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

> <gs_chandy@yahoo.com> 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

> carry)

>

> 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 ...

>

> Problems...

>

> Division, as the reverse of multiplying, whole

> results only, recall your multiplication facts ...

>

> Problems...

>

>

> 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.

GSC

("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)