
Unsolved problems with the common core - rflrob
https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/
======
beneater
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
I interpreted it as a joke.

------
viraptor
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
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.

~~~
viraptor
I think I didn't use the right phrase. I was thinking of something more
general than discussion - meaningful interaction? But the main point was that
I don't think those kids would understand map colouring as something beyond
map colouring. It's just trivia - you can do it with 4 colours. A huge
majority of people of any age doesn't know why that fact is in any way
significant.

On the other hand if anyone wants to prove me wrong about kindergarten kids,
I'd be glad to hear that.

~~~
lkozma
I thought that example was quite nice, and it might work with kindergarten
kids. It has to be kept easy and not overwhelming, though. I'd do it like
this:

First take some simpler map that can be 3-colored and ask the child to 3-color
it. Make sure it is always correct (no neighbors have the same color).

Then give a simple map that cannot be 3-colored. Let the child try it for a
while, try arguing together or let them try to argue why it is not possible.

Maybe even start with 2-colorable first, and move up slowly to 3 and then to
4.

Mentioning that 4 always works is optional.

Depending on the child's patience, they can even try coming up with the
3-counterexample themselves. It can be a back-and-forth of challenges and
solutions between the child and adult.

------
DanBC
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
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.

~~~
cokernel
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
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/4colp...](http://research.microsoft.com/en-
us/um/people/gonthier/4colproof.pdf)

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

