211–220 of 356 posts
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It's almost never explained with such lucidity. If you find a hundred people who can perform long division, I would be surprised if twenty of them have any idea how it works.
You're never going to find 100 people who can perform long division.
My ex taught grade 6 geography. Is a principal now. Thought the seasons were due to earth tilting back and forth. Didn't get that the tilt was the same, but the position around the sun was different
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Try again? Long division, dividing A by B, is a for loop that maintains the invariant A = QB + R, where Q is the number written on top (initially 0) and R is the number written on the bottom (initially A), by subtracting values of the form B(C)(10^N) from R and adding values (C)(10^N) to Q to compensate. func Longdiv(A, B) { // Assume A > 0, B > 0. N = floor(log10(A)) Q = 0 R = A while (N >= 0) { // Invariant: A = B…
This is just re-stating the algorithm for the umpteenth time. I can read the algorithm. I just can't intuit why it works or why it has been designed this way. I mean your very first line has a logarithmic operation that was not mentioned anywhere in your text or your comment! Just pops in there out of nowhere. I guess it's about the number of decimal digits? But why? Why are we doing things in decimal? A dense ten-li…
I think this is maybe the crucial part that you might be missing. All this long division (and long multiplication) stuff works on base-n represenations of numbers. I.e. if you have an number like 3376, it is actually a short hand for
3*10^3 + 3*10^2 + 7*10^1 + 6*10^0.
(3*10^3 + 3*10^2 + 7*10^1 + 6*10^0) / 4 ==
3*10^3/4 + 3*10^2/4 + 7*10^1/4 + 6*10^0 / 4 ==
(3*10^3 + 3*10^2)/4 + 7*10^1/4 + 6*10^0 / 4 ==
(33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(32+1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(32)/4*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
8*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
8*10^2 + (17)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + (16+1)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + 4*10^1 + (1)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + 4*10^1 + (16)/4*10^0 / 4 ==
844
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Sure you can! Just look at a typical fifth- or sixth-grade classroom. Most of them can perform long division, briefly, before the test. They may even remember it for several days after the test.
What none of them can tell you is why they've been taught long division, when short division is faster. And it's because you need long division to do division in algebra
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I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.
Well if long division is unintuitive, the procedure to compute square-roots by hand is even more mysterious, as if it has transcended dealing with numbers and instead become a symbolic manipulation. Thankfully it is something I never had to apply, even once in adult life till now.
My ex taught grade 6 geography. Is a principal now. Thought the seasons were due to earth tilting back and forth. Didn't get that the tilt was the same, but the position around the sun was different
Your ex was correct. https://spaceplace.nasa.gov/seasons/en/ What Causes the Seasons? The Short Answer: Earth’s tilted axis causes the seasons. Throughout the year, different parts of Earth receive the Sun’s most direct rays. So, when the North Pole tilts toward the Sun, it’s summer in the Northern Hemisphere. And when the South Pole tilts toward the Sun, it’s winter in the Northern Hemisphere. It's all about Earth's…
> But many elementary students don’t understand the meaning of the equal sign. Some people might not have made the connection, but by and large programming languages make this mistake too. All the major ones are using the = glyph for assignment.
What mistake is that? The left hand variable equals the right hand value assigned to it. Foo = bar and 1 = 1 are equivalent in my mind.
Depending on which programming language you're using, "foo = bar" can mean either "make foo now equal to what bar is" or "is foo equal to bar now?". Neither one is the same as in mathematics, which is "foo is categorically equal to bar".
Consider:
foo = 1
bar = 2
foo = bar
In programming, this is perfectly valid, and means "ok, ignore what I wrote just 2 lines ago, and now make foo equal to 2, also".
You say you think it means "the LH var equals the RH value assigned to it", but you don't say what you think it means to assign to the literal 1.
If we are serious about trying to "break the cycle", one obvious solution is to pay teachers more. Imagine if teaching paid as well as software development.
Are they going to spend their extra cash on mathematics lessons? In their copious free time, I presume. Just like these rich software developers spend all their extra cash on more school. I see a lot of Porsches and BMWs in my neighborhood but curiously none parked at the community college.
Are you going to hire different people, who chose to go into computers just because it paid better? I don't see many software developers who would make good elementary teachers. Or would want to try.
If we are serious about trying to "break the cycle", one obvious solution is to pay teachers more. Imagine if teaching paid as well as software development.
This would be true if we had any idea what makes a good teacher, or could reliably determine who good teachers are. Neither of those are true. TEACHER QUALITY https://faculty.smu.edu/millimet/classes/eco7321/papers/hanu... > Perhaps most remarkable is the finding that a master’s degree has no systematic relationship to teacher quality as measured by student outcomes. This immediately raises a number of issues for pol…
Perhaps base 10 is the problem. In base 2, you can learn the "multiplication table" in 10 seconds (0 X n=0, 1 X n=n, we're done.) The long multiplication algorithm for n and m is just multiply by n by the least significant bit of m, add to the accumulator, left shift, repeat. You can do subtraction by adding the two's complement (just flip every 1 to a zero and every zero to a one - trivial for a child to learn) inst…
The real world depends on base 10, and so it's way more practical to learn calculations in base 10. I said calculations, not math. I think there is a big confusion between math and calculations. Kids learn calculations, not math. Learning the multiplication table is just a way of how to do quick calculations inside your head. I need to pay 35.15 euros, so how much do I get back? That is calculation not math. In my op…