Date: Oct 10, 2012 9:18 AM
Author: GS Chandy
Subject: Re: Topic 5 - The Beginning of the End
Robert Hansen posted Oct 10, 2012 1:59 PM (GSC's remarks follow):
> On Oct 9, 2012, at 7:25 PM, GS Chandy
> <firstname.lastname@example.org> wrote:
> > "What, in your opinion, constitutes this wonderful
> 'systematic approach to teaching arithmetic'that you
> now seem to applaud?"
> Counting to 10, to 20, to 100
> 1+1 = 2, 1+2 = 3, 1+3 = 4, ...
> 11+11 = 22, 11+12 = 23, ... (multi digit without
> 15+7 = 22, ... (multi digit with carry)
> Jane has 3 apples, Troy has 6 apples, how many apples
> do they have together...
> Subtraction, as the reverse of adding, recall your
> addition facts (no negative numbers) ...
> Sinclair has 12 dollars, Mary has 3 dollars, Sinclair
> gives 6 dollars to Mary, how many does he have left?
> How many does Mary now have? ...
> Multiplication, starting with repeated addition, then
> the tables ...
> Division, as the reverse of multiplying, whole
> results only, recall your multiplication facts ...
> During this phase we focus on number, operation and
> context (problems).
> Numbers are made of digits that occupy places
> (columns). Learn to recognize them, say them, write
> them (notationally and with words).
> The four basic operations are addition, subtraction,
> multiplication and division.
> Use visualizations only when they are contextual and
> developmentally equal to what they represent and what
> your are doing, like the number line, rows of
> objects, arrays of objects.
> Do not use visualizations that are reconstructions or
> proofs, they will only confuse and hide deficiencies
> in understanding.
> The same thing goes for problems. Problems at this
> stage are language training, not problem solving.
> They teach how to map (not model) a literal
> situation, involving numbers and operations, to an
> arithmetic expression. They start straightforward at
> first but then reverse the order as you go on. Jane
> is two years older than Mary. Mary is two years
> younger than Jane.
> Do not teach things that involve fluency in these
> operations before that fluency is obtained, like Time
> or Money. Reading a basic clock is ok, but don't
> start into the number of minutes in an hour etc.
> Whole dollars or whole cents are ok, but decimals,
> even just a written example of them, is not.
> Do not teach estimation directly, let it build
> through familiarity. How can you teach a student to
> estimate addition if you have not yet finished
> teaching them addition?
> Do not teach shortcuts in the path before they know
> the path first. Talk to them as they arise naturally
> but stay on target.
> Do not teach algebra! Math is simply not reasonable
> enough yet. Fill in the blank problems are good (3 x
> __ = 12), but reasoned solutions are premature at
> this stage. As you get further along set aside time
> for reasoned discussion but don't conflate it with
> the developmental tasks at hand. There will be
> students that see further ahead and you should
> recognize it and encourage it, but stay on track.
> They are not going to see all of the baggage ahead.
> Fractions should be introduced gently at first, in
> literal form, "What is half of six?" They shouldn't
> be dealt with technically until the students are
> fluent with whole number multiplication and division,
> probably 4th grade, decimals some time after. And tie
> them to arithmetic, not pictures. The pictures are
> too algebraic. We are still talking about the number
> line here, not algebra.
> Fifth and sixth grade should be used to solidify
> fluency in all of these things, technically and
> operationally. By sixth grade, a student should look
> at a common fraction or decimal pretty much as we do.
> They should be able to perform the four basic
> operations on any pair of numbers (with finite
> digits), with and without a calculator. Mental math
> is developed and stressed during this stage, along
> with data sense (charts etc). The problems are still
> more technical than reasonable although they now have
> a hint of algebra. They can involve multiple steps or
> simultaneous conditions.
> Later I will explain my theory as to how it fell
> apart and became what we struggle with today. It has
> to do with college and I think people are just now
> realizing it.
> Bob Hansen
Interesting - and revealing.
The above probably represents a pretty good way to reach what you wish to ***teach*** as beginning arithmetic - assuming your charges are well prepared and ready. But in the real world, you get all kinds, the ready and the unready; the willing and the unwilling; those who have good 'learning situations' at home and those who do not. Thus, what you are discussing is approximately one-half of a 'system'.
Revealingly, no attention is paid to the fundamental ***learning needs*** of those you wish to ***teach***. The above will probably work for those students who're primed, so to speak, to take benefit of what you intend to ***teach***. It's not a system. A "system" is somewhat different, as indicated in some of the attachments I put up from time to time.
("Still Shoveling Away!" - with apologies if due to Barry Garelick for any tedium caused; and with the humble suggestion that the SIMPLE way to avoid such tedium
is simply to refrain from opening any message purported to originate from GSC)