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Unsolved problems with the common core

75 points by rflrob 11y ago

beneater 11y ago
That article has a terrible title. It implies it discusses deficiencies with the Common Core standards. A much better title would be something like "Open questions in mathematics that align to Common Core standards"
- benkuykendall 11y ago
  I interpreted it as a joke.
DanBC 11y ago
I love this, and I'm going to try some of these with my child. > The four color theorem states (informally) that “given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.” (from wikipedia). The mathematics statement is that any planar graph can be colored with four colors. Thus, the first part of the “warm up” has a solution; in fact the world map can be colored with four colors. The four color theorem is deceivingly simple- it can be explored by a kindergartner, but it turns out to have a lengthy proof. In fact, the proof of the theorem requires extensive case checking by computer. Not every map can be colored with three colors (for an example illustrating why see here). It is therefore natural to ask for a characterization of which maps can be 3-colored. Of course any map can be tested for 3-colorability by trying all possibilities, but a “characterization” would involve criteria that could be tested by an algorithm that is polynomial in the number of countries. The 3-colorability of planar graphs is NP-complete. This is the background and context for the problem for kindergarten. It would be good if educators and writers could provide a version of this for young children and their teachers.
- whoopdedo 11y ago
  I've never understood why the proof had to be so complicated. Instead of color filled areas, use nodes and arcs. One and two nodes are trivial. When you add a third node it either connects to one other node and then can be just a second color, or it connects to both and you have an enclosed region (two actually). When you add the fourth node it will have a fourth color if it has three arcs. You now have 6 arcs which is the maximum possible with four nodes. To reiterate, a map the requires three colors has three nodes and three arcs; a map the requires four colors has four nodes and 6 arcs. So a map that would require five colors should have five nodes and ten arcs. But that's not possible.
  - johntb86 11y ago
    > a map the requires four colors has four nodes and 6 arcs That's not the only case - there are an infinite number graphs that require 4 colors. It's possible that all of them have a subgraph with four nodes and 6 edges connecting them, but I don't know either way. > So a map that would require five colors should have five nodes and ten arcs How do you know there isn't a graph with 35 nodes and 127 edges that requires 5 colors?
  - cokernel 11y ago
    The standard proof does use nodes and arcs (or vertices and edges, which I am taking to mean the same thing). A proper coloring must assign distinct colors to neighboring vertices, but this doesn't mean that vertices with the same color are identified. Here's an overview of the simplest version of the proof as of 1998: http://www.ams.org/notices/199807/thomas.pdf http://www.ams.org/notices/199807/thomas.pdf . This should shed some light on why the proof is as complicated as it is. (See especially the section on equivalent formulations.)
    - cokernel 11y ago
      I'm way out of date. The proof (discretized using hypermaps instead of graphs) was completely formalized in Coq in 2005: http://research.microsoft.com/en-us/um/people/gonthier/4colproof.pdf http://research.microsoft.com/en-us/um/people/gonthier/4colp...
  - 11y ago
viraptor 11y ago
I can't tell if this post is serious, or making fun of the topic. I got to the point where the author thinks it's a good idea to talk about graph k-colouring with kids in kindergarten. Does anyone think this can have any meaningful discussion beyond a "you can colour any map with 4 colours" trivia? Just as a reminder, these are points from the grade k overview: - Know number names and the count sequence. - Count to tell the number of objects.
- Rusky 11y ago
  Getting kids to think about and experiment with graph coloring is probably a more reasonable and productive goal than trying to have a "meaningful discussion" about it with kindergarteners. It's certainly not something that stood out as ridiculous to me.
  - aaron695 11y ago
    http://www.pbs.org/parents/childdevelopmenttracker/five/mathematics.html http://www.pbs.org/parents/childdevelopmenttracker/five/math... "At age five, some children may still be gaining an understanding of the number words up to "four" (e.g., distinguishes one-four items from "many"; can identify collections of up to four items with a corresponding number; asks for up to "four" of something; knows age; can put out "one," "two," "three," or "four" items upon request)." So rather than setting a childs education backward and depress them with abstracts before nouns, when I read about it and understood the question at age 10 is was kinda a cool and interesting problem. That's if you want to take the ages and article seriously, which the OP was asking if the original was. It being more a thought piece I suspect.
  - baldfat 11y ago
    I work for Head Start and we actually have "meaningful" conversations with children about what interest them. Making math, engineering or science "interesting" is very natural for child. The hardest part is getting adults to think through those lens and break them down into simple interconnecting ideas to the child's world and help them learn new vocabulary. Kids are natural learners.
  - 11y ago
vilhelm_s 11y ago
Heh, their "Kindergarten-level" problem is proving P≠NP? :)

75 points by rflrob 11y ago

That article has a terrible title. It implies it discusses deficiencies with the Common Core standards. A much better title would be something like "Open questions in mathematics that align to Common Core standards"

benkuykendall 11y ago
I interpreted it as a joke.

DanBC 11y ago

I love this, and I'm going to try some of these with my child. > The four color theorem states (informally) that “given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.” (from wikipedia). The mathematics statement is that any planar graph can be colored with four colors. Thus, the first part of the “warm up” has a solution; in fact the world map can be colored with four colors. The four color theorem is deceivingly simple- it can be explored by a kindergartner, but it turns out to have a lengthy proof. In fact, the proof of the theorem requires extensive case checking by computer. Not every map can be colored with three colors (for an example illustrating why see here). It is therefore natural to ask for a characterization of which maps can be 3-colored. Of course any map can be tested for 3-colorability by trying all possibilities, but a “characterization” would involve criteria that could be tested by an algorithm that is polynomial in the number of countries. The 3-colorability of planar graphs is NP-complete. This is the background and context for the problem for kindergarten. It would be good if educators and writers could provide a version of this for young children and their teachers.

whoopdedo 11y ago
I've never understood why the proof had to be so complicated. Instead of color filled areas, use nodes and arcs. One and two nodes are trivial. When you add a third node it either connects to one other node and then can be just a second color, or it connects to both and you have an enclosed region (two actually). When you add the fourth node it will have a fourth color if it has three arcs. You now have 6 arcs which is the maximum possible with four nodes. To reiterate, a map the requires three colors has three nodes and three arcs; a map the requires four colors has four nodes and 6 arcs. So a map that would require five colors should have five nodes and ten arcs. But that's not possible.
- johntb86 11y ago
  > a map the requires four colors has four nodes and 6 arcs That's not the only case - there are an infinite number graphs that require 4 colors. It's possible that all of them have a subgraph with four nodes and 6 edges connecting them, but I don't know either way. > So a map that would require five colors should have five nodes and ten arcs How do you know there isn't a graph with 35 nodes and 127 edges that requires 5 colors?
- cokernel 11y ago
  The standard proof does use nodes and arcs (or vertices and edges, which I am taking to mean the same thing). A proper coloring must assign distinct colors to neighboring vertices, but this doesn't mean that vertices with the same color are identified. Here's an overview of the simplest version of the proof as of 1998: http://www.ams.org/notices/199807/thomas.pdf http://www.ams.org/notices/199807/thomas.pdf . This should shed some light on why the proof is as complicated as it is. (See especially the section on equivalent formulations.)
  - cokernel 11y ago
    I'm way out of date. The proof (discretized using hypermaps instead of graphs) was completely formalized in Coq in 2005: http://research.microsoft.com/en-us/um/people/gonthier/4colproof.pdf http://research.microsoft.com/en-us/um/people/gonthier/4colp...
- 11y ago

viraptor 11y ago

I can't tell if this post is serious, or making fun of the topic. I got to the point where the author thinks it's a good idea to talk about graph k-colouring with kids in kindergarten. Does anyone think this can have any meaningful discussion beyond a "you can colour any map with 4 colours" trivia? Just as a reminder, these are points from the grade k overview: - Know number names and the count sequence. - Count to tell the number of objects.

Rusky 11y ago
Getting kids to think about and experiment with graph coloring is probably a more reasonable and productive goal than trying to have a "meaningful discussion" about it with kindergarteners. It's certainly not something that stood out as ridiculous to me.
- aaron695 11y ago
  http://www.pbs.org/parents/childdevelopmenttracker/five/mathematics.html http://www.pbs.org/parents/childdevelopmenttracker/five/math... "At age five, some children may still be gaining an understanding of the number words up to "four" (e.g., distinguishes one-four items from "many"; can identify collections of up to four items with a corresponding number; asks for up to "four" of something; knows age; can put out "one," "two," "three," or "four" items upon request)." So rather than setting a childs education backward and depress them with abstracts before nouns, when I read about it and understood the question at age 10 is was kinda a cool and interesting problem. That's if you want to take the ages and article seriously, which the OP was asking if the original was. It being more a thought piece I suspect.
- baldfat 11y ago
  I work for Head Start and we actually have "meaningful" conversations with children about what interest them. Making math, engineering or science "interesting" is very natural for child. The hardest part is getting adults to think through those lens and break them down into simple interconnecting ideas to the child's world and help them learn new vocabulary. Kids are natural learners.
- 11y ago

vilhelm_s 11y ago

Heh, their "Kindergarten-level" problem is proving P≠NP? :)