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

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Robert Hansen

Posts: 11,345
From: Florida
Registered: 6/22/09
Re: Topic 5 - The Beginning of the End
Posted: Oct 10, 2012 4:29 AM
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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 ...


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

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