Is there anything good about Common Core?

[Left_Hand_of_Dorkness]

@Nelson_Pike

I do not understand. What does “result” mean? In what sense is the result of 7-2=5 the same as the result of 7-5=2?

Nelson, I’m afraid it’s I who lacks clarity. Are you unclear on the relationship between 7, 5, and 2, as demonstrated in these problems, and would like me to explain this third-grade mathematical concept to you? Or are you clear on the principle I’m talking about and are simply trying to score points by nitpicking my word choice in that post? Please clarify, and I’ll respond accordingly.

[Nelson_Pike]

@Raygun99

You can add in place holders as necessary. To wit: 137.6 - 124.33 can become 13760-12433, with the two decimal places added back in afterwards. It’s the method actually preferred when doing long division involving decimals. i.e. divide 156 by 3.4. You change the problem to 1560 divided by 34 to make it “easier”. Note that it’s not impossible to do it with the decimal, but that it becomes cumbersome.

OK- I think I can go along with you to here.

@Raygun99

My point however is that the decimal places are just another metaphor used in math that can be used or discarded as desired,

I would prefer to say that decimals are a system of notation which economically express non-integer values. The modern world employs a common mathematical notation as a universal language.

@Raygun99

and that understanding how the numbers operate in relation to one another is what’s important. Seeing beyond the metaphors is what creates real understanding in math.

I am skeptical of the utility of these truisms.

@Raygun99

I wouldn’t particularly recommend this method to everyone as it’s more of a stepping stone, but then again, there’s scant few methods that I would recommend to everyone.

What few methods would you recommend to everyone?

[Nelson_Pike]

@Left_Hand_of_Dorkness

Nelson, I’m afraid it’s I who lacks clarity.

Yes.

@Left_Hand_of_Dorkness

Are you unclear on the relationship between 7, 5, and 2, as demonstrated in these problems, and would like me to explain this third-grade mathematical concept to you?

Yes and yes.

@Left_Hand_of_Dorkness

Or are you clear on the principle I’m talking about and are simply trying to score points by nitpicking my word choice in that post?

No and no.

@Left_Hand_of_Dorkness

Please clarify, and I’ll respond accordingly.

Go for it.

[Nelson_Pike]

@Thudlow_Boink

In the sense that both equations express the same (true) relationship among the numbers 7, 5, and 2.

The word “result” can refer to the numerical value you get at the end of a calculation, but that’s not the only way the word is used, and LHoD already clarified that he wasn’t using the word to mean “the number on the right side of the equation.” In higher math especially, the word “result” is also used to refer to a fact, like the equation “seven minus five is equal to two” as a whole, rather than to the number 2. That’s how the word is used in, for example, Wikipedia’s "list of mathematical jargon:

Baloney. LHoD had two chances to provide this (or any other explanation) her/himself but failed to do so. And no, just saying that my interpretation of the jargon was wrong and leaving it at that is not a form of explanation. No doubt she/he is grateful to you for your suggestions, though.

[Raygun99]

@Nelson_Pike

I would prefer to say that decimals are a system of notation which economically express non-integer values. The modern world employs a common mathematical notation as a universal language.

I am skeptical of the utility of these truisms.

The point of my examples is to show the fungibility of math. Decimals are often a sticking point for learners and removing the complexity to show that it’s the same process they’ve already learned can be helpful in breaking down mental barriers to math. Hopefully those steps (and there’s similar things for just about every may concept) can eventually be discarded, but not everyone is going to get to that point. We’re taking about teaching people how math works. The more intuitive feel you have for operations, the more sense more difficult problems are going to make.

What few methods would you recommend to everyone?

That was more a matter of being non specific, but I suppose everyone should try “chunking” problems - ie rounding to create easier problems, and I would recommend everyone take a stab at memorizing multiplication tables to 12*12 in order to create familiarity with numbers.

[Kearsen]

@DSeid

And that is precisely the problem. Not anything to do with Common Core: one teacher who does not understand the concepts (s)he is supposed to be teaching and mistakenly thinks that there is functional difference between 3x5 and 5x3.

Common Core Standard for Grade 3 is to understand and apply properties of operations as strategies to multiply and divide. The grading of that test is teaching the exact opposite of the correct information and in direct opposition to the Common Core Standard.
What goes on is that anything bad is being blamed on Common Core. Odd and different grading system? Common Core. A bad teacher doing the job poorly? Common Core. Too much homework? Common Core. Too little homework? Common Core …

How long ago were the teachers versed in teaching Common Core? College level course focus on what exactly?
If the teachers are teaching something wrong that they have had thrust on them, then I’d not fault the teachers. I’d fault the administration for not providing an environment where the teachers could succeed.

[Nelson_Pike]

@Raygun99

The point of my examples is to show the fungibility of math. Decimals are often a sticking point for learners and removing the complexity to show that it’s the same process they’ve already learned can be helpful in breaking down mental barriers to math. Hopefully those steps (and there’s similar things for just about every may concept) can eventually be discarded, but not everyone is going to get to that point. We’re taking about teaching people how math works. The more intuitive feel you have for operations, the more sense more difficult problems are going to make.

This is not what I was skeptical about. It was “understanding how the numbers operate in relation to one another is what’s important. Seeing beyond the metaphors is what creates real understanding in math.” that strikes me as a slogan.

That was more a matter of being non specific, but I suppose everyone should try “chunking” problems - ie rounding to create easier problems, and I would recommend everyone take a stab at memorizing multiplication tables to 12*12 in order to create familiarity with numbers.

Rounding sounds good to me, and Praise The Lord there is someone around here who advocates memorizing the multiplication tables. Most people now think that multiplication tables are best learned by a sort of osmosis which takes place with a few years of contemplating bar models, number arrays, and the like. Rote memorization and the drill associated with it are heresy!

[Left_Hand_of_Dorkness]

@Nelson_Pike

Yes.

Yes and yes.

No and no.

Go for it.

Okay. I’ll grant you exactly one post in which I give you the benefit of the doubt.

Imagine that I’m holding a tower of connected blocks in my hand. Five of the blocks are red. Two of them are green.

What equation can you come up with that represents the sum of the red and green blocks in this stack?

Did you say 5 + 2 = 7? Excellent!

Now I flip the tower over. What equation can you come up with that represents the sum of the green and red blocks in this stack?

Did you say 2 + 5 = 7? Very good! Notice that all I did was to change the position of the red and green blocks. The number of blocks doesn’t change. If all I want to know is the number of blocks–the sum–I can add the red and green blocks together in whichever order I like. I can change the position of the smaller numbers, the addends, and still have a true equation.

Now I take the red blocks away from the tower. What equation can you come up with that represents the subtraction of the red blocks from the tower?

Did you say 7-5=2? Perfect!

Notice that I’m still working with the same numbers that I used for 5 + 2 = 7. Do you see how, if you know that 5 + 2 = 7, you can use that to figure out that 7 - 2 = 5?

If you’re having trouble seeing that, watch this: I put the red blocks back onto the tower. Taking them away from the tower led to the equation 7 - 5 = 2; putting them back leads to 2 + 5 = 7. Same numbers!

Now I take the green blocks away. What’s the equation?

That’s right! 7 - 2 = 5! Again, we’re working with the same numbers!

In all these cases, you can transpose the two smaller whole numbers and still have a valid equation.

You may be asking why this is important. Well, it’s because mathematicians are lazy. If you already know the answer to one question, and you need to solve a related question, sometimes you can use the solution you already know to solve the second question in a much easier way.

For example, I’m going to tell you right now that 7,346 + 6,128 = 13,474. Without breaking out pencil and paper, how quickly can you solve this subtraction problem: 13,474 - 7,346?

[Left_Hand_of_Dorkness]

@Nelson_Pike

This is not what I was skeptical about. It was “understanding how the numbers operate in relation to one another is what’s important. Seeing beyond the metaphors is what creates real understanding in math.” that strikes me as a slogan.

Worst. Slogan. Ever.

[Raygun99]

@Nelson_Pike

This is not what I was skeptical about. It was “understanding how the numbers operate in relation to one another is what’s important. Seeing beyond the metaphors is what creates real understanding in math.” that strikes me as a slogan.

My point here is that, even though I do promote the idea of some memorization and repeated rpactice, that’s the not how you develop numeracy, and making it just that is a great way to turn people off it forever. Solving a bunch of nearly identical math problems is not really math education. And things like showing how the decimal place is just a metaphor is a good way to help kids understand that (for instance) a billion divided by a million is the same as 10^-6/10^-9.

[Nelson_Pike]

[COLOR=“Blue”]
Sorry about the blue letter formatting. I just can’t see any better way to clearly distinguish my commentary from the numerous quotes.

I think at this point it has become necessary to repost some earlier replies.

From reply #64:

(highlighting and numbers in parentheses added):[/COLOR]

[Quote=Left Hand of Dorkness]

The important thing for kids to know, though, is that in whole-number addition, subtraction, multiplication, and division, in which the equation has three numbers, **you can always switch the places of the two smaller numbers and get the same result **.

So:

(1)
2+5=7
5+2=7

(2)
7-2=5
7-5=2

(3)
2x5=10
5x2=10

(4)
10/2=5
10/5=2
[/quote]

Reply #71:

[Quote=Nelson Pike]

@Left Hand of Dorkness

…in whole-number…subtraction…and division, in which the equation has three numbers, you can always switch the places of the two smaller numbers and get the same result.

ahem:
7-2=5
7-5=2

ahem:
10/2=5
10/5=2
[/quote]

It was equation pairs (2) and (4) of reply #64 that I was solely concerned with, because they contradict the remarks highlighted. Those remarks state as universal a property which applies only to addition and multiplication, as in equation pairs (1) and (3). The property does not apply to subtraction and division, as in (2) and (4). The commutative property discussed below does not apply to subtraction and division either.

[Quote=Left Hand of Dorkness]

Okay. I’ll grant you exactly one post in which I give you the benefit of the doubt.

Imagine that I’m holding a tower of connected blocks in my hand. Five of the blocks are red. Two of them are green.

What equation can you come up with that represents the sum of the red and green blocks in this stack?

Did you say 5 + 2 = 7? Excellent!

Now I flip the tower over. What equation can you come up with that represents the sum of the green and red blocks in this stack?

Did you say 2 + 5 = 7? Very good! Notice that all I did was to change the position of the red and green blocks. The number of blocks doesn’t change. If all I want to know is the number of blocks–the sum–I can add the red and green blocks together in whichever order I like. I can change the position of the smaller numbers, the addends, and still have a true equation.

[/quote]

This is a satisfactory exposition on the commutatative property of mathematics, using the addition equations 5 + 2 = 7 and 2 + 5 = 7.

However, the commutative property is irrelevant to my objection in reply #71, which concerned only equation pairs (2)(subtraction) and (4)(division) of reply #64.

[Quote=Left Hand of Dorkness]

Now I take the red blocks away from the tower. What equation can you come up with that represents the subtraction of the red blocks from the tower?

Did you say 7-5=2? Perfect!

Notice that I’m still working with the same numbers that I used for 5 + 2 = 7. Do you see how, if you know that 5 + 2 = 7, you can use that to figure out that 7 - 2 = 5?

If you’re having trouble seeing that, watch this: I put the red blocks back onto the tower. Taking them away from the tower led to the equation 7 - 5 = 2; putting them back leads to 2 + 5 = 7. Same numbers!

Now I take the green blocks away. What’s the equation?

That’s right! 7 - 2 = 5! Again, we’re working with the same numbers!

(highlighting and numbers in parentheses added):
In all these cases, you can transpose the two smaller whole numbers and still have a valid equation.
[/quote]

This is a satisfactory exposition of the principle that the integrity of an equation will be preserved if the same operation is performed on both sides.

However, that is irrelevant to my objection to the incorrect claim that the result will be the same. Valid, yes, the same, no.

[Quote=Left Hand of Dorkness]

You may be asking why this is important. Well, it’s because mathematicians are lazy. If you already know the answer to one question, and you need to solve a related question, sometimes you can use the solution you already know to solve the second question in a much easier way.

For example, I’m going to tell you right now that 7,346 + 6,128 = 13,474. Without breaking out pencil and paper, how quickly can you solve this subtraction problem: 13,474 - 7,346?
[/quote]

Quick enough, and you seem to have forgotten that in a reply to you in that other Common Core thread I mentioned I got a B in college algebra 101. That was in 1967-68 before electronic calculators, and at an institution where I did not see a single multiple choice test in any subject the whole four years. Break out pencil and paper, and show your calculations in the Blue Book! I made out like a complete innumerate here in this thread partly in jest, and partly to see if it would elicit a somehow satisfactory explanation of the issues I raised.

Such shortcuts as you mention above are fine within limits; here is a more elaborate one taken by Gauss in his student days:

The Great Gauss Summation Trick

One of the most famous mathematicians of all times was named Karl Gauss. One day,
as the story goes, his teacher gave the class an assignment to keep them busy so
that he could take a nap in the back of the class. The problem he assigned would
keep most of us busy for at least a half an hour, if not more. However, to his
teacher’s surprise, young Mr. Gauss solved it in seconds.

Here is the problem the teacher assigned. Students were told to add all the whole
numbers from one to one hundred. That is, 1+2+3+4+5 …98+99+100.

In less time than it took most students to write out this one hundred number addition problem,
Gauss got the answer. The sum is 5,050 he told his teacher confidently, and so it was.
But how did he arrive at this answer in so short a time?

Gauss was a genius, and geniuses sometimes see things differently than most of
us non genius types. But that doesn’t mean that after being shown the way that
we can not solve a problem like a genius would, having first been shown the way.

Here is how young Gauss arrived at his answer so quickly. He observed that in
the series of numbers 1 +2 + 3 +4 …97 + 98+ 99 + 100, the sum of pairs of numbers
from each end, and working in toward the middle summed to the same value,101.
In other words, 1 + 100, 2 +99, 3 + 98, 4 + 97 etc. all sum to 101! Since there are
fifty pair of numbers in the series 1 to 100, Gauss reasoned that the sum of all the numbers would be 50 times 101 or 5,050.

But math students should learn to be at ease solving such problems as 13,474 - 7,346 by brute force.

Here is an example of brute force displayed by Isaac Newton, who was not lazy.

Some avoid brute force calculation as though it were cyanide. If there’s no easy way to do it then don’t do it at all. That is sad because few worthwhile challenges are easy, especially in education. Encouraging avoidance of such realities is not going to do anyone any favors.

[Left_Hand_of_Dorkness]

If anyone else can figure out what your central thesis is, perhaps they’d care to explain. You seem to have a problem with my pointing out that, in a valid division equation comprising only whole numbers, the divisor and quotient may be switched without affecting the equation’s validity. I’m glad you clarified that in all that blue text, and wish you’d clarified that earlier when I asked you to clarify what your objection is. I’m still not understanding what your objection is to what I’m saying.

[Nelson_Pike]

@Left_Hand_of_Dorkness

If anyone else can figure out what your central thesis is, perhaps they’d care to explain. You seem to have a problem with my pointing out that, in a valid division equation comprising only whole numbers, the divisor and quotient may be switched without affecting the equation’s validity. I’m glad you clarified that in all that blue text, and wish you’d clarified that earlier when I asked you to clarify what your objection is. I’m still not understanding what your objection is to what I’m saying.

My central thesis should be obvious to and anyone else. It is that this comment by by you: “The important thing for kids to know, though, is that in whole-number addition, subtraction, multiplication, and division, in which the equation has three numbers, you can always switch the places of the two smaller numbers and get the same result.” does not apply to subtraction and division.

There is no reasonable interpretation for the word “result” except as a synonym for “answer” and “solution”. I suspect you meant it that way, but made a careless error, such as we all do. It would have been best to correct the error, rather than embark on an exercise in face-saving.

[Thudlow_Boink]

@Nelson_Pike

There is no reasonable interpretation for the word “result” except as a synonym for “answer” and “solution”.

I thought I already disabused you of this notion in Post #80.

[swampspruce]

@BigT

I disagree. The array method with one digit numbers is just a completely different problem than doing it with two digit numbers. I don’t think they are transferable except in the general process of using area for multiplication.

It’s really not useful until you are at least doing 2 x 1 digits. Then you can start to see that you’re putting different rectangles together. Then, when you get to 2 x 2, you can see you’re putting together more rectangles.

And then, if you get what the rectangles represent, you can basically derive the FOIL method by yourself. 2854 = 20 * 50 + 204 + 50 * 8 + 4 * 8 = 2500 + 80 + 400 + 32 = 3012.

In fact, I’m pretty sure that’s exactly how I learned using the area method at Montessori, where I maxed out their math curriculum by fourth grade and coasted in public until seventh grade when I finally got to learn something new–pre-algebra. And be fed up with a teacher that made us do 30 problems every night for no good reason.

And that, BTW, is my problem with Common Core. It still is adhering to the crappy “tons of homework” method of teaching. It’s busy work. At home you’ll either get it quickly and not need to practice a whole lot, or you won’t get it and will be stuck.

Arithmetic is the one subject that should be online now. And streamlined to each student and how they learn. We were doing that in fifth and sixth grade in the computer lab. What happened? Why did we revert to new math but old methods of giving the kids busywork?

BTW, i also learned multiplication using the times table square. That was the last step before you were to have your times tables memorized.

I generally don’t like to nitpick but your example is wrong. 28*54 is not equal to 3012 unless you’re Winston Smith.

3054=1620, so to get 2854 I subtract 108 (or 2*54)=1620-108= 1512. To be fair, I actually went 1620-110+2 to simplify the subtraction in my head.

You get 1/2 marks for using a suitable method though…

[Left_Hand_of_Dorkness]

@Nelson_Pike

My central thesis should be obvious to and anyone else. It is that this comment by by you: “The important thing for kids to know, though, is that in whole-number addition, subtraction, multiplication, and division, in which the equation has three numbers, you can always switch the places of the two smaller numbers and get the same result.” does not apply to subtraction and division.

There is no reasonable interpretation for the word “result” except as a synonym for “answer” and “solution”. I suspect you meant it that way, but made a careless error, such as we all do. It would have been best to correct the error, rather than embark on an exercise in face-saving.

Ah, so this is about scoring points over particular word choice; you have no substantial disagreement with my point, and I was too generous in giving you the benefit of the doubt. As I thought. Carry on then.

[BrainGlutton]

Just to keep this in its proper perspective, CC, regardless of its actual merits, has become one of 10 right-wing conspiracy theories that have slowly invaded American politics.

At their most benign, attacks on the Common Core have portrayed the standards as either a key step in a federal takeover of public education or yet another reform attempt that overemphasizes testing and standardization. But in the hands of radical groups like the John Birch Society and a whole array of far-right groups and politicians, the proposed program has morphed into what former Fox News host Glenn Beck characterized as “Communism. We are dealing with evil.”

Forty-five states adopted the initially uncontroversial standards in a bid to improve the competitiveness of the American work force. But the brouhaha has had consequences: Indiana, Oklahoma and South Carolina have withdrawn from the standards, and at least a dozen other states have seen similar legislation introduced.

Virtually all the radical claims about the standards are false. They do not mandate any particular texts — other than the Declaration of Independence, the Preamble to the Constitution, the Bill of Rights and Lincoln’s Second Inaugural Address. They do not promote homosexuality. They are not critical of Christianity, nor do they promote Islam. They do not require the collection of individual data that will then be sold to private interests. They don’t push the idea of global warming or hector students about “social justice” or being good “global citizens.” They are not part of a plot to impose a global government known as the “New World Order.”

Although more and more outlandish conspiracy theories are part of mainstream American political culture, wildly untrue claims about the Common Core have far more “mainstream” proponents than most. Politicians from around the country and all levels of government, pundits, a large number of Christian Right and anti-LGBT groups, and many others are part of the paranoid chorus.

[Left_Hand_of_Dorkness]

@BrainGlutton

Just to keep this in its proper perspective, CC, regardless of its actual merits, has become one of 10 right-wing conspiracy theories that have slowly invaded American politics.

And the problem is that IMO there are legitimate and substantive critiques of Common Core, ranging from the structural (there’s a legitimate argument that a nation of 300 million doesn’t actually need to teach math and science in exactly the same order from state to state) to the specific (not everyone agrees that it makes sense to teach perimeter and area in the same year, or that third graders need to concern themselves with literary themes).

But the crazy shouty complaints about common core make it difficult to hear legitimate objections, and it’s tempting for non-crazy people to suppport common core reflexively, just so that other people don’t think they’re a tinfoil hat wearer.

[John_Bredin]

@Left_Hand_of_Dorkness

(there’s a legitimate argument that a nation of 300 million doesn’t actually need to teach math and science in exactly the same order from state to state)

And a legitimate argument the other way. In a nation of 300 million which tends to be one labor market, people move from state to state for work, etc., and their children end up in a different school district in a different state. There’s a value in a kid transferred from New Jersey to North Dakota over the summer between 3rd and 4th grade being at essentially the same place in the curriculum instead of a hodge-podge of behind in some subjects and ahead in others for no particular reason.

[even_sven]

@John_Bredin

And a legitimate argument the other way. In a nation of 300 million which tends to be one labor market, people move from state to state for work, etc., and their children end up in a different school district in a different state. There’s a value in a kid transferred from New Jersey to North Dakota over the summer between 3rd and 4th grade being at essentially the same place in the curriculum instead of a hodge-podge of behind in some subjects and ahead in others for no particular reason.

Another important factor it’s kind of silly that people who develop innovative curriculum design and teaching materials are often limited to an audience of a single state. The teacher in California has a rich network of teachers to share with and get ideas from, while the teachers in Montana are stuck with what their small cohort can produce.