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"
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.
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.
> 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?
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 .
This should shed some light on why the proof is as complicated as it is. (See especially the section on equivalent formulations.)
"I've never understood why the proof had to be so complicated."
Others are trying to explain, but I'd point out, as the article observes, this is very accessible math. If you really want to know, sit down and try it.
The only warning I'd give is that this is an extraordinarily well-examined problem, so I'd encourage you as strongly as I can that if you think you found a simple proof that you leap to the conclusion that you must be missing something, and then go searching for that something, rather than believing that you've found the simple proof.
I'm being serious, not sarcastic at all. This sort of thing is really quite educational, with counterexamples literally producible with a bit of doodling and four colors of pen, and can be quite fun, and this is far more accessible than many such mathematical exercises.
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:
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.
"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.
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.
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.
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.
You don't have to call it "graph k-coloring". Especially when so much of math is taught as a solved thing, it's exciting to know there are problems which nobody knows how to solve. You can learn a lot by trying to solve these problems yourself.
>it's exciting to know there are problems which nobody knows how to solve
Not sure this fits in with the systemic cultural goal of authoritarianism and conformity at all costs. I mean we both know its a good goal in an abstract sense, but its directly at odds with what we pay them money to indoctrinate into kids. "Sometimes your leaders can't tell you what to think because they don't know" isn't exactly standard authoritarian conditioning. And "think and imagine" isn't standard conformity-inducing conditioning. Furthermore "here's something important, but we can't graph a metric result on a graph" is inherently insubordinate to existing leadership dogma.
Who do you know who pays money explicitly to indoctrinate "authoritarianism and conformity at all costs" into kids? It's true that schools tend to emphasize these values, but I think that's a side-effect of the structure of the system, not an explicit goal. Individual teachers can and often do encourage "thinking and imagining".
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[ 154 ms ] story [ 667 ms ] thread> 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.
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.
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?
Here's an overview of the simplest version of the proof as of 1998: 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.)
Others are trying to explain, but I'd point out, as the article observes, this is very accessible math. If you really want to know, sit down and try it.
The only warning I'd give is that this is an extraordinarily well-examined problem, so I'd encourage you as strongly as I can that if you think you found a simple proof that you leap to the conclusion that you must be missing something, and then go searching for that something, rather than believing that you've found the simple proof.
I'm being serious, not sarcastic at all. This sort of thing is really quite educational, with counterexamples literally producible with a bit of doodling and four colors of pen, and can be quite fun, and this is far more accessible than many such mathematical exercises.
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.
"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.
On the other hand if anyone wants to prove me wrong about kindergarten kids, I'd be glad to hear that.
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.
Not sure this fits in with the systemic cultural goal of authoritarianism and conformity at all costs. I mean we both know its a good goal in an abstract sense, but its directly at odds with what we pay them money to indoctrinate into kids. "Sometimes your leaders can't tell you what to think because they don't know" isn't exactly standard authoritarian conditioning. And "think and imagine" isn't standard conformity-inducing conditioning. Furthermore "here's something important, but we can't graph a metric result on a graph" is inherently insubordinate to existing leadership dogma.