Author Topic: You won't believe the method that Common Core is using to teach our kids subtraction  (Read 913 times)

0 Members and 1 Guest are viewing this topic.

Offline rangerrebew

  • America defending Veteran
  • TBR Contributor
  • Hero Member
  • *****
  • Posts: 74,482
"Of all the dispositions and habits which lead to political prosperity, religion and morality are indispensable supports. In vain would that man claim tribute to patriotism who should labor to subvert these great pillars of human happiness -- these firmest props of the duties of men and citizens. . . . reason and experience both forbid us to expect that national morality can prevail in exclusion of religious principles."
George Washington

"Only a virtuous people are capable of freedom. As nations become more corrupt and vicious, they have more need of masters."
Benjamin Franklin

Offline Oceander

  • Technical
  • Hero Member
  • ****
  • Posts: 47,439
  • Chief Dork
More vapid anti-common core nonsense.  Why is it common core critics keep getting lost in the weeds, ignoring the parts that really deserve criticism and going after those things that don't?

Let's take a stab at some of these things the author doesn't like.  Here's the graphic for the first "issue":


That example is both sophisticated and elegant, and kids who have been taught to see mathematics in different ways, both old and new, will benefit tremendously.  The only real criticism here is the use of the labels "old" and "new" which both raise rhetorical hackles and imply (falsely) that the method on the top is to be replaced with the method on the bottom.

Why do I say this is sophisticated and elegant?  Because what the second version is doing is computing the difference between 32 and 12 by working out what number you have to add to 12 in order to get 32.  That is a simple arithmetic manipulation of the standard algebraic form of the equation.

Here's the standard algebraic equation:   32 - 12 = x

Here's a perfectly logical variant of that equation:  12 + x = 32

To see that, here it is step by step (--> means "implies"):  32 - 12 = x  -->  (32 -12) + 12 = x + 12  -->  32 -12 + 12 = x + 12  -->  32 + 12 - 12 = x + 12  -->  32 + (12 - 12) = x + 12  -->  32 + 0 = x + 12  -->  32 = x + 12 -->  32 = 12 + x  -->  12 + x = 32  QED


Now, how to go about determining what x is?  Well, if your forte in math to-date is addition, why not keep adding small numbers to 12 until you reach 32?  Then you can add up all of those little numbers by themselves, and - voila - you have the value of x.

And that is precisely what the second method on that page does, and it does it by building on what the kids have learned before; namely, adding by 5s and adding by 10s.  How?  First you add a small number to get you to the closest number with a 5 in the ones place -- adding 3 to 12 does quite nicely because that gets you to 15 -- now you add by 5s and, voila, you get to 20 after adding the first 5 -- this is lucky because now we can add by 10s, which is super simple -- we add a 10 to 20 and - voila - we get 30 -- when we try to add another 10 we get 40, which is too big, when we try to add another 5 we get 35, which is also too big, so we drop back to adding numbers smaller than 5, well, by simple inspection, or by the judicious use of one's fingers (on only one hand, even!), we quickly see that we need to add 2 to 30 in order to get 32.

Great!  But we still don't know the value of the single number to add to 12.  Well, that's easy enough to figure out.  Again, we go with our mathematical strength - addition - and add up those little numbers, like so:  3 + 5 = 8; 8 + 10 = 18; 18 + 2 = 20.  And now we know the answer:  20.  So, since adding 20 to 12 gives us 32, we know by simple manipulation of the number sentence that 32 - 20 = 12.

Ta-daa!

The breathtaking thing here is that there is a lot of fundamental algebraic understanding implicit in doing the problem that way and by doing it this way - the kids get walked through it by the teacher many times, they aren't just thrown in the deep end without swimmies - they will unconsciously pick up that understanding, which should make it a little easier for them when they finally get to their first algebra class in high school.

Sorry, but this isn't some parody or bizarre oddity, this is very sophisticated mathematics demonstrated to young kids in a way they can understand.


The problem here isn't the math that's being taught, it's the adults who cannot understand that math - perhaps because they were only taught the "old" method - and the primary criticism this example raises viz. common core is the fact that the proponents thereof have utterly failed to help parents by providing them with materials that explain what's going on and that walk them through things like this.

« Last Edit: April 22, 2014, 12:43:37 PM by Oceander »

Offline Oceander

  • Technical
  • Hero Member
  • ****
  • Posts: 47,439
  • Chief Dork
Here's the one that went viral, from the frustrated parent who claims - without much evidence - to have a BS in electronics engineering and extensive experience with differential equations and other higher math applications:



I, too, have had experience with differential equations - yuck! - but I don't need anything more than a basic understanding of subtraction to figure out what's going on.  Here's the answer, in a nutshell:  Jack got the wrong answer because he treated each of the small skips as representing the subtraction of 1 from the prior number.  What he should have done was to treat each skip as representing the subtraction of 2 from the prior number until he got to 117, at which point he should have treated the last skip as representing the subtraction of 1 from the prior number, giving him the result 111, not 121.

Here is what is going on:  the exercise is building on the fact that it is easier to subtract by simple numbers, such as 100, 2 or 1, than it is by a complicated number like 316.  So, the idea behind the problem is this:

316 can be represented by its individual place values, like so: 300 + 10 + 6.  It can also be represented as the following sum:  100 + 100 + 100 + 10 + 1 + 1 + 1 + 1 + 1 + 1.  It can also be represented by this sum: 100 + 100 + 100 + 2 + 2 + 2 + 2 + 2 + 1.  Using that information, you can work out the difference 427 - 316 like so:  427 - 100 - 100 - 100 - 2 - 2 - 2 - 2 - 2 - 1 = 111.  It's a lot easier to subtract by 100s, 2s and 1s for little kids than it is to do the column subtraction right off the bat.

Again, I didn't need differential equations, I didn't even need algebra, all I needed was some basic information on the various ways of representing numbers using only 10 digits (0 to 9, inclusive), and addition.


Now, here is the criticism that should have been leveled at this question because, unless there is more context that we're not being given (a not unreasonable possibility, btw), it's too complicated, and strips the problem of its didactic strength, because it requires way too much figuring to come up with the values for those little skips.  Instead, building off my discussion of the first problem, the first little skip should have represented the subtraction of 10 from 127, giving you 117.  The next skip should have represented the subtraction of 5 from 117, giving you 112 - why 5?  because 10 is too large here; we've already subtracted 100 + 100 + 100 + 10 = 310, so subtracting another 10 would give us 320, which is bigger than 316; that means we need to subtract a number smaller than 10 and the next logical number down is 5 - so we do the subtraction, and get 117 - 5 = 112.  We then check to see how much we've subtracted by, and we see that we've only subtracted away 100 + 100 + 100 + 10 + 5 = 315, which means we still have more to subtract because 315 is less than 316.  We consider another 5, but 315 + 5 = 320, which is too big - 320 > 316 - so we have to use a number smaller than 5.  The next logical number down is 1, the easiest number of all to subtract by, and we know that we won't have to subtract more than 4 1s because we already know that subtracting 5 1s would be too much (because 5 1s = 5).  We inspect what we've subtracted so far and we see that we've subtracted 315; we then check to see what happens if we subtract 1 from 112 by adding that 1 to 315 - if the result is too small we know we need to subtract more 1s).  When we do this, we see that subtracting just one 1 gives us a total of 316, which is the number we have to subtract from 427.  That means that our answer must be 112 - 1 = 111.  And therefore that 427 - 316 = 111.

Had the problem been written this way, Jack's incorrect answer would have had to have been changed, perhaps to 106, on the basis that Jack subtracted two 10s, then one 1, on his number line.

Now, some may say that this is awfully complicated, but it's actually not.  It's simplifying because it's breaking the problem down into a number of small steps each of which is very easy to do:  subtracting 100 from a number three times, then a 10, then a 5 and finally a 1, is substantially easier than subtracting 316 all at once.  The reason why it might seem complicated is because, again, the adult is viewing it from the adult perspective, where it's obvious how to do three-digit subtraction, and not from the perspective of a young kid who is trying to figure out how to do it in the first place at all.


This is a perfect example of what I said earlier about common core critics getting lost in the weeds and missing the real issues for which the common core materials should be criticized.  "Frustrated parent" is criticizing the valid part of the question - breaking three-digit subtraction problems down into a set of simple steps - and completely missing the real problem with the problem (pun intended):  the set of simple steps it uses is not at all intuitive based on what kids would most likely have learned to-date and requires so much trial and error to work it out that the didactic value of the question is missed altogether.



Offline Oceander

  • Technical
  • Hero Member
  • ****
  • Posts: 47,439
  • Chief Dork
Now, speaking of helping parents to understand what's going on, this "letter to parents" giving them a "cheat sheet" is very helpful - if you're willing to read it with an open mind and think about it:




All this is saying is that, as part of teaching addition, the class will be separating each addend into its component place values, adding like places together, then putting the place values back together to get one final number - the answer.

Why is this helpful?  Because it simplifies the addition into adding single-digit numbers, which is much easier than trying to add multi-digit numbers all at one go.  How so?  Well, if you expand 62 into 60 + 2 - i.e., one number with a nonzero value in the tens place and another number with a nonzero value in the ones place, and similarly expand 26 into 20 + 6, the addition boils down to this:  add 6 to 2, then tack a zero on the end of the result - i.e., 6 + 2 = 8, tack a zero on the end and you get 80, then add 2 to 6 without tacking on a zero because these two numbers are in the ones place; this gives you two new numbers:  80 and 8.  Adding these two numbers together is easy as pie because 8 + 0 = 8, thus, 80 + 8 = 88.  Bingo!!!!

Furthermore, by introducing the concept of expanding numbers in a very simple way, this problem starts to get kids comfortable with the concept itself because when they get to algebra, for example, where expanding numbers happens all the time, they'll "get it" more easily because they've done it before.  To see this, consider the equation  x2 - 1 = 48.  (I'm using a simple equation for didactic purposes here).  That can be solved one way:  x2 - 1 = 48  -->  x2 - 1 + 1 = 48 + 1  -->  x2 = 49, so x = +7 or -7.  Alternatively, the left side could be expanded into (x + 1)(x - 1) = 48.  A little trial and error shows that x must equal +7 or -7 because (7 + 1)(7 - 1) = 6 * 8 = 48 and (-7 + 1)(-7 - 1) = (-6)(-8) = 48.

That was handy little cheat sheet the teachers provided, although perhaps they erred in assuming that parents generally understand what it means to expand a number into its component place values.

Offline Oceander

  • Technical
  • Hero Member
  • ****
  • Posts: 47,439
  • Chief Dork
I'm not going to go through the "ten dumbest common core problems" article by Alec Torres on National Review Online, partly because it would be repetitive; suffice it to say, however, that he's about as clueless as the parents who complained about the two problems I've dissected above.  Some of the comments to that otherwise grade F article are good because they demonstrate that there are at least some people out there who do "get it."


Share me

Digg  Facebook  SlashDot  Delicious  Technorati  Twitter  Google  Yahoo
Smf