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Math for seven-year-olds

jdh.hamkins.org

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Re: Math for seven-year-olds

#31

Post by slackpad »

kaitai wrote: post #29
slackpad wrote: post #28

This is the kind of experience would never be allowed with the Common Core type approach. I can't imagine people spending a classroom day on something like this (even though it's awesome and super valuable), but alas it's not content that will appear on the standardized tests.

Why wouldn't this be allowed with the Common Core? The Common Core is a set of standards for what students should have learned by the end of a year, and for fourth grade, for instance, the Core consists of things like "Use the four operations with whole numbers to solve problems," "Understand decimal notation for fractions, and compare decimal fractions, " and "Generate and analyze patterns." One can fault the Common…

That's a fair point - his content is certainly compatible with the Common Core's standard. I was really venting about your point [2] and shouldn't have implicated the standard for that. Thanks for the references.

Re: Math for seven-year-olds

#32

Post by lifeofanalysis »

Interesting co-incidence. Just last week, I sent out an email to my friends, saying that I want to teach mathematics to their kids by a math-by-email service, kinda like the chess-by-email of the old days.

The idea is quite simple: I will send out a daily email with a grade appropriate set of math questions and/or games. Your child provides the answers back by email. I check the answers, provide corrections/feedback. And the next day's worksheet is customized to the child's history. If there is an interest, I could follow the child all the way from pre-K to graduate level Math subjects. Think about that -- wouldn't it be awesome if when you are getting your PhD, you could look back over 20 years of daily problems you solved and how you progressed in your conceptual understanding?

Naturally, you want to balance the gaming aspect with the rigorous aspects. You can start learning Graph Theory with diagram filling, but as you get more serious, there is no substitute to solving several hard problems to get a deeper/intuitive understanding of the concepts. There is no question that people learn different ways, some visually, others through games, and others through mental modeling. I am convinced that if we could tailor math teaching to each kid, we could get rid of the stigma that "Math is hard", or, worse, "Girls can't do Math".

Math, as I say, is a contact sport, not a spectator sport. You have to grab a pencil and a piece of paper to work on 20-30-40 harder and harder problems to master each concept. But to learn new concepts, you also have to cross the significant hurdle of climbing the first few rungs of each concept, so to speak. So let's learn by balancing games and theory.

Re: Math for seven-year-olds

#34

Post by j2kun »

pbhjpbhj wrote: post #26
j2kun wrote: post #22

Earlier quoted context omitted.

I have given lectures to students of this age range (13-18 years), but I don't have a booklet. See this blog post: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...

That's great, an enjoyable and inspiring read. Can you - or another reader - recommend a resource that's perhaps just a little more advanced though, please? What's the next step after formalising the language used, understanding connectivity and path traversal ... perhaps, what would the subheadings of a follow up lesson to this be? I'm not asking to be spoonfed, really!, there are just so many resources on the web a…

I think a good answer is going to depend heavily on your goals (applied or theoretical). Maybe you could elaborate on that?

Re: Math for seven-year-olds

#35

Post by j2kun »

kaitai wrote: post #29
slackpad wrote: post #28

This is the kind of experience would never be allowed with the Common Core type approach. I can't imagine people spending a classroom day on something like this (even though it's awesome and super valuable), but alas it's not content that will appear on the standardized tests.

Why wouldn't this be allowed with the Common Core? The Common Core is a set of standards for what students should have learned by the end of a year, and for fourth grade, for instance, the Core consists of things like "Use the four operations with whole numbers to solve problems," "Understand decimal notation for fractions, and compare decimal fractions, " and "Generate and analyze patterns." One can fault the Common…

At a high level you're right, Common Core philosophy does encourage this sort of lesson. The unfortunate part is that Common Core still is largely a list of facts that students need to know, and when they need to know them. This is why graph theory is not "compatible" with the standards, because in the huge list of topics they pretend these things don't exist. And since the Common Core is really a document for non-teachers, I agree with the original comment that this wouldn't fly. It's just due to opposition by administrators, test writers, and standards-compliance checkers who read the document too thoroughly and, paradoxically, too shallowly. They don't look at this lesson to see whether it fits with the philosophy of learning espoused by the standard; rather, they check off which standards it covers and complain when the answer is zero. I put some fault in the document for this.

Re: Math for seven-year-olds

#36

Post by j2kun »

sgustard wrote: post #30
kirsebaer wrote: post #27

Earlier quoted context omitted.

Here is TEDx talk, "Why Math Education is Unnecessary", by a math teacher arguing that compulsory math classes should be replaced with voluntary math-based games: https://www.youtube.com/watch?v=xyowJZxrtbg Here is an essay by a mathematician, A Mathematician's Lament by Paul Lockhart (2002), arguing that math education is completely stupid, like teaching art only through paint-by-number exercises, and should be repl…

Music is taught by the equivalent of paint by number, which is reading sheet music. Or play, which is improvisation. There's room for both.

I agree, but I think the amount of "paint by number" in mathematics should be minimized, and most of it (learning multiplication tables, isolating variables in simple equations) is already in the form of games. I'd be interested to hear of a topic in math that you would argue can't be turned into a non-superficial game.

Re: Math for seven-year-olds

#37

Post by Someone »

vanderZwan wrote: post #4

> Next, we considered Eulerian paths and circuits, where one traces through all the edges of a graph without lifting one’s pencil and without retracing any edge more than once. We started off with some easy examples, but then considered more difficult cases. Wait, there's math devoted to that? I used to do that as a kid for fun!

Yes, but it is somewhat dull math because the problem is relatively simple. See http://en.wikipedia.org/wiki/Eulerian_graph, and compare it with http://en.wikipedia.org/wiki/Hamiltonian_path and http://en.wikipedia.org/wiki/Hamiltonian_path_problem

Re: Math for seven-year-olds

#38

Post by quadrangle »

sgustard wrote: post #30
kirsebaer wrote: post #27

Earlier quoted context omitted.

Here is TEDx talk, "Why Math Education is Unnecessary", by a math teacher arguing that compulsory math classes should be replaced with voluntary math-based games: https://www.youtube.com/watch?v=xyowJZxrtbg Here is an essay by a mathematician, A Mathematician's Lament by Paul Lockhart (2002), arguing that math education is completely stupid, like teaching art only through paint-by-number exercises, and should be repl…

Music is taught by the equivalent of paint by number, which is reading sheet music. Or play, which is improvisation. There's room for both.

Music as paint-by-number is one of the worst failings of music education. The most robust and life-long music in the world is in cultures that actually use notation only as appropriate if at all and don't think of it as the music. Most of the world's music is not so notation-focused.

Re: Math for seven-year-olds

#39

Post by kaitai »

j2kun wrote: post #35
kaitai wrote: post #29

Earlier quoted context omitted.

Why wouldn't this be allowed with the Common Core? The Common Core is a set of standards for what students should have learned by the end of a year, and for fourth grade, for instance, the Core consists of things like "Use the four operations with whole numbers to solve problems," "Understand decimal notation for fractions, and compare decimal fractions, " and "Generate and analyze patterns." One can fault the Common…

At a high level you're right, Common Core philosophy does encourage this sort of lesson. The unfortunate part is that Common Core still is largely a list of facts that students need to know, and when they need to know them. This is why graph theory is not "compatible" with the standards, because in the huge list of topics they pretend these things don't exist. And since the Common Core is really a document for non-te…

I'd disagree that it's a list of facts; the high school modeling standards are skills. I've read some good discussions of whether the earlier-year standards are useful for their age groups and am not sure (I think they're pretty reasonable). The high school modeling and geometry components are great.

But what does Common Core have to do with graph theory not being taught in elementary school? It never has been, and not because of new standards. It's not taught because too many teachers don't know it, and because too many "administrators, test writers, and standards-compliance checkers" have their noses where they shouldn't! It is terrible that in the US we have this attitude that teachers must be told exactly what to teach and when. We have it because we don't trust teachers, and swapping out NCTM standards for Common Core standards doesn't change that.

Re: Math for seven-year-olds

#40

Post by kaitai »

slackpad wrote: post #31
kaitai wrote: post #29

Earlier quoted context omitted.

Why wouldn't this be allowed with the Common Core? The Common Core is a set of standards for what students should have learned by the end of a year, and for fourth grade, for instance, the Core consists of things like "Use the four operations with whole numbers to solve problems," "Understand decimal notation for fractions, and compare decimal fractions, " and "Generate and analyze patterns." One can fault the Common…

That's a fair point - his content is certainly compatible with the Common Core's standard. I was really venting about your point [2] and shouldn't have implicated the standard for that. Thanks for the references.

There is so much to talk about re: point [2]...!

(Evaluating teachers using student test scores has been a push of the Bill & Melinda Gates Foundation, but it's not well-supported by evidence. One more link: http://www.latimes.com/opinion/editorials/la-ed-teacher-eval... )

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