Calculus for 9 year olds
Locked
 
|
Hi, everybody, I have a granddaughter on vacation who is showing some interest in maths. We have been fooling around with graph paper, you know, “the squaw upon the hippopotamus is equal to the suns of the squaw’s on the other two hides”, etc., and playing race track on graph paper (which didn’t grab her (used squares that were too small) but that’s about all I have in my repertoire. Any suggestions for really nifty stuff on the web (or that I could learn from the web quick enough) for 9 year olds. I;ve been told that early childhood is the best time to teach calculus, but not by anybody who actually knew how to do it. She is quick on a computer. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Calculus for 9 year olds
|
|
On Tue, Mar 29, 2016 at 04:15:25PM -0600, Nick Thompson wrote:
> Hi, everybody, > > > > I have a granddaughter on vacation who is showing some interest in maths. > We have been fooling around with graph paper, you know, "the squaw upon the > hippopotamus is equal to the suns of the squaw's on the other two hides", > etc., and playing race track on graph paper (which didn't grab her (used > squares that were too small) but that's about all I have in my repertoire. > > > > Any suggestions for really nifty stuff on the web (or that I could learn > from the web quick enough) for 9 year olds. I;ve been told that early > childhood is the best time to teach calculus, but not by anybody who > actually knew how to do it. She is quick on a computer. > Not sure about the web, but you would need to get in algebra first. A bright 9yo should easily be able to handle the concept that letters can stand abstractly for a number. Lack of algebra prevent the ancient Greeks from getting calculus. I'd avoid trig, though, it's not necessary for getting the concepts of differentiation and integration (unless Norm Wildberger's approach helps?). Then once you have algebra to hand, you need to teach the concept of limits. eg If x->0 and y->0 twice as fast, what is the limit of y/x? The answer is 1/2, not 0/0. With limits and algebra on hand, you can tackle differentiation and integration of polynomial functions. If she's any good at computer programming (eg perhaps using Scratch or Alice*), then get her to write a program printing out the value of something like (x+1)/x as x->infinity. Its a really good way (IMHO) of grokking limits. Then you can write a program to estimate the area of some random shape by tiling it with rectangles and then letting the tile size go to zero. That will give an excellent introduction to integration. * It might be possible to use a spreadsheet for this as well, with the added advantage of being easily able to graph the results. Cheers -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
|
|
Thanks, Russ,
I appreciate the help. Myself, I never got the "primary directive" of the calculus, or whatever it is called (that integration is the inverse of differentiation) until I graphed it. I hope you are well, Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: Friam [mailto:[hidden email]] On Behalf Of Russell Standish Sent: Tuesday, March 29, 2016 4:45 PM To: Friam <[hidden email]> Subject: Re: [FRIAM] Calculus for 9 year olds On Tue, Mar 29, 2016 at 04:15:25PM -0600, Nick Thompson wrote: > Hi, everybody, > > > > I have a granddaughter on vacation who is showing some interest in maths. > We have been fooling around with graph paper, you know, "the squaw > upon the hippopotamus is equal to the suns of the squaw's on the other > two hides", etc., and playing race track on graph paper (which didn't > grab her (used squares that were too small) but that's about all I have in my repertoire. > > > > Any suggestions for really nifty stuff on the web (or that I could > learn from the web quick enough) for 9 year olds. I;ve been told that > early childhood is the best time to teach calculus, but not by anybody > who actually knew how to do it. She is quick on a computer. > Not sure about the web, but you would need to get in algebra first. A bright 9yo should easily be able to handle the concept that letters can stand abstractly for a number. Lack of algebra prevent the ancient Greeks from getting calculus. I'd avoid trig, though, it's not necessary for getting the concepts of differentiation and integration (unless Norm Wildberger's approach helps?). Then once you have algebra to hand, you need to teach the concept of limits. eg If x->0 and y->0 twice as fast, what is the limit of y/x? The answer is 1/2, not 0/0. With limits and algebra on hand, you can tackle differentiation and integration of polynomial functions. If she's any good at computer programming (eg perhaps using Scratch or Alice*), then get her to write a program printing out the value of something like (x+1)/x as x->infinity. Its a really good way (IMHO) of grokking limits. Then you can write a program to estimate the area of some random shape by tiling it with rectangles and then letting the tile size go to zero. That will give an excellent introduction to integration. * It might be possible to use a spreadsheet for this as well, with the added advantage of being easily able to graph the results. Cheers -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Calculus for 9 year olds
|
|
In reply to this post by Nick Thompson
5-Year-Olds Can Learn Calculushttp://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/On Tue, Mar 29, 2016 at 5:15 PM, Nick Thompson <[hidden email]> wrote:
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Calculus for 9 year olds
|
|
In reply to this post by Nick Thompson
On Tue, Mar 29, 2016 at 08:42:35PM -0600, Nick Thompson wrote:
> Thanks, Russ, > > I appreciate the help. > > Myself, I never got the "primary directive" of the calculus, or whatever it > is called (that integration is the inverse of differentiation) until I > graphed it. Haha - its the fundamental theorem of calculus. And if you try to differentiate a Riemann sum, the fundamental theorem is pretty obvious. Not sure that graphing it would be convincing, however, even though I'm generally a fan of pictures. -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
|
|
In reply to this post by Nick Thompson
Nick,
Do NOT try to teach your 9 year old granddaughter calculus. Instead give her challenging problems that can be solved by elementary methods. If you try to teach her something she is not ready for, math will become a chore --possibly a chore for which she is greatly praised, but still a chore. Something bad happens when people see math as a chore. They try to memorize a lot of methods and go further than their understanding will normally take them. Doing math by memorizing without understanding is a dead end street. Many high schools, maybe most, now teach calculus and think they have taken a big step forward in their math education. But if they rush through algebra and geometry to do this, they have ruined their teaching of math. Instead of turning out people who understand algebra and geometry, they produce people who get by on memory. A Clark U., we have, for quite some time, required that students pass placement exams in algebra, geometry and trig before being admitted to Calculus. The first year we did this (in the 1980s) a student came to my office saying there was a serious mistake in his math placement because he was placed in remedial algebra when, in fact, he had already taken calculus and gotten an A in it, so was ready for second year calculus. I thought Clark had made a mistake, but, to be sure, I asked him some elementary algebra questions, such as to expand (x+1)^2 (IN THIS email I use ^2 for "2 in the exponent" when I talked to the student I simply wrote the 2 as a superscript.) The student produced x^2 + 4. When I suggested that this was incomplete, he seemd to feel that I was asking him an unfair question because that sort of math problem was what he had studied 4 years ago and it was unreasonable for me to expect him to remember things from a long ago. --I have heard similar stories from colleagues in other schools. Give her some simple word problems and allow her to solve them by trying out values. (Algebra teachers disallow this approach --for the seemingly good reason that it shortcircuits the learning of algebra) but trying out values allows the student to develop intuitions about how a problem works. Tell your granddaughter to make a note of any patterns she sees in the different numbers she tries. When she can detect patterns and express what they are coherently, that is the time to teach her algebraic notation. --John ________________________________________ From: Friam [[hidden email]] on behalf of Nick Thompson [[hidden email]] Sent: Tuesday, March 29, 2016 10:42 PM To: 'The Friday Morning Applied Complexity Coffee Group' Subject: Re: [FRIAM] Calculus for 9 year olds Thanks, Russ, I appreciate the help. Myself, I never got the "primary directive" of the calculus, or whatever it is called (that integration is the inverse of differentiation) until I graphed it. I hope you are well, Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: Friam [mailto:[hidden email]] On Behalf Of Russell Standish Sent: Tuesday, March 29, 2016 4:45 PM To: Friam <[hidden email]> Subject: Re: [FRIAM] Calculus for 9 year olds On Tue, Mar 29, 2016 at 04:15:25PM -0600, Nick Thompson wrote: > Hi, everybody, > > > > I have a granddaughter on vacation who is showing some interest in maths. > We have been fooling around with graph paper, you know, "the squaw > upon the hippopotamus is equal to the suns of the squaw's on the other > two hides", etc., and playing race track on graph paper (which didn't > grab her (used squares that were too small) but that's about all I have in my repertoire. > > > > Any suggestions for really nifty stuff on the web (or that I could > learn from the web quick enough) for 9 year olds. I;ve been told that > early childhood is the best time to teach calculus, but not by anybody > who actually knew how to do it. She is quick on a computer. > Not sure about the web, but you would need to get in algebra first. A bright 9yo should easily be able to handle the concept that letters can stand abstractly for a number. Lack of algebra prevent the ancient Greeks from getting calculus. I'd avoid trig, though, it's not necessary for getting the concepts of differentiation and integration (unless Norm Wildberger's approach helps?). Then once you have algebra to hand, you need to teach the concept of limits. eg If x->0 and y->0 twice as fast, what is the limit of y/x? The answer is 1/2, not 0/0. With limits and algebra on hand, you can tackle differentiation and integration of polynomial functions. If she's any good at computer programming (eg perhaps using Scratch or Alice*), then get her to write a program printing out the value of something like (x+1)/x as x->infinity. Its a really good way (IMHO) of grokking limits. Then you can write a program to estimate the area of some random shape by tiling it with rectangles and then letting the tile size go to zero. That will give an excellent introduction to integration. * It might be possible to use a spreadsheet for this as well, with the added advantage of being easily able to graph the results. Cheers -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
|
|
Thanks, John,
One would have thought that the word "expand" would have been a clue. What you describe seems a lot what they are doing in her school. So maybe I will leave it alone. I did fool around with "Alice" a bit. Have you ever known a child to run with "Alice"? Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: Friam [mailto:[hidden email]] On Behalf Of John Kennison Sent: Wednesday, March 30, 2016 11:24 AM To: The Friday Morning Applied Complexity Coffee Group <[hidden email]> Subject: Re: [FRIAM] Calculus for 9 year olds Nick, Do NOT try to teach your 9 year old granddaughter calculus. Instead give her challenging problems that can be solved by elementary methods. If you try to teach her something she is not ready for, math will become a chore --possibly a chore for which she is greatly praised, but still a chore. Something bad happens when people see math as a chore. They try to memorize a lot of methods and go further than their understanding will normally take them. Doing math by memorizing without understanding is a dead end street. Many high schools, maybe most, now teach calculus and think they have taken a big step forward in their math education. But if they rush through algebra and geometry to do this, they have ruined their teaching of math. Instead of turning out people who understand algebra and geometry, they produce people who get by on memory. A Clark U., we have, for quite some time, required that students pass placement exams in algebra, geometry and trig before being admitted to Calculus. The first year we did this (in the 1980s) a student came to my office saying there was a serious mistake in his math placement because he was placed in remedial algebra when, in fact, he had already taken calculus and gotten an A in it, so was ready for second year calculus. I thought Clark had made a mistake, but, to be sure, I asked him some elementary algebra questions, such as to expand (x+1)^2 (IN THIS email I use ^2 for "2 in the exponent" when I talked to the student I simply wrote the 2 as a superscript.) The student produced x^2 + 4. When I suggested that this was incomplete, he seemd to feel that I was asking him an unfair question because that sort of math problem was what he had studied 4 years ago and it was unreasonable for me to expect him to remember things from a long ago. --I have heard similar stories from colleagues in other schools. Give her some simple word problems and allow her to solve them by trying out values. (Algebra teachers disallow this approach --for the seemingly good reason that it shortcircuits the learning of algebra) but trying out values allows the student to develop intuitions about how a problem works. Tell your granddaughter to make a note of any patterns she sees in the different numbers she tries. When she can detect patterns and express what they are coherently, that is the time to teach her algebraic notation. --John ________________________________________ From: Friam [[hidden email]] on behalf of Nick Thompson [[hidden email]] Sent: Tuesday, March 29, 2016 10:42 PM To: 'The Friday Morning Applied Complexity Coffee Group' Subject: Re: [FRIAM] Calculus for 9 year olds Thanks, Russ, I appreciate the help. Myself, I never got the "primary directive" of the calculus, or whatever it is called (that integration is the inverse of differentiation) until I graphed it. I hope you are well, Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: Friam [mailto:[hidden email]] On Behalf Of Russell Standish Sent: Tuesday, March 29, 2016 4:45 PM To: Friam <[hidden email]> Subject: Re: [FRIAM] Calculus for 9 year olds On Tue, Mar 29, 2016 at 04:15:25PM -0600, Nick Thompson wrote: > Hi, everybody, > > > > I have a granddaughter on vacation who is showing some interest in maths. > We have been fooling around with graph paper, you know, "the squaw > upon the hippopotamus is equal to the suns of the squaw's on the other > two hides", etc., and playing race track on graph paper (which didn't > grab her (used squares that were too small) but that's about all I > have in > > > > Any suggestions for really nifty stuff on the web (or that I could > learn from the web quick enough) for 9 year olds. I;ve been told that > early childhood is the best time to teach calculus, but not by anybody > who actually knew how to do it. She is quick on a computer. > Not sure about the web, but you would need to get in algebra first. A bright 9yo should easily be able to handle the concept that letters can stand abstractly for a number. Lack of algebra prevent the ancient Greeks from getting calculus. I'd avoid trig, though, it's not necessary for getting the concepts of differentiation and integration (unless Norm Wildberger's approach helps?). Then once you have algebra to hand, you need to teach the concept of limits. eg If x->0 and y->0 twice as fast, what is the limit of y/x? The answer is 1/2, not 0/0. With limits and algebra on hand, you can tackle differentiation and integration of polynomial functions. If she's any good at computer programming (eg perhaps using Scratch or Alice*), then get her to write a program printing out the value of something like (x+1)/x as x->infinity. Its a really good way (IMHO) of grokking limits. Then you can write a program to estimate the area of some random shape by tiling it with rectangles and then letting the tile size go to zero. That will give an excellent introduction to integration. * It might be possible to use a spreadsheet for this as well, with the added advantage of being easily able to graph the results. Cheers -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Calculus for 9 year olds
|
|
On Wed, Mar 30, 2016 at 01:13:32PM -0600, Nick Thompson wrote:
> Thanks, John, > > One would have thought that the word "expand" would have been a clue. > > What you describe seems a lot what they are doing in her school. > > So maybe I will leave it alone. > > I did fool around with "Alice" a bit. Have you ever known a child to run > with "Alice"? > Not Alice, but I got my son to fool around with Scratch when he was about 8 or 9. He picked it up and did quite a few things in it, as well as coding lego robots in Robo C later on, but didn't end up being a coder. However I have hired him as a software tester (now that he's 18) as he has a good logical brain, even though his thing is more literature and history. -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow [hidden email] Economics, Kingston University http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
Re: Calculus for 9 year olds
|
|
In reply to this post by John Kennison
Dear Nick
I firmly agree with John Kennison. Develop their basic arithmetic, geometry and algebra at this age while boosting their basic mental computational abilities (mental maths) . Develop their foundations / concepts firmly. All this should be done in a way so that it is not imposed on the child. Calculus especially is a subject which the child has to be slowly eased into and correlated to physical and biological phenomena. Some non-US boards do this very well,eg. the IGCSE and the IB. Checkout www.desertacademy.org in Santa Fe. Sarbajit Roy http://www.mathspro.in/ On 3/30/16, John Kennison <[hidden email]> wrote: > Nick, > > Do NOT try to teach your 9 year old granddaughter calculus. Instead give her > challenging problems that can be solved by elementary methods. If you try to > teach her something she is not ready for, math will become a chore > --possibly a chore for which she is greatly praised, but still a chore. > Something bad happens when people see math as a chore. They try to memorize > a lot of methods and go further than their understanding will normally take > them. Doing math by memorizing without understanding is a dead end street. > > Many high schools, maybe most, now teach calculus and think they have taken > a big step forward in their math education. But if they rush through algebra > and geometry to do this, they have ruined their teaching of math. Instead of > turning out people who understand algebra and geometry, they produce people > who get by on memory. A Clark U., we have, for quite some time, required > that students pass placement exams in algebra, geometry and trig before > being admitted to Calculus. The first year we did this (in the 1980s) a > student came to my office saying there was a serious mistake in his math > placement because he was placed in remedial algebra when, in fact, he had > already taken calculus and gotten an A in it, so was ready for second year > calculus. I thought Clark had made a mistake, but, to be sure, I asked him > some elementary algebra questions, such as to expand (x+1)^2 (IN THIS email > I use ^2 for "2 in the exponent" > when I talked to the student I simply wrote the 2 as a superscript.) The > student produced x^2 + 4. When I suggested that this was incomplete, he > seemd to feel that I was asking him an unfair question because that sort of > math problem was what he had studied 4 years ago and it was unreasonable for > me to expect him to remember things from a long ago. --I have heard similar > stories from colleagues in other schools. > > > Give her some simple word problems and allow her to solve them by trying out > values. (Algebra teachers disallow this approach --for the seemingly good > reason that it shortcircuits the learning of algebra) but trying out values > allows the student to develop intuitions about how a problem works. Tell > your granddaughter to make a note of any patterns she sees in the different > numbers she tries. When she can detect patterns and express what they are > coherently, that is the time to teach her algebraic notation. > > --John > ________________________________________ ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com |
«
Return to Friam
|
1 view|%1 views
| Free forum by Nabble | Edit this page |