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 > 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)