The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Education » math-teach

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Topic 5 - The Beginning of the End
Replies: 12   Last Post: Oct 11, 2012 12:34 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
GS Chandy

Posts: 8,307
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05
Re: Topic 5 - The Beginning of the End
Posted: Oct 10, 2012 9:18 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Robert Hansen posted Oct 10, 2012 1:59 PM (GSC's remarks follow):
> On Oct 9, 2012, at 7:25 PM, GS Chandy
> <> 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.

("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)

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.