Use anzan and other high-tech math methods

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John Moser

2018. május 6. 12:57

Khan Academy should use higher-tech mathematics methods, such as soroban and anzan math for basic arithmetic. That includes rapid, registered addition (soroban uses the same operation to compute each digit) and memorized look-up tables (two and four pairs for addition, 9x9 multiplication table), as well as the basic procedural algorithm.

Building on the base algorithms enables third-grade students of all aptitudes to essentially compute square roots and other complex computations in their heads in record time. This eliminates the arithmetic component of mathematics from the effort applied during algebraic operations.

It is likely possible to begin instruction of algebra in third grade. I don't have any advanced algebra teaching methods on-hand; I can think of a few, but they're by my own design, and not tested in international circles.

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Admiral Betasin

2018. május 6. 16:12
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Hi John Moser,

Thanks for your suggestion!

The content specialists are busy adding new content all of the time. Currently, they're focused on adding more K12 subjects (which may include your suggestion, so I'm sure they'll consider it.)
mtschulz72

2018. május 16. 12:46
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this site helps me learn more about math
John Moser

2018. május 16. 15:40
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Sure, but how well can you do it? Single-digit addition? How about keeping up with third graders? There's a video out there of kids playing word games while paying little attention to the computations being given them, and then reciting the computation.

High school doesn't prepare you to keep up with a Japanese seven-year-old's mathematical prowess.

How fast can you perform the following computation: 2 / .04

The real question: How did you approach it?
Regina Litman

2018. május 16. 17:53
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The answer is fifty, I got it by converting zero point zero four to the fraction one over twenty-five and then multiplying the original numerator two by the resulting reciprocal of twenty-five.

It took a whole lot less time to do it than to explain it. I spelled out the number so that a quick read of my answer would possibly not yield spoilers.
John Moser

2018. május 16. 19:15
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Think about these numbers: 2.00 and 0.04

The problem is merely 2 / 4.

4 isn't a factor of 2; the next digit is 0, thus 20 / 4. We know that's 5 (multiplication table).

The 2 was two columns left; the 20 was one column left, so you get 0.5.

The answer is 50: decimal points are meaningless and shift in the same direction in division, so just move it two columns right (200 / 4).

My point here is that you're looking at the operation as a sort of complex thing which requires an approach strategy because there's a decimal point. People do this all the way down to addition, and will convert (6 + 7 + 4) to (5 + 5 + 7) to (10 + 7) to (17)—having carried out three additions, one subtraction, and a strategic analysis instead of only one subtraction and one addition (7-3 and 3+4).

There's no "ones place" or "tens place". You could say that the 5 was one column left of the 4 and the decimal point was two columns left, so the 5 shifts to three columns left to give 50; that's valid, and it's required when carrying out multiplication. Applying this prior to division happens to apply it to every operation ahead of time.

Teaching this early (instead of the Friendly Numbers thing) by digit-wise columnar addition and subtraction (Soroban and Anzan use this strategy) avoids developing the perception that math is complex and hard.

That said, division with multiple-digit numbers is…not simple. I have never encountered any mathematical approach to division which was simple and linear. Wikipedia doesn't suggest one exists. Multiplication, on the other hand, can be done digit by digit—which is why we memorize single-digit multiplication tables.

There are two addition sets. One is {(1,4),(2,3)} and the other is {(1,9),(2,8),(3,7),(4,6),(5,5)}. Addition is rather simple, and you eventually just see (7 + 8) and immediately run 7-2 to get 15 (check the second set: (2,8)). That's why people get so fast at addition and subtraction in particular: they run through single-digit isometric operations.

With addition, you can see there are no "friendly numbers" or other algorithms to reduce to easily-recognized sets such as additions of 5s and 10s; instead, you make every simple addition operation fast and easy. Multiplication and division come down to being able to keep a lot of numbers in your head at once or using a physical register (beads, paper, a soroban, etc.). Computing square roots is relatively-simple.

It's easy to demonstrate a single, simple algorithm for addition and subtraction, and thus convince people that there's only one correct way to add numbers (it's "correct" because attempting to use other methods only adds complexity, introduces human error, slows you down, and creates frustration). Multiplication and division are harder to argue—and, indeed, multiple algorithms exist for human use both on paper and mentally.

Paper division has strategies like lattice division, which is just long division; or the partial quotient method, which is the only sane mental division method.

I'm trying to work out if there's an absolute lowest-effort, highest-comprehension set of basic math (and algebra) approaches; the only ones that stand out at the moment are Anzan addition and subtraction, which are far-advanced compared to how we teach introductory arithmetic in the US now.
John Moser

2019. október 9. 15:04
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I've thought about this more, and considered a rough curriculum for absolute early math—the earliest, learning counting and basic arithmetic. Roughly:
First introduction to Counting is a brief evolution of counting methods. Small children seem to easily move from one point of reasoning to the next. This is purposely expanded to maximize future plasticity, but brief and focused.

Show several objects, demonstrate counting, and explain the tally method. Describe subitization to explain tallying (we can't subitize 5, but we can instantly recognize 4 or fewer marks once we learn to count).

Explain that tallying is not space-efficient, so we invented NUMERALS. Show the 1-9 numerals, and the concept that 10 is 1-0, to prep the mind for reaching this stage.

Break down to binary. Show that one digit counts 0 or 1 thing, but runs out; so we move one left, 1-0 is one more than 0-1, and so forth. Do this for four digits (1111 = 15).

Move back to the 1-9 and show that these four digits represent our "decimal" numerals—"Deci" meaning "ten"—and that in the span of two digits we can count 0-9, then just as in binary we move over one more. Ten. The reasoning from there comes easily.

Introduce the 4/1 Abacus—the Soroban—to students now familiar with digit placement and representation. The lower four beads represent 1 as a count, and so a tally 1 2 3 4; the upper bead is the tally of 5; and each bar is a decimal digit of 0-9.

Briefly mention that the Suan Pan, the Chinese one, uses five beads at the bottom and two at the top, so each bar represents four of those "binary" digits or 0-15—and that this is something we can explore later, because it becomes relevant with computers.

From here, indicate counting on the Soroban. One, two, three, four, five+1, 5+2, 5+3, 5+4, 1/0. Liken it to the tally method, but with each rod clearly being a decimal digit.

Introduce mathematical relationships.

Adding 3 to 2 doesn't work, and gives you 5; adding 4 to 1 gives 5. Therefor, {1,4} and {3,2} are the complements on 5.

Likewise, {1,9}, {2,8}, {3,7}, {4,6}, and {5,5}.

Rote memorization of the above look-up tables. Briefly discuss the role of rote memorization as a learning tool, and indicate that these rote tools can be worked out by their understanding of how they work if they forget them, and that they will need to be immediately available to do arithmetic quickly (hence memorization by rote).

From here, we use the mathematical relationships to discuss addition, and then subtraction. These are easy: 7 + 4? {7,3} so subtract 3, 4-3=1, you're adding and so you overflow one to the left, the answer is 11. No friendly numbers, because everything is mechanical and direct and thus a lot faster than preprocessing.

The Soroban will be the calculator of choice here.

The concept of a decimal place is inherent, and so fractional components are introduced immediately at counting. Thus 100 + 15 is just 1 moved left two bars, plus 15 based at the base, and it's 115. 11.5 is the same operation with a different position for the decimal place.

Introducing decimals requires introducing division, but not introducing the computation of division. It is sufficient to show that a whole is made up of 10 parts and that 3 and 7 parts is 0.3 and 0.7 of one whole. We can also show that of five parts you have to double that to get the decimal number, and indicate to the student that we'll cover the methods for this in further and more-complex arithmetic. This avoids the student's mind basing on a solid world view of everything being nice, even parts.

After solidifying counting on decimal—after sufficient study and practice on addition and subtraction—we return to the Suan Pan to discuss Hexidecimal.

This is useful for computer science.

The 1111 binary count—0-15—is the same as 0-9A-F in a single Hexidecimal digit. Likewise, 111 is the same as 0-7 in Octal (not covered in this phase).

Demonstrate expansion of Soroban arithmetic to the Suan Pan's 5/2 structure. One math lesson, and then return to decimal.

Later classes broach multiplication and division.

This requires memorizing the multiplication table, another rote tool.

Order of operations must occur here, which means exponent must be introduced. Reverse-polish notation is also useful here.

Division is interesting, as you can show that 1 / 2 is also 10 / 2 and then shift the decimal point to the right, or 1 / 0.2 and then shift the decimal point to the left. Either way, your multiplication chart contains 10 at the intersection of 5 and 2, and so 10/2 = 5, and you get 0.5 equal to one half.

Division introduces fractions, and we explore fractions first because division is fractions. Computing the fractions and then computing the division on the final fraction is more-efficient than computing many divisions in most cases.

Advanced arithmetic, such as exponents and logarithms, builds on this basis.

A math course such as the above has several useful characteristics.
First, its brief movement through counting introduces the students to a reasoned method of thinking: why are things as they are? This must be brief, as the new topic must not create confusion. The tally won't create confusion because it is unlike numerals; binary will create confusion, so numerals 0-9 are introduced first, as well as the concept of 10, to affix that basis before breaking down to binary. The Soroban is examined as a way to represent Decimal, so does not introduce confusion.

Second, the use of Soroban promotes Anzan: the student will learn an isometric register method to perform addition and subtraction, which means all addition and subtraction of infinite digits with any number of decimal places is learned at once. This method is fast and efficient, and can be carried out mentally (Anzan) after short practice. This maximizes confidence with math.

Third, the student becomes aware of how number systems work, and thus more flexible with things like binary and hexidecimal when facing computer science concepts in the future.

Fourth, the Soroban method makes the numerical concept of division obvious, in that 1/2 or 3/7 has an obvious and meaningful result. 3/7 is just 30/7, or 300/7, or so forth, with the decimal place moved about; thus 0.4285… is not a mystery, but an obvious thing (also I did that in my head…I've never actually tried to do that kind of division in my head before).

A lot of this tries to wedge in just enough deviation into further topics to keep the student's mind from solidifying around an ideal world where everything is as it is because it can be no other way, while avoiding confusion and distortion. It is an attempt to maximize future learning capability as much as it is to teach the topic at hand.

Any thoughts on the above?

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Use anzan and other high-tech math methods

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