makes all the difference.
> By the way, Geri, the math section in the new book from NAP "How
> Students Learn: History, Mathematics, and Science in the Classroom"
> is all the ammunition you will ever need to be convinced of the
> richness of the path you chose.


Ihor, no disrespect, but you seem to be dwelling on your personal experiences
rather than on what might be best for the larger majority of students.  Please
don't think for a moment that I taught only arithmetic, and that by rote method.
I taught every topic, every grade, every level.  There is much more to any
subject as we both know.  I have had students in my classes who I'd say were
genius, or nearly so; exceptionally skilled to be sure.  They did not need, and
were not fed rote methods, but were given what they needed at a much higher
level of competency.  Any student seen to have those qualities, learning easily,
would not be put through any sort of torture.  My concern is those who have
difficulty or great difficulty.

As I write this, I am constantly correcting my typing errors on the fly, and
then in retrospect.  It would have done me well to have taken a
well-disciplined course in typing.  It will not make me an acomplished writer,
just an accomplished typist.  However, in what I do need to accomplish, that
task could have been made much, much easier.  You must have faced horrendous
difficulties getting your degree if you still did not have your tables
memorised.  That is not to say that you need them in order to understand systems
of differential equations and their application to chemical rates of  reaction.
It does mean to say that you would have had difficulty with what had necessarily
gone before: simplification of fractions, rational algegraic expressions,
ratio, proportion, variation, and trig and similarity in general; structures,
and similarity of structure, I could see at a glance, so having seen one, it was
as easy to see all.  That would make later studies the more difficult also.  How
much easier it would have been for you had you learned them earlier I can not
say, for everyone is different.  However, seeing, over many years, students
evolve from grade 9 to grade 13 [now grade 12 graduation] and seeing their
difficulties through an endless series of tests marked, I can say without any
doubt at all that those who had learned earlier skills benefitted later on,
sometimes by a large degree.  I just recently was tutoring a student in her
graduating year, and am constantly amazed [or perhaps no longer] at what was to
her a stunbling block in solution to some problems.  I could not help but think,
"But you should know that."  The reason was constant:  She had not had enough
*practice* on variation of type of problem.  As a result, she could not see, or
readily see the broader applications.  This was a 90% student.

The more enlightened super-structure of a study in mathematics is of course
what it really is all about.  However, the large majority will never come close
to that as they plod through their daily lives, either shovelling gravel or
writing newspaper articles, or studying law.  And it is they who are my concern.
There will be some minimal amount that they will need for their daily chores,
and knowledge of their times tables is paramount.  Just considering any basic
skill in arithmetic, you don't want to wait ten minutes while the clerk tries to
figure out your bill.

An addendum:  I got my early education during and just after WW2 [Yes, I'm that
old.] in a British boy's grammar school, finishing in a mixed school [where
other things got my attention].  Although most of the young teachers were at war
and all we had for the large part were those too old, or even brought out of
retirement, I learned two things:  A person's education is very personal.  If
there are thiry people in a class, then what they receive from the one
presentation by one teacher can be divided thirty different ways.  How they
receive and interpret what they receive is entirely up to them:  Their
individual capability, how intently they study, and so on.  Secondly, skills in
fundamentals are paramount to later success or failure.  The concert pianist
does not think how to place fingers.  Long, long hours,leading to years of
practice did that.  Now s/he can think of the music, not the notes, and again
after much practice, that music comes from the heart.  I remember the concert
violinist.  After the concert, a lady said, "I'd give my life to play like
that."  He responded .."I did."


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