Topic (follow-up to Thursday, 14th October 2010):
Does the nature of mathematics change as students get older or is it only the teaching methods that change?
As it happened on Monday, 18th October 2010, 19:30 - 21:00 GMT
7:31 pm
ColinTGraham:
Hello and welcome to Monday's #mathchat
7:32 pm
ColinTGraham:
If it's your first time, don't try to keep up with everything, just interact with one or two people. Don't forget the hashtag --> #mathchat
7:32 pm
ianaddison:
@ColinTGraham #mathchat fancy finding teachers in your area? try http://bit.ly/teachmap :-)
7:32 pm
ColinTGraham:
Tonight's topic is: "Does the nature of mathematics change as students get older or is it only the teaching methods that change?" #mathchat
7:34 pm
ColinTGraham:
So, let's start with a focus: When, or if, you think about mathematics, what do you actually imagine mathematics is? #mathchat
7:34 pm
maxmathforum:
@ColinTGraham #mathchat can you tweet the link with our "homework" again?
7:35 pm
ColinTGraham:
@maxmathforum http://mathschat.wikispaces.com/!+mathchat+-+Preparation is the preparation page... if that's what you mean ;-) #mathchat
7:36 pm
ColinTGraham:
@maxmathforum http://bit.ly/a7h7Ae may work better... TweetGrid is temperamental about URLs sometimes! #mathchat
7:38 pm
ColinTGraham:
Implied in the topic question: mathematics has a nature, which may or may not change; we may/may not change our teaching methods #mathchat
7:39 pm
pavkahu:
RT @ColinTGraham: Tonight's topic is: "Does the nature of mathematics change as students get older or is it only the teaching methods that change?" #mathchat
7:40 pm
Matt_Arguello:
One's perception of math changes, not the nature of math itself. As people learn more their knowledge expands. #mathchat
7:41 pm
ColinTGraham:
@Matt_Arguello So you see it as an adjustment of the definition of mathematics - maybe an expansion to include new concepts etc. ? #mathchat
7:42 pm
maxmathforum:
#mathchat The nature of math changes, slowly. Proving the incompleteness theorem changed math. Mathematical philosophies go in&out of style.
7:43 pm
republicofmath:
Odd - if mathematics doesn't change what are all those journal articles about? @Matt_Arguello#mathchat
7:43 pm
Caro_lann:
#mathchat As ss get older they seem to think maths is harder - even if it's the same maths
7:43 pm
maxmathforum:
Like @republicofmath says, math itself is developmental (e.g. it develops and changes and we get better at dealing w/ that) #mathchat
7:44 pm
ColinTGraham:
@republicofmath The point here Gary, is that Matt may be looking at content rather than 'essence' #mathchat
7:45 pm
NicolRHoward:
Agreed! @Matt_Arguello One's perception of math changes, not the nature of math itself. People learn more, knowledge expands. #mathchat
7:45 pm
republicofmath:
Of course math changes - Greeks didn't know complex numbers, Newton didn't know matrix algebra. #mathchat
7:45 pm
ColinTGraham:
@Caro_lann Is this because of forgetting what they were taught or do you think it is because of realizing abstraction is greater? #mathchat
7:46 pm
ColinTGraham:
@maxmathforum But does the way in which mathematics develops change too or is it part of a fundamental process inherent to itself? #mathchat
7:47 pm
republicofmath:
The content - even of school math - changes. Calculus wasn't taught in 19th C. AP calculus didn't used to be taught. #mathchat
7:48 pm
maxmathforum:
#mathchat @NicolRHoward @Matt_Arguello are you saying matrix algebra and complex numbers were there all along, we recently discovered them.
7:48 pm
ColinTGraham:
To me, expansion of content is not the same as changing the fundamental procedures on which mathematics operates (the 'nature') #mathchat
7:48 pm
maxmathforum:
@ColinTGraham #mathchat yeah I think I'm saying how we do math and what it means to do it changes
7:49 pm
maxmathforum:
@ColinTGraham #mathchat example: using computers to prove the 4-color theorem? Is that really a proof? How does computing change math?
7:49 pm
NicolRHoward:
@republicofmath Content changes, people learn more - that's math. But, the nature of it doesn't change, only perceptions about it. #mathchat
7:51 pm
ColinTGraham:
@maxmathforum The same could be said of visual 'proofs', since mathematics 'requires' proof to be deductive reasoning...or does it #mathchat
7:51 pm
maxmathforum:
@ColinTGraham #mathchat Computers are changing the way modeling and statistics work, the way we use data to think (think google adsense)
7:52 pm
maxmathforum:
@ColinTGraham Does software like Geometer's Sketchpad or Geogebra change how we do math? #mathchat
7:52 pm
ColinTGraham:
@maxmathforum the ability to produce geometric representations (graphs) of algebra can speed up conceptual awareness #mathchat
7:53 pm
suburbanlion:
It's difficult to separate #math into "content" & "essence". Both very much intertwined. #mathchat
7:53 pm
ColinTGraham:
@maxmathforum I think the software changes how we approach solutions/investigations and what we feel able to tackle... #mathchat
7:55 pm
ColinTGraham:
@suburbanlion Not sure I agree. The idea of going back to first principles and axioms on which theories are built, for example #mathchat
7:57 pm
ColinTGraham:
@suburbanlion It's is 'essentially' mathematical to build from an 'accepted' fact, eg arithmetic facts (content) are the fundament #mathchat
7:57 pm
Caro_lann:
@ColinTGraham #mathchat They seem to focus in on specific things e.g. fractions but lose sense of connection between areas of maths
7:58 pm
ColinTGraham:
@Matt_Arguello Do you change your teaching to accommodate changes of definition, or review concept knowledge, at all? #mathchat
8:00 pm
republicofmath:
My point is a simple one: we lose a lot by ignoring the very real developmental aspect of math & developmental aspect of kids. #mathchat
8:01 pm
Matt_Arguello:
@ColinTGraham If the definition changes, yes. It doesn't do students any good to teach outdated definitions. Yes we review. #mathchat
8:01 pm
Matt_Arguello:
RT @suburbanlion: It's difficult to separate #math into "content" & "essence". Both very much intertwined. #mathchat
8:02 pm
ColinTGraham:
@republicofmath Developmental aspect here means introducing increasing levels of abstraction... as appropriate? #mathchat
8:02 pm
maxmathforum:
@ColinTGraham I'm not convinced that math is fundamentally axiomatic either #mathchat
8:03 pm
maxmathforum:
@ColinTGraham Sometimes it's just calculating debts and figuring out how many turnips you should have if you planted 100 last week #mathchat
8:03 pm
Matt_Arguello:
@republicofmath I think you're right. It's very difficult (at least to me) to transition from developmental to pencil+paper math #mathchat
8:03 pm
maxmathforum:
@ColinTGraham Sometimes we use math (induce things) before we mathematize is (deduce things) #mathchat
8:04 pm
ColinTGraham:
@maxmathforum Yey! Glad someone rose to the bait, Max! So what defines mathematical approaches as opposed to, say, scientific ones #mathchat
8:04 pm
maxmathforum:
#mathchat (I meant last spring, not last week! turnips don't grow that fast)
8:04 pm
suburbanlion:
@ColinTGraham but this idea of "going back to principals & axioms" can be traced to a change in content: Al-Khwarizmi's "Algebra" #mathchat
8:05 pm
erinfrog:
RT @republicofmath: My point is a simple one: we lose a lot by ignoring the very real developmental aspect of math & developmental aspect of kids. #mathchat
8:05 pm
ColinTGraham:
@maxmathforum "Doing sums" as Joe Public calls it, which is basic arithmetic, a subset of mathematics as a whole... #mathchat
8:06 pm
ColinTGraham:
@suburbanlion Or introduction of new content to assist in, or new ways to approach, problem solving... #mathchat
8:07 pm
maxmathforum:
@ColinTGraham when kids do a bunch of problems and intuit a pattern and use it, are they doing math? or science? #mathchat
8:08 pm
ColinTGraham:
Let's try to home in on what mathematical methods are... and then we can discuss whether these are teachable from the beginning #mathchat
8:09 pm
ColinTGraham:
Does it matter? RT @maxmathforum: @ColinTGraham when kids do... problems and intuit a pattern... are they doing math? or science? #mathchat
8:09 pm
Matt_Arguello:
@maxmathforum Good question. I think both. @ColinTGraham #mathchat
8:12 pm
ColinTGraham:
Maybe the difference between mathematics and science is the preferred investigative process. Science=experiment. Maths=reasoning #mathchat
8:14 pm
maxmathforum:
@ColinTGraham I do lots of experimenting when I solve maths problems. But I can't stop there. #mathchat
8:14 pm
maxmathforum:
seems like there's mathematical methods for problem posing, problem solving, and proving, all different #mathchat
8:15 pm
maxmathforum:
and, I think, all accessible at many ages! #mathchat and they all start with p ;)
8:17 pm
ColinTGraham:
I don't think anyone would disagree that there are huge overlaps between mathematical and scientific approaches to problem solving #mathchat
8:20 pm
ColinTGraham:
One of my 14yo students: Physics is measuring, Chemistry is mixing, Biology is cutting, Maths is writing the science stuff. #mathchat
8:23 pm
ColinTGraham:
@maxmathforum You say you can't stop at experimenting... what comes next? #mathchat
8:23 pm
maxmathforum:
#mathchat Is it true that we tend to teach problem-solving first, then proving, and finally problem-posing? Is that the best order?
8:24 pm
maxmathforum:
@ColinTGraham I'm not satisfied until I can use reasoning or algebra to say, yeah, this always works this way. #mathchat
8:25 pm
maxmathforum:
@ColinTGraham And then the cycle continues because I wonder what else happens if this is always true #mathchat
8:25 pm
ColinTGraham:
@maxmathforum In WCYDWT @ddmeyer is throwing out a situation and basically eliciting the questions or problems to solve #mathchat
8:26 pm
ColinTGraham:
@maxmathforum So, mathematician: "I want to know why this works, then I'll use it to help me understand why something else works" #mathchat
8:28 pm
ColinTGraham:
@maxmathforum Scientist: "I think if I do x, y will happen. Let's see if it does, and if not let's explore why." #mathchat
8:28 pm
suburbanlion:
I sometimes think of math as experiment in communication. i.e. Can present proof based on intuition, but it has to convince people #mathchat
8:29 pm
maxmathforum:
Yeah! RT So, mathematician: "I want to know why this works, then I'll use it to help me understand why something else works" #mathchat
8:30 pm
ColinTGraham:
@suburbanlion My immediate reaction of the proof of both Fermat's last theorem and the 4-colour map was "I can see it might be..." #mathchat
8:30 pm
NicolRHoward:
@republicofmath Solid points! #mathchat
8:31 pm
maxmathforum:
@ColinTGraham Do students need to be good problem solvers already to play #WCYDWT? Or can they pose problems before they solve 'em #mathchat
8:31 pm
ColinTGraham:
@suburbanlion Convincing people is not the same as convincing mathematicians! Visual 'proofs' are often the most effective. #mathchat
8:32 pm
ColinTGraham:
@maxmathforum The idea is that Ss ask the questions and then come up with approaches to finding the answer, which is checkable #mathchat
8:33 pm
suburbanlion:
@ColinTGraham aye, but trying to squeeze into 140 chars. point is that I don't think social factors should be ignored #mathchat
8:34 pm
maxmathforum:
@ColinTGraham So if students can do that, why don't we let 'em make up problems 'til late in grad school?! #mathchat
8:34 pm
ColinTGraham:
@suburbanlion Introducing social factors smacks of making things 'real' ;-) I am all for anything which helps convey the concepts #mathchat
8:36 pm
ColinTGraham:
@maxmathforum That raises the whole point of tonight's topic. If Ss are to be successful at mathematics, they need to be doing it! #mathchat
8:39 pm
ColinTGraham:
@suburbanlion's A Rebel Math Curriculum http://bit.ly/caWArr offers one way that we could actually teach mathematics, rather than... #mathchat
8:40 pm
ColinTGraham:
Playing Devil's Advocate: Are there certain techniques or methods which are always successful and so should not be changed? #mathchat
8:41 pm
iteachyear4:
RT @ColinTGraham: Playing Devil's Advocate: Are there certain techniques or methods which are always successful and so should not be changed? #mathchat
8:42 pm
suburbanlion:
RT @ColinTGraham: @suburbanlion's A Rebel Math Curriculum http://bit.ly/caWArr offers one way that we could actually teach mathematics, rather than... #mathchat
8:45 pm
ColinTGraham:
Do you think there is an age/school year when the focus/content changes too suddenly and Ss lose sight of what mathematics is? #mathchat
8:49 pm
BE_PURE:
RT @ColinTGraham: @suburbanlion Convincing people is not the same as convincing mathematicians! Visual 'proofs' are often the most effective. #mathchat
8:50 pm
Caro_lann:
Sorry #mathchat Got caught up in "Is seeing believing" Horizon. Fascinating.
8:50 pm
maxmathforum:
@suburbanlion @ColinTGraham I'd teach the Rebel curriculum. Especially if we added in some pure math challenges/games too. #mathchat
8:51 pm
suburbanlion:
@ColinTGraham "methods which should not be changed?" adapting to students needs -- i.e. @RepublicOfMath's http://bit.ly/cBPqdV #mathchat
8:51 pm
BE_PURE:
Attention Parents...#mathchat is happening now. Educators are talking math/math strategies in classroom. Worth taking a look at, you think!
8:52 pm
mrsenorhill:
RT @BE_PURE: Attention Parents...#mathchat is happening now. Educators are talking math/math strategies in classroom. Worth taking a look at, you think!
8:52 pm
maxmathforum:
@suburbanlion @ColinTGraham like this game: http://mathforum.org/pows/submission-flow.do?publicationId=3721 #mathchat
8:52 pm
ColinTGraham:
@Caro_lann Try iPlayer, Carolann! Welcome back! #mathchat
8:53 pm
suburbanlion:
@maxmathforum that link requires a login, have another? #mathchat
8:53 pm
ColinTGraham:
@suburbanlion Adapting to student needs is not necessarily in conflict with "this is the best way to teach..." #mathchat
8:55 pm
maxmathforum:
@suburbanlion It's a free account, and get's you a bunch of problems with applets or other tech, plus tons of kenken puzzles #mathchat
8:56 pm
ColinTGraham:
OK, last five minutes.... any thoughts or comments on the nature of mathematics & how we adapt, or not, teaching methods to Ss age #mathchat
8:58 pm
ColinTGraham:
@suburbanlion MathForum allows you to sign up for a free account... with problem of the week etc. #mathchat
9:01 pm
ColinTGraham:
Right, it's that time again! Thanks everyone. Vote for the next topic: http://bit.ly/c9KL7s Add some ideas: http://bit.ly/c9KL7s #mathchat
9:02 pm
ColinTGraham:
Remember to follow @mathchat to keep updated/reminded about events or changes #mathchat
9:03 pm
maxmathforum:
Thanks all! Nice chatting with you! #mathchat
9:03 pm
ColinTGraham:
The archive from tonight will be online in about 30 minutes. See you next time! #mathchat