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Re: Concrete to abstract in dividing fractions?
Posted:
Aug 23, 2014 5:49 PM
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On Sat, Aug 23, 2014 at 2:27 PM, Robert Hansen <bob@rsccore.com> wrote:
>
> On Aug 23, 2014, at 4:45 PM, kirby urner <kirby.urner@gmail.com> wrote:
>
> > Confining "fractions" to mean "rational numbers" is a worthwhile step in
> at least some lesson plans, as then we're getting back into the fact that
> numbers come in types, as in "types of number".
>
> No one is *confining* anything. It has to do with developmental stages. It
> takes time and exercises to build up the foundational stuff. Before you can
> introduce irrational well, the student needs a lot of familiarity with
> decimal fractions. And whether it is a rational expression involving
> rational or irrational values, the rules still apply.
>
A lot of us believe in spiraling. I often cite John Saxon as a master
spiral-er and promulgator of the concept.
So early on, the grand scheme of N < Z < Q < R < C should be unveiled. See
the staircase before you climb all the stairs. Preview. Heads up.
This is versus "heads down" curricula which just say: trust us, you'll see
a bigger picture someday, in the meantime just listen to us and focus on
that next step.
>
> How to divide a number by a fraction is a 4th or 5th grade exercise. Most
> of what you are describing is 8th or 9th grade. And all of my times are
> based on a decent curriculum. Today, you are more apt to find a student in
> 11th or 12th grade AP calculus that can?t divide by a fraction than can.
>
> Bob Hansen
Again, we're looking at a spiral stair case in cross section, and even if
we're talking about the early grades, we should be thinking ahead as to how
we'll build consciousness in the directions I've outlined.
The status quo by the way is to say nothing about continued fractions ever,
until high school is over. A computer console where we might have typed
one: not usually contemplated.
So yes, I'm using this thread to suggest new forms of lesson planning that
are CS-friendly, as I've called it e.g. lets look at non-numeric
algorithms, along with semi-numeric (a related thread).
The lesson plans I'm suggesting around "type consciousness" (awareness of
"types of number") might come quite early as a step, if you design the
staircase to go the distance.
Kirby
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