Common Core Math: Interesting Article

[wapiti]

I have a lot of thoughts on this topic though it's not easy to untangle them.

 

It seems to me that - from my vague understanding - the type of writing about math that is being requested from early elementary students is painfully pedantic for average students and even moreso for advanced math students.

 

Perhaps we should consider the two-column proofs in the older algebra 1 texts (even though they too may have that painfully pedantic quality).  There's a reason the two-column format came to be, I assume in part for easier grading purposes, among other reasons.  More generally, why did proofs in algebra 1 fall out of favor?  Why doesn't CC bring them back (or alternatively, paragraph-style proofs) for algebra 1?  Is the writing in early elementary math akin to proofs previously asked of algebra 1 students and is that developmentally appropriate, i.e., is there a reason that for many many years, proofs were not required prior to algebra 1?  Is there a middle ground, e.g., proofs in middle school?  Are written proofs, formal or informal, the only way to gauge concept understanding?  I haven't had enough coffee yet, but maybe some of you can see where I'm going with this...

[Farrar]

I think pedantic doesn't quite describe it. I think it's just confusing for a lot of younger kids. I mean, once you grasp that 2+2 is 4, justifying it when you're trying to solve something slightly more complex, like 22+4 feels redundant. It's like trying to define the most basic vocabulary. I think even little kids chafe when asked to define really easy words. It only begins to make sense to have definitions when you start getting to more complex words. And it only makes sense to really get into proofs when you get into more complex math, like, multistep word problems and eventually to algebra.

 

Instead, a culture of discussion and manipulatives and just having problems that are complex enough to challenge thinking in this way seems like a vastly superior solution.

 

And then, yeah, maybe algebra should go back to proofs. Like I said, I am really finding Jousting Armadillos, with its emphasis on "Notes to Self" and opening chapters about logic, really solid as an introduction for kids.

[Spy Car]

If a 1st Grader had X + 5 = 9 they could explain that 9 is the "whole value" (or the sum) and that 5 is one "part" (or addend) of the whole (sum).

 

When we have the "whole" we find the missing part (or missing addend) by finding the "difference"between  the whole and the known part.

 

9 (the whole) - 5 (the known part) = 4 (the missing or unknown part or addend).

 

X = 4

 

With C Rods you would have a 9 value rod (Blue) with a 5 value rod (yellow) lined up above it, and find the rod that makes up the difference: the 4 rod ( pinkish-purple).

 

X = 4

 

Understanding this is central to good math education, and a focus of programs like Miquon, MEP, and Primary Mathematics (Singapore).

 

Bill

 

 

[Roadrunner]

If a 1st Grader had X + 5 = 9 they could explain that 9 is the "whole value" (or the sum) and that 5 is one "part" (or addend) of the whole (sum).

 

When we have the "whole" we find the missing part (or missing addend) by finding the "difference"between  the whole and the known part.

 

9 (the whole) - 5 (the known part) = 4 (the missing or unknown part or addend).

 

X = 4

 

With C Rods you would have a 9 value rod (Blue) with a 5 value rod (yellow) lined up above it, and find the rod that makes up the difference: the 4 rod ( pinkish-purple).

 

X = 4

 

Understanding this is central to good math education, and a focus of programs like Miquon, MEP, and Primary Mathematics (Singapore).

 

Bill

Understing this is, but my point is writing all of what you just did, isn't. I can't imagine most elementary school students articulating and writing out what you just did. 

 

X+5 = 9

X=9-5

X=4

draw a line and write a check under

4+5 =9

 

This should be enough. No paragraph necessary. Nobody is arguing that understanding is not necessary. Nowhere does SM ask for what CC is asking kids to do. Yet I would argue SM does an outstanding job teaching concepts. 

[Renai]

IMACS linked this article from The Atlantic criticizing aspects of Common Core math standards.  I think it's an interesting read.

 

It is interesting, but the tests I had to mark did not show the level of explanation in words shown in that article (Need, know, do) and could still be considered correct.

[Farrar]

Right, a child should be able to orally explain that. Or show it with C-rods. But they haven't come up with a good way to test that, they don't trust the teachers to teach it that way (and... some of that is justified, but it's also a vicious circle) so they put it on the test. But to put it on the test means they have to come up with a formulaic way to ask for it (aka more algorithm memorization) or they have to ask first graders to write sentences about it where they're inevitably being grading as much if not more on their ability to express themselves in writing. Not okay. And the way that CC is written, I don't know any way around this.

 

[Spy Car]

Understing this is, but my point is writing all of what you just did, isn't. I can't imagine most elementary school students articulating and writing out what you just did. 

 

X+5 = 9

X=9-5

X=4

draw a line and write a check under

4+5 =9

 

This should be enough. No paragraph necessary. Nobody is arguing that understanding is not necessary. Nowhere does SM ask for what CC is asking kids to do. Yet I would argue SM does an outstanding job teaching concepts. 

 

I simply disagree about what PM is expecting children to understand and be able to articulate. The whole-parts method is at the heart of the Singapore Math Model, including the bar-diagram method of word problem solving. If they don't expect the method in my first post on this, they certainly do expect what's in the second. This is fundamental to the Primary Mathematics approach.

 

Bill

[Roadrunner]

I simply disagree about what PM is expecting children to understand and be able to articulate. The whole-parts method is at the heart of the Singapore Math Model, including the bar-diagram method of word problem solving. If they don't expect the method in my first post on this, they certainly do expect what's in the second. This is fundamental to the Primary Mathematics approach.

 

Bill

 

But that is not what I am stating. I am talking about written output, not understanding. Are you implying that written output required by CC and SM is similar? Nobody is here arguing understanding. 

[Spy Car]

But that is not what I am stating. I am talking about written output, not understanding. Are you implying that written output required by CC and SM is similar? Nobody is here arguing understanding. 

 

If PM is done correctly, yes, I'm saying there is no difference. If PM is limited to just filling in the blanks in the Workbook, or treated as if that is considered sufficient by the authors of the program, maybe. But that is not how PM is supposed to be used. Making sure the students understand and can express their understanding is the role of the educator, including home-educators. The Home Educators Guides were written to make that abundantly clear.

 

I see nothing different between the objectives of PM and CC, same some fiddling on scope and sequence. But being able to articulate mathematical reason is, quite properly, a top objective of both the Singapore standards and the Common Core standards. 

 

Bill

[Roadrunner]

If PM is done correctly, yes, I'm saying there is no difference. If PM is limited to just filling in the blanks in the Workbook, or treated as if that is considered sufficient by the authors of the program, maybe. But that is not how PM is supposed to be used. Making sure the students understand and can express their understanding is the role of the educator, including home-educators. The Home Educators Guides were written to make that abundantly clear.

 

I see nothing different between the objectives of PM and CC, same some fiddling on scope and sequence. But being able to articulate mathematical reason is, quite properly, a top objective of both the Singapore standards and the Common Core standards. 

 

Bill

 

 I don't think you are reading what I am writing.

 

Who said PM is filling in workbooks? I have been homeschooling with SM for a while now with HIG, IPs, CWPs, workbooks and the like, so I know what SM requires. Yet I will repeat this again. Nowhere does SM require written output similar to CC. Once again, we aren't talking out understanding. 

[Spy Car]

 I don't think you are reading what I am writing.

 

Who said PM is filling in workbooks? I have been homeschooling with SM for a while now with HIG, IPs, CWPs, workbooks and the like, so I know what SM requires. Yet I will repeat this again. Nowhere does SM require written output similar to CC. Once again, we aren't talking out understanding. 

 

There is a whole bar-diagram method that requires problem solving to be shown representationally. Student learn to solve work and concretely, pictorially, and using mathematical symbols. The means of explanation are very little different between the PM materials and the math texts we've had through school. 

 

It was certainly my expectation that my child could (and did) justify his answers when using PM (or now in AoPS) in exactly the same fashion CC expects. I think the ability to explain and show ones work is fundamental to sound education. My old posts are searchable. I've always said "the right answer" isn't enough, and that students should be able to explain their reasoning in mathematical terms. I still believe that is the case.

 

I'm happy to grant the fact that it is easier (and more productive) to engage in Socratic dialogue with a student to probe their understanding, that giving a test. Especially at a young age. But the aim is the same. 

 

Bill

[Roadrunner]

Another critical piece on CC. This is what we are seeing in local schools.

http://www.wsj.com/articles/marina-ratner-making-math-education-even-worse-1407283282

[Spy Car]

Another critical piece on CC. This is what we are seeing in local schools.

http://www.wsj.com/articles/marina-ratner-making-math-education-even-worse-1407283282

 

Should we be surprised that excellence in math education will come under attack? It has been the same story for more than 50 years. Going back to nothing but memorization and drill, and the "plug and chug" approach of putting supplied numbers into a supplied algorithm will not advance us as a nation in a fashion that meets the needs of the 21st Century world we're living in.

 

If tests and teaching methods can be improved, great! But going backwards is not an option.

 

Bill

[Roadrunner]

Should we be surprised that excellence in math education will come under attack? It has been the same story for more than 50 years. Going back to nothing but memorization and drill, and the "plug and chug" approach of putting supplied numbers into a supplied algorithm will not advance of as a nation is a fashion that meets the needs of the 21st Century world we're living in.

 

If tests and teaching methods can be improved, great! But going backwards is not an option.

 

Bill

I assure you that a Russian trained professor of   mathematics at UC Berkeley isn't advocating "plug and chug" approach. Those accusations aren't based on any evidence. 

 

CC isn't the only answer to memorization. There is plenty of land in-between. 

[Tsuga]

Understing this is, but my point is writing all of what you just did, isn't. I can't imagine most elementary school students articulating and writing out what you just did.

 

X+5 = 9

X=9-5

X=4

draw a line and write a check under

4+5 =9

 

This should be enough. No paragraph necessary. Nobody is arguing that understanding is not necessary. Nowhere does SM ask for what CC is asking kids to do. Yet I would argue SM does an outstanding job teaching concepts.

In first grade, that, plus ten rods and one dots, would suffice as a perfect answer in our schools.

 

Just that answer would be correct but a lack of drawing would not get you the additional "use pictures or words". In BA they have the same matrices, even more, than my daughter gets in her Math Expressions books. It's more conceptual, not less. They don't say "draw a picture" but the drawing has helped her attack problems that are too difficult.

 

As for the subsequent articles... I won't discuss this with anyone who hasn't read the standards which are appropriately flexible. I am not going to defend Pearson or the defunding of American public schools or crappy teaching.

 

I will defend a set of comprehensive common standards that help us measure how well or poorly privatization and defunding are helping children.

[Tsuga]

I assure you that a Russian trained professor of mathematics at UC Berkeley isn't advocating "plug and chug" approach. Those accusations aren't based on any evidence.

 

CC isn't the only answer to memorization. There is plenty of land in-between.

CC is not even an answer to that. I have a hard time believing you have read the standards. Have you read them, line by line? There is room for a million different ways to teach concepts. And there is room for procedural fluency and math facts. But you do need to connect procedures to some other way of describing the relationships between numbers, other than line by line procedures, yes. However in my kids' classes dots and sticks or "quick arrays" (5 marks by six makes to represent a 5x6 matrix) have always been acceptable. My stepson and stepdaughter don't even do that much anymore.

[Ray]

Is Common Core making a claim that it'a adoption by Schools will improve something about Math for the learner? And if so can someone post links to the research behind the claimed benefits. Thank you.

[Roadrunner]

Is Common Core making a claim that it'a adoption by Schools will improve something about Math for the learner? And if so can someone post links to the research behind the claimed benefits. Thank you.

If it isn't meant to improve learning, then what would be the point of adopting? 

[Ray]

If it isn't meant to improve learning, then what would be the point of adopting?

Well yes that seems logical, but I went to the CC web site looking for a plain statement about its intent to better wrap my brain around the debate going on here. So if we assume the CC is meant to improve math education then we can ask how this is a result of Common Core.

[sassenach]

There are no fees. It is 100% donations and many employees get corporate matching up to 100% in some cases. There is no public record of membership. Only the PTSA knows and many donations are made anonymously. But they fundraise hard and try to get big money from families that can give. From what I have seen as a non member, member and board member, my kids were treated the same everywhere. Nobody gave us any issues when we did not pay (it was actually an oversight I learned of at year's end).

 

Please note it's not the "base" teachers, but assistants, after school programs like homework helpers, etc. and technically this is all the district, but fundraising emphasizes "freeing up funds for X people in the classroom y hrs / week". So the assistants, math help under the guise of "enrichment", whatever can be done legally to support teachers.

 

I know this is happening in at least five districts, not all wealthy, because I heard about it at a state level meeting. Whoever can raise that much is absolutely raising it for man hours. I know it is happening in Seattle as well because I know people on the PTSA who were raising for addidional English instruction for ESL kids, beyond what the state pays for.

This is pretty much standard with all of the districts in our county. How much they ask for depends on the district. Our local district asks for $500 per child. My friend's school had its own foundation, which asked for $800 per child. Everyone just shrugs and says it's cheaper than private school. It's not mandatory, but it is highly encouraged.

[Ray]

 

oops double post

[Tsuga]

Is Common Core making a claim that it'a adoption by Schools will improve something about Math for the learner? And if so can someone post links to the research behind the claimed benefits. Thank you.

 

No. That is a gross misunderstanding of what kind of policy this is.

 

There is a government initiative to set benchmarks nationally so that students across the country can see how their schools compare to others before the SATs; so that children can move from school to school more fluidly and missing less material; so that schools cannot abrogate their obligation to provide each child a full education by skimping on education. This might seem silly, because wouldn't every community want the same for their children? But it was increasingly obvious that even comparing students from the same race, socio-economic background, and so on, that they were NOT all getting even close to equal opportunity in education.

 

This is a government standard policy and NOT a curriculum nor a tool to achieve the standards. That is left to implementing school districts. You can adopt any standard you want but it should be clear that unless you provide the necessary investment in achieving that goal, it won't happen.  It's like if in the 60s they said we're going to fly to the moon... and then stopped and let every state try to make their own rocket if they had enough money. What CC will really do, amusingly enough, is just highlight what a crappy job some schools are doing by providing objective benchmarks for education.

 

Ironically, all the complaints about CC would be impossible to make if we didn't have the objective standard whereby to measure achievement.

 

Pearson and some private companies and sub-contractors have, as usual, taken advantage of particularly corrupt systems in some areas, and less corrupt but still venal systems in others, to use the initiative for their own gain by selling their goods to schools even if they are not the best.

 

On the other hand, in the top districts, many have found that this only barely keeps up to their own curricula and that they haven't had to adapt much at all. At our district, the continuum is the same but they do have more work shown and more proofs, which is fine, since all the upper-class kids were paying to do that in the tutoring centers.

 

Now the public schools must provide the same education to all.

[Ray]

No. That is a gross misunderstanding of what kind of policy this is.

 

There is a government initiative to set benchmarks nationally so that students across the country can see how their schools compare to others before the SATs; so that children can move from school to school more fluidly and missing less material; so that schools cannot abrogate their obligation to provide each child a full education by skimping on education. This might seem silly, because wouldn't every community want the same for their children? But it was increasingly obvious that even comparing students from the same race, socio-economic background, and so on, that they were NOT all getting even close to equal opportunity in education.

 

This is a government standard policy and NOT a curriculum nor a tool to achieve the standards. That is left to implementing school districts. You can adopt any standard you want but it should be clear that unless you provide the necessary <quote  sniped>

 

So it is a measuring device to ensure schools are providing a timely education for students with a component of the benchmarks being a judgment of the students understanding of the why behind the math?

 

Question: If a student were asked "explain why 6984 is a multiple of 18" would an answer of 6984/18 = 388 be good?

[Spy Car]

Question: If a student were asked "explain why 6984 is a multiple of 18" would an answer of 6984/18 = 388 be good?

I suppose it would depend on the age. By Pre-Algebra or Algebra one would hope a student could see right away using the divisibility rules for 18 that 6984 is both divisible by 2 and the sum of the digits is divisible by 9, so 6948 meets the divisibility rules for 18 (being divisible by both 2 and 9).

 

Then if the student needed to explain why the divisibility rules work, they could show you using prime factorization.

 

The goal is to develop higher level math skills.

 

Bill

[Tsuga]

 

So it is a measuring device to ensure schools are providing a timely education for students with a component of the benchmarks being a judgment of the students understanding of the why behind the math?

 

 

No. I suggest you read it. It is a list of standards, benchmarks, metrics.

 

It is not an instrument whereby to measure progress.

 

It is not a method whereby to achieve the end.

 

It is simply a list of desired outcomes by grade.

 

That's it.

 

States and districts must come up with plans, inputs, outputs, and resources to achieve those, themselves. They must also come up with the means to measure whether the outcomes are achieved, and whether the achievement of outcomes can be related to the plans, inputs, outputs and resources, or not, or to what degree.

 

Common Core needs a little sign: "Don't Blame the Messenger". It's world-class standards. Meeting them in rural school districts with a history of shooting down property tax levies? LOL good luck with that.

 

And yes, this IS a sneaky way to get people to realize how crappy their schools are and to demand better even if that means the federal government funding education.  :svengo: The first nationwide comparison of schools with the PISA was a GW Bush initiative. They didn't have national standards so they used international standards, the PISA.

 

Compare schools here:

 

http://globalreportcard.org/map.html

 

 

I suppose it would depend on the age. By Pre-Algebra or Algebra one would hope a student could see right away using the divisibility rules for 18 that 6984 is both divisible by 2 and the sum of the digits is divisible by 9, so 6948 meets the divisibility rules for 18 (being divisible by both 2 and 9).

 

Then if the student needed to explain why the divisibility rules work, they could show you using prime factorization.

 

The goal is to develop higher level math skills.

 

Bill

 

I agree.

 

For my daughter who is in third, they are still looking to make sure they understand the relationship between the numbers and concrete objects / amounts. So they would be asked to draw a "fast array" showing an array of 18 * 388 which looks like this:

 

__________________________388_____________________________

|

|

18

|

|

 

When you have a 388 x 18 array, you have exactly 6,948 with no remainder. That's a good enough explanation in our schools (I'm not going to take a picture of my kids' schoolwork--you'll have to believe me). In second grade, you could have said "with none left over" but now they are focusing on "math vocabulary" so they get extra points for using words like "remainder" and "matrix" and "array".

 

My daughter is not verbally gifted (last verbal IQ tested her at slightly below average). She is not mathematically gifted (math IQ above average and that's with a lot of support at home). She is in the third grade of an excellent but open-enrollment neighborhood public school not a charter, not a magnet. Standard-issue, standard-salary PS teachers but with a lot of support from parents.

She does this every week.

 

If my kids can do it through hard work and diligence then most kids should be able to do it if they have proper instruction.

 

DD2 also does work like this. I make them. School makes them. Bright, not gifted. For both, they developed English in a foreign language environment and while DD1 sounds precocious, you can tell on tests that she is not highly advanced. Like their mom, they score about average for a well-educated boy, verbally. So I don't think this is a "girl thing". Yes, it's a verbal thing. So... work harder.

 

I get how this is hard for some kids, but school is their job. It's supposed to be hard. English should also be harder.

 

Another way to deal with it was as Spy Car said--I suppose that's the fourth grade or fifth grade answer.