Is "rise over run" confusing?
LFS
the following this week.
Every year I have the same teaching problem that gets worse and worse.
First - to give you some perspective - I teach mathematics to college
students who consider themselves reasonable at math - not great, but
not bad either.
Here is a screencast of my question. http://www.screencast.com/t/vWfJdZin4L
Brief text of Q: Frequently - when graphing an ordered pair, they will
graph backwards. That is, they will graph the first coordinate on y
and the second on x. I kept asking myself - how can they possibly
forget? After 30 years, I think one reason might be because we teach
slope as "rise over run" and they get confused. What do you think? How
can we avoid this confusion?
Jacqueline Barbour
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Jacqueline Barbour
Sarah Ives
Sarah E. Ives,
Ph.D.
Assistant Professor of Mathematics
Department of Mathematics and Statistics
Texas A&M University - Corpus Christi
Corpus Christi, TX 78412
Office: 361.825.2151
Fax: 361.825.2795
Mike South
kirby urner
I wanted to reply to the other discussions, but I was frustrated by
the following this week.
Every year I have the same teaching problem that gets worse and worse.
First - to give you some perspective - I teach mathematics to college
students who consider themselves reasonable at math - not great, but
not bad either.
Here is a screencast of my question. http://www.screencast.com/t/vWfJdZin4L
Thanks for making a video out of your question, that was most excellent,
and a great exposure to the pedagogical tool of our day: the screen.
Bye bye blackboards and whiteboards (or hello "screen as blackboard").
Brief text of Q: Frequently - when graphing an ordered pair, they will
graph backwards. That is, they will graph the first coordinate on y
and the second on x. I kept asking myself - how can they possibly
forget? After 30 years, I think one reason might be because we teach
slope as "rise over run" and they get confused. What do you think? How
can we avoid this confusion?
Like Sarah was saying below (above?), adding contextual richness may help.
The word "slope" is part of the problem as it assumes a geographic orientation
for the XY plane, such that gravity is "down" and "rise" is therefore "up". Just
saying "rise over run" may not be sufficient to fix the mental picture of a
gradient vis-a-vis gravity. I'd talk about trucks and those "steep slope ahead"
warning signs, gradients as percentages.
However, that same XY plane might be a farmer's field seen from above,
perhaps a circular field given the irrigation system. So in what sense is the
word "slope" even appropriate? It's a "slant" through a plain, and this
tends to be measured relative to some arbitrary X-axis, starting from the
positive side (typically on the right, though I believe it's a valuable test
of understanding to relabel, use oblique axis, flip signs, then plot points).
If students see that "m" as implying a degree measure, with 1 representing
45 degrees from positive X radius, then they're (a) linking to trig, which is
good and (b) developing an intuitive sense of flat = 0, 1 = 45 degrees, with
a 90 degree vertical a whole other discussion, as now "m" would appear
to be infinity (another source of confusions).
Having students think gravitationally, in terms of physical hiking trails
or roads, and how > 1 means really quite steep, more like stairs. Something
like 90% is gonna be pretty close to vertical. That's 9/10 or 90/100 or...
Students should understand intuitively that negative m is a mirror of m
around the y axis. --m == m. 1/7 is pretty shallow (in terms of gradients)
or runs close to the x axis (not parallel of course). 6/7 is steep to the
right, while -6/7 is steep to the left (dropping quickly). Roller coasters
would be good to talk about too. How steep do they get? Some contain
loops, so all angles apply. It's good to get them on roller coasters (at
least mentally) because that keeps us in space/volume. Flatlander
math is always verging on irrelevance at the gut level, because nothing
is really flat. Even the ocean has ripples (and wraps around a ball).
Plane geometry should be contextualized within spherical geometry,
even if marketed as "Euclidean" (but I digress, plus many would
disagree).
Yes, I'm being very literal about both deltas being distances, whereas
in most of science, x will very likely be a time axis, with positive time
to the right (by convention -- too stuck in a rut on that one).
After dwelling on purely distance-oriented meanings, then I'd switch
to these other relationships, as most of the plots students are going
to see will involve direct and inverse relationships, rates of change that
themselves change, with non-linear taking precedence. Transferring
the notion of "slope" to curved lines and surfaces (manifolds) is of
course a bridge to "higher" math.
In sum: I think the very words "rise" "run" and "slope" are verging
on meaningless until strongly linked to "in the bones" experiences
of climbing and/or descending "against gravity". It pays to get
geographic. Then it pays, after laying this groundwork, to explicitly
leave the gravitational and/or distance context and start working with
more arbitrary quantities and relationships. However, by this time,
students should have an intuitive sense of how "m" relates to
steepness, so in reading a linear equation, they'll automatically
see "something is wrong" if it's a lot less than 1, yet positive,
yet appears to slope. The reality checks will already be there.
When talking about steep gradients, show pictures of funiculars
why not?:
http://www.google.com/images?hl=en&q=funicalar
Kirby
Colleen King
I haven't run into this difficulty with the students I've taught. Generally, students plot and identify points as early as grade 4 and continue to practice this skill for another two to three years before encountering slope. When slope is introduced, they've had quite a lot of experience with coordinates. I know they are given the "rise over run" explanation of slope at school but I don't reinforce that at the math center. But I do make sure kids know they don't have to rise before they run...they can run first when locating the next point.
I think my students would be confused by the "unit slope" explanation. If the slope is 5/3, it's much easier for them to understand and locate a vertical change of 5 and a horizontal change of 3. Converting to a vertical change of 1.67 might make things more difficult. I don't know, though. I might give it a try it with my algebra students.
Had your college students taken a break from math courses before coming to your class? If so, I would probably spend a little time reviewing coordinates. I'm not convinced unit slope will solve the backward coordinates problem. The formula for slope still places y first.
What are you using these days for annotation software?
Colleen
Maria Droujkova
I usually teach it through function families.
Make the line y=x, point-by-point if needed. Boring, but clear and obvious. Hold on!
Make y=2x. AHA! I see how it goes!
Make y=3x.
Make y=4x. See it?
Make y=2x again.
Make y=2x+1
Make y=2x+2
Make y=2x+3 AHA!
Make y=2x-5. Tricked you? No? Good!
I want kids to understand y=mx as a statement of PROPORTION (visually) and y=mx+b as a translation of it by b.
Then we can do extensions like negative, fractional, irrational coefficients, 3d, etc.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
LFS
Wow - Thanks everybody for responding and with such care and thought.
First the easy question – I annotate using smartboard software (the old 9.** version – I love the floating toolbar). I print to pdf and then put my classes online. I am thinking of running a tiny workshop on mathfuture some Saturday on this. Would anybody be interested?
Below I explain my order of teaching and why, but mostly I was just interested in thinking about (a) how could they not know order of graphing, (b) whether this could be related to how we explain slope and (c) whether we could better explain slope without using “rise over run”.
BTW1: I do get a much better reaction with the deltax=1 => deltay=m and in fact as Mike pointed out, it works well for a multiplier and they seem to get that.
Just reading Maria’s response. I think that’s it – that’s why they liked it. Proportions – they could understand slope as proportions.
BTW2 Kirby et.al. I tried explaining km/hr as a rate of change thinking that for sure they would know that time is x-axis and distance is y-axis. OMG – the blank faces. That was to be my next topic of discussion – where should we put “time” when we have virtual manipulatives – e.g. do kids connect: http://standards.nctm.org/document/eexamples/chap5/5.2/index.htm#APPLET …. But mustn’t stray off my point.
BTW3: I am ashamed to say that I have given up on fractions. It takes up too much time. They can’t plot (5/3,0). But they can plot (1.7,0).
Finally, I invite you to try this on your kiddies before doing slope. Draw a line with a negative slope and ask them whether it is increasing or decreasing. I get lots of blank faces and some kids will actually say “it is increasing to the left”. So about 5 years ago – I started to ask. “Would you put your money in this bank?” pause “So is this line increasing or decreasing?” Much better result on negative slope.
I do think that by the time kids get to college they should know how to graph a point without any review. Most have not had a break. But here they “cover” really hard stuff in 11th and 12th grade so that might count as a break. And indeed I have faced facts. (BTW: My class size is 65-120.)
-----My order of teaching ----
If I want them to understand anything - instead of going directly into engineering functions as the curriculum requires - I spend 1/5 of my semester – that is, the first 3 weeks (9 classes) reviewing (a) solving a linear equation in one unknown, (b) graphing points and graphing lines by finding 2 points using substitution (c) calculating an expression using a calculator by correctly determining the order of operations, (d) finding the function value of a function using a calculator and (e) – no kidding- finding the area of a rectangle attached to either a semi-circle or a triangle using one application of Pythagoras’ theorem.
I submit that this non-functioning with basic skills is not atypical and before Alex jumps in and says that they should know this or they shouldn’t be in college – I totally agree. However, I will selfishly say that I would like to draw a decent pension one day and for this I need more than 10% of the work force to finish college and be employed, which is about the % that does know these skills without review.
… In almost every class after that I require them to graph a line by finding 2 points even though we are actually working through matrices and basic word problems.
Then – 6 weeks into the semester, I spend 2 weeks (6 classes) teaching them about polynomials of degree n where I try to introduce the idea of “rate of change” by showing no change for n=0, constant change=m for n=1 and variable change for n>=2. I have given up teaching them to actually graph a line using slope-intercept as it takes up too much time and I am so happy if they can graph a line by finding 2 points and graphing them correctly :).
Finally, we move onto derivatives and I am ecstatic if they actually can connect the function to its derivative and the slope of the tangent line at a point on the function to the value of the derivative at that point and find the equation of the tangent and the normal at a given point and graph these lines correctly. (I am going to try these applets this year: http://geogebramath.org/lms/nav/activity.jsp?sid=__shared&cid=emready@linda_s_calculus_class&lid=2 – we shall see how it goes.)
Sorry to go on and on and AGAIN – THANKS EVERYONE SO MUCH FOR RESPONDING! I really like this group.
Phil Wagner
Phil Wagner
Math,Physics, and Robotics (VEX/FRC)
Education Resources Blog: www.brokenairplane.com
LinkedIn 
TwitterContact me:
brokenairplane
kirby urner
Wow - Thanks everybody for responding and with such care and thought.
First the easy question – I annotate using smartboard software (the old 9.** version – I love the floating toolbar). I print to pdf and then put my classes online. I am thinking of running a tiny workshop on mathfuture some Saturday on this. Would anybody be interested?
Below I explain my order of teaching and why, but mostly I was just interested in thinking about (a) how could they not know order of graphing, (b) whether this could be related to how we explain slope and (c) whether we could better explain slope without using “rise over run”.
BTW1: I do get a much better reaction with the deltax=1 => deltay=m and in fact as Mike pointed out, it works well for a multiplier and they seem to get that.
Just reading Maria’s response. I think that’s it – that’s why they liked it. Proportions – they could understand slope as proportions.
I'd say you've provided an interesting new slant on things, plus are reminding students that a/b == (a/b)/1 where a/b might be written as some decimal or floating point number -- or "real number" if you believe in those :).
I really think it pays to go to trig, even if just on a detour, and show right triangles with "proportional legs" (funny slide goes here). The slope-intercept treatment fairly screams "unit circle" and given all right triangles of legs a/b are similar (proportional), a unit circle version (h = 1) is always possible.
However, given my pronounced aversion to "flatland", I'm always biased to turning that unit circle into a unit sphere, getting back to the circle in "cut away". I might project this poster:
http://www.rwgrayprojects.com/synergetics/plates/figs/plate02z.html
(originally printed in Singapore for a folio as I recall -- I took one to Cairo for math teachers there to use).
BTW2 Kirby et.al. I tried explaining km/hr as a rate of change thinking that for sure they would know that time is x-axis and distance is y-axis. OMG – the blank faces. That was to be my next topic of discussion – where should we put “time” when we have virtual manipulatives – e.g. do kids connect: http://standards.nctm.org/document/eexamples/chap5/5.2/index.htm#APPLET …. But mustn’t stray off my point.
Yes, helps to put an iconic clock on x, have it tick away as the slider moves to the right.
Where your students might have seen this, or would like to, is when a music file (or speech file) goes by, and the vertical line is the play head, meaning it's giving us what we hear "right now". Segue to 'Who is Fourier?' (another math book I like).
In the meantime, some spiky thing is moving across the screen.
Might be a video (successive frames) in which case the representation convention is different (thinking of movie-editing software, time axis positive to the right).
Lots of software they're likely to encounter (including free stuff like Audacity) fits this description.
BTW3: I am ashamed to say that I have given up on fractions. It takes up too much time. They can’t plot (5/3,0). But they can plot (1.7,0).
Finally, I invite you to try this on your kiddies before doing slope. Draw a line with a negative slope and ask them whether it is increasing or decreasing. I get lots of blank faces and some kids will actually say “it is increasing to the left”. So about 5 years ago – I started to ask. “Would you put your money in this bank?” pause “So is this line increasing or decreasing?” Much better result on negative slope.
I can see where there'd be some confusion here, as in the wild, a slope (of a mountain) is neither strictly "up" nor "down" -- it all depends. Up *is* down.
So they look at your slanted line and don't want to commit as to whether it's positive or negative, as why should the left side of a mountain be treated differently from its right side?
Indeed, left and right are relative to the observer (camera position) and have no absolute meaning vis-a-vis the geographic scene or vista. In this way, always showing x going positive to the right seems suspicious. What makes "east" so special? There's a hidden bias here.
I do think that by the time kids get to college they should know how to graph a point without any review. Most have not had a break. But here they “cover” really hard stuff in 11th and 12th grade so that might count as a break. And indeed I have faced facts. (BTW: My class size is 65-120.)
I empathize. Indeed, I'd really hope to be beyond XY and well into XYZ by college.
They should have been using a 3D Turtle (swims in a tank, like Logo on steroids) since around 5th grade, played some flight simulator (and ridden a roller coaster if lucky **).
Polyhedra should be no mystery to an 8th grader. If you hold up a cube and they say "square" or if you ask how many sides and they say "four", then you know this must be the USA, the most backward nation on Planet Earth. No fault of the people, who tend to be above average as Garrison Keillor likes to say. I prefer to tell them conspiracy theories as this makes 'em more likely to become activists. That's part of me job as a buckaneer, yar! (how we talk in Portland):
http://mybizmo.blogspot.com/2006/09/yar.html
-----My order of teaching ----
If I want them to understand anything - instead of going directly into engineering functions as the curriculum requires - I spend 1/5 of my semester – that is, the first 3 weeks (9 classes) reviewing (a) solving a linear equation in one unknown, (b) graphing points and graphing lines by finding 2 points using substitution (c) calculating an expression using a calculator by correctly determining the order of operations, (d) finding the function value of a function using a calculator and (e) – no kidding- finding the area of a rectangle attached to either a semi-circle or a triangle using one application of Pythagoras’ theorem.
I'm always fighting the calculator industry, trying to get some words in edgewise for using an industrial strength programming language coupled to some graphical I/O. In my summer classes this year, that meant Python + Vpython as usual. We do XY graphs, but they're always fully rotatable, viewable from multiple angles, with the Z axis never suppressed.
It's been my position since the 1990s that we waste too much time on the plane, should be starting spatially, at least conceptually, and using the plane as a special case. It's the spherical planet that's real, whereas Farmer John's wheat field is only locally planar (and probably square, given the guy's ethnic background -- though circles have tended to grab market share, pure this screen shot over Libya:
http://www.flickr.com/photos/17157315@N00/5112444674/in/set-72157625315757940/
).
I submit that this non-functioning with basic skills is not atypical and before Alex jumps in and says that they should know this or they shouldn’t be in college – I totally agree. However, I will selfishly say that I would like to draw a decent pension one day and for this I need more than 10% of the work force to finish college and be employed, which is about the % that does know these skills without review.
In Oregon, we've been lobbying for a Digital Math track that would parallel the standard algebra geometry algebra2 precalc calc sequence. DM inherits from Discrete Math and is offered for credit towards fulfilling the 3-year requirement for an HS diploma in this state. My blogs lay out the plans, starting with a workshop on August 7 of last year.
Kids on our spanking new track get exposure to SQL, RSA, other computer science like topics, but it counts as math, not just an elective. I'm not saying its widespread yet, just that the Silicon Forest is behind it and has considerable political clout.
Students are getting the message that if they want to work for Intel someday, it might be wise to check out their options. Plus you can still take calculus if you want to (= analog math).
One of the textbooks: Mathematics for the Digital Age and Programming in Python by Gary and Maria Litvin (Skylit Publishing). This is already being used in some pre-college curricula. Professors getting this new breed from the feeder schools will find a different mix of strengths and weaknesses than they've been getting -- a different kettle of fish.
… In almost every class after that I require them to graph a line by finding 2 points even though we are actually working through matrices and basic word problems.
Then – 6 weeks into the semester, I spend 2 weeks (6 classes) teaching them about polynomials of degree n where I try to introduce the idea of “rate of change” by showing no change for n=0, constant change=m for n=1 and variable change for n>=2. I have given up teaching them to actually graph a line using slope-intercept as it takes up too much time and I am so happy if they can graph a line by finding 2 points and graphing them correctly :).
Finally, we move onto derivatives and I am ecstatic if they actually can connect the function to its derivative and the slope of the tangent line at a point on the function to the value of the derivative at that point and find the equation of the tangent and the normal at a given point and graph these lines correctly. (I am going to try these applets this year: http://geogebramath.org/lms/nav/activity.jsp?sid=__shared&cid=emready@linda_s_calculus_class&lid=2 – we shall see how it goes.)
Sorry to go on and on and AGAIN – THANKS EVERYONE SO MUCH FOR RESPONDING! I really like this group.
We might get to this in digital math but keep it discrete. Floating point numbers are never infinitely close together i.e. there's a lower bound to our epsilon. That doesn't keep us from looking over the cliff and talking about jumping off (into analog math world, where so-called "real numbers" take over).
Kirby
** a picture I think about when writing about roller coasters sometimes (also First Person Physics):
http://worldgame.blogspot.com/2005/09/powells-on-hawthorne.html
Melanie
-Melanie
Melanie Stein
phone: (607) 256 7368
Yin Yang Accent
helping non-native speakers use English with confidence
LFS
Hiya Melanie,
Much more info than you want :)
I teach math1 and math2. They are required courses in the first and second semesters for students in the (now European standard) 3 year course leading to a Bachelor’s of Science Degree in Applied Computer Science in the state university UKLO in Bitola, FYR Macedonia. (They can elect to take Finite Math and Math Modeling with Computers.) All of these are what are called 3+2 courses for 6 european credits = 3 us credits. 3+2 means that in the 15 week semester they will have 14*3*45 minute lectures with professor and 14*2*45 minute exercise classes with T.A.s. with one week for a midterm exam (final exam is after semester end). (This means they typically get more class time than in the USA, but since in my experience the average total time a student will actually study in a week is about the same everywhere, you will usually get less self-study from them than in the USA. So you trade them seeing more examples for them actually trying to work through more examples, but I digress.)
They do not take a placement test (like many colleges and universities in the world we are hard pressed for engineering and natural sciences students), but there is a maximum number accepted and they are ranked and in some sense pay tuition according to their grades from high school (a whole other story).
By engineering functions, I mean the group of functions that engineers typically use including polynomial, exponential, trigonometric, … functions and their inverses. Nothing fancy and btw – in order to find a place for matrices and some basic probability theory (absolutely essential to programming), differential equations were the first to go…
(And – in case someone is reading from other engineering programs in EU and saying “our kids do real engineering math and are real engineers” – I would point out that their “average” graduates are no better in math and programming than ours – I have them in my master’s program. Their top kids are better, but again we are talking about 10%. “Covering” complex material is absolutely no guarantee of “understanding”.)
----
The first 1/5 that I mentioned below is not listed in the curriculum, but I found that without it, we were just “covering” material. By spending this time “reviewing”, I have had significant success in getting them to understand the “real” material. They have a pass/fail test on this material which they must pass in order to take the colloquia/exams. They must get 4 out of 5 questions completely correct (they are allowed some “rounding” errors in the calculator questions) and every answer must be decimal. They cannot circle 9p/2 because this means nothing to them. They must write A=14.1 cm^2. I hope that they will look at the sketch and see if this makes sense J.
When I began teaching 36 years ago in the USA and then 32 years ago in MK , the level of understanding of mathematics of incoming students was much, much higher and their ability to “problem solve” or “think logically” was decent. This applies to both USA and MK (I have returned several times to teach in the USA). That is, they understood 3/5 as three-fifths and “knew” that 3/5*x = 60%*x = 0.6x and 3/5+2/5=1 both mathematically and “inside themselves”. Most could easily graph lines, quadratics and they understood the relationship between a function and its graph. They had a much better intuitive grasp of what was happening and math was not the enemy. (My opinion is that at all levels, we mathematicians abused this understanding by requiring ridiculous math knowledge – e.g. solving trigonometric identities in trig. and solving lim (x->0) (tanx)^(1-cosx) in calculus and so educators took over from mathematicians and we have the current situation. Now the mathematicians are trying to take it back again with “calculus for everybody”. Our poor kiddies.)
Now – a good percentage of my kids have no intuitive knowledge of math; they calculate 3/5+2/5 and say 3 over 5 plus 2 over 5… They have no rigor. To them expressions and equations are the same thing …, forgetting a sign is a “small mistake”,…
Just yesterday we were working through basic functions. They did not know the graph of 1/x. They know that they cannot put x=0 into the machine, but they didn’t know that this means that you cannot touch the y-axis with the graph (hence my comment in the screencast about “y-intercept”). They only know to make a table of points with whole numbers so they couldn’t get the vertical asymptote. They have seen limits in hs, but have no idea what they mean. They have no intuitive understanding of “small positive number”. To them – if a number is “getting smaller”, it must be going negative – Kirby’s points here are very valid. They don’t know the graph is a hyperbola. To them a hyperbola is a conic section with huge formulas. When I told them of the cheap experiment we did in 1970 in physics class with 2 vibrating nails hitting a pool of water to generate 2 sets of waves and noticing that where the waves eliminated each other was a hyperbola, they were astounded.
The connections are missing. Again, I go on and on. My apologies.
Edward Bujak
I am a public charter high school teacher in an urban city Philadelphia, PA, USA.
Your previous email is revealing. Interest in STEM and Computer Science (CS) seems to be extrememly low. My teaching friends and I have theories along your line. I do not necessarily agree that the kids are not learning because the teaching is incorrect or could be better. I do believe most kids are average or smart, but the educators have taken control away from rigor and moved to "touchy and feely" aspects. Personally I do not care how you feel about the math answer of 3/2. Is the answer 3/2 adn why. Justify your answer with appropriate work.
Somewhere before the kid gets to high school (grades 9-12), their natural inquisitivenss and engineering mindset is destroyed. Was it in middle school years or elementary school years? The kids I see are mostly apathetic and lack skills to do longer problems that involved multiple steps of logical thinking. Most just want a formula for everything. Critical thinking and understanding is not part of their vocabulary.
Grant it that this is not true for all kids, but I find there is a strong correlation with parental/guardian involvement and tudent success in academics and life.
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From: alicet...@gmail.com
To: mathf...@googlegroups.com
Subject: RE: [Math 2.0] Is "rise over run" confusing?
Date: Sat, 6 Nov 2010 10:32:47 +0100
Sue VanHattum
But I'm so glad you shared these details. I have no personal knowledge of education in other countries, and really appreciate hearing this kind of detail. You've painted a vivid picture.
I've been teaching at community college level since the early 80's, and haven't personally seen the decline you speak of. My impression is that most students have always been as bad at math as you describe. Is it possible that a smaller percentage of the population was attending college when you started out?
Warmly,
Sue
Edward Bujak
You bring up an interesting thought: "Is it possible that a smaller percentage of the population was attending college when you started out? "
At least in the USA there is still the notion that anyone can learn anytime at any age at any cost. I graduated undergraduate college 30 years ago and not everyone went to college. It seems now that almost everyone is admitted to come college somewhere. This is having a bad impact in many places:
[1] colleges are now offering non-credit remediation course to get the student proficient at that missing pre-requisite knowledge
[2] students are dropping out of college within the first year because they lack basic knowledge and are subsequently stuck with huge school loans they cannot pay without a higher paying job requiring a college diploma.
In some countries, students are tested at multiple times and placed (sorted) into levels. Academically good students go one direction and the other go another direction. This does NOT happen in the USA. Everyone is lumped together. The USA moved away from tracking where, for example, smarter students had 1st track Biology and others had 3rd track Biology. Now we have inclusion classes or immersion classes where students from all levels are placed into one class. This could be another reason why students are not learning as much. We have the slower students being frustrated and we have the smarter students not learning enough and being bored.
From: suevan...@hotmail.com
LFS
Hiya Ed and Sue,
Actually no Sue, the enrollment has gone down here in both engineering and natural sciences. Also the drop in quality might not be as drastic as I perceive. Perhaps I was paying less attention to their understanding and just looking for “correct” solutions. Still I am sure that both their basic skill level was higher and they did not perceive e.g. an incorrect sign as an insignificant error.
Generalizing very substantially, I still might say that both quality and interest of students in STEM subjects has gone down here.
---
Quality –
1. Maybe “touchy-feely” math has replaced not just rigor, but the ability to make “connections”. Rigor gives you confidence and connections give you the “ah ha” moment.
2. But I most definitely think that the direction now of “touchy-feely” in grades 1-4 followed by intensive “covering” of skills required by standardized testing in 5-12 is an even worse idea.
Interest –
3.
I most definitely agree with
you Edward about the impact of the home. But more than that. We now believe
that a good life is an easy life, but (at least when life gets tough) I try to remember
this story I heard from an old man when I was young. “We only got
oranges at Christmas. I can still vividly remember the smell, taste, and feel
of the orange in my mouth on Christmas morning. You don’t get that memory
when you always have oranges.” (And when asked about marriage: “We
expected less and got more.”.)
STEM subjects take time to learn and the problem is that until you reach the
“ah ha” moment you have nothing but maybe some ability to
manipulate numbers. But can we wait?
4. Also, I must admit that here it is now very difficult to get employment in a STEM profession so why bother with something more difficult if you cannot find work.
Edward – I do hope to make it to ISTE2011 so I do hope to meet you there!
LFS
Hi Phil,
I love that you are doing GeoGebra on the fly in your classroom. Both I and my students have learned so much from doing that and then they will try it at home.
Your blog was great. I made a really simple math art thing with geogebra. You got any good ones?
http://geogebramath.org/lms/nav/activity.jsp?sid=__shared&cid=emready@mathematical_art&lid=1
Linda
From: mathf...@googlegroups.com
[mailto:mathf...@googlegroups.com] On Behalf Of Phil Wagner
Sent: Friday, November 05, 2010 11:36 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Is "rise over run" confusing?
My student's moment always comes in Geogebra when I show them how to create a function and a slider and change the value.
Edward Bujak
Your comments about "connections" is right on. Cognitive psychologists and studies indicate that the human brain memory works best by association. We try to make connections to stuff we already know and extend our knowledge. I think the real problem is that you must have working knowledge to extend it and that basic knowledge is hard to acquire.
A big thing now (it changes every year) is relevance and rigor. I agree with the rigor, but not necessarily with the relevance. This is not a math issue. As a child we learned things at an amazing rate and some had no real direct purpose. We learned them because it was simply something we needed to get to the next step. Teaching about relevance in a high school class steals valuable time from teaching, learning, and understanding the real material. Do our textbooks need 25 colors and other distractions to try to bring relevance into the material. That is the job of the teacher to try to make the material interesting. Case in point: if we only taught material which was relevant now, that's Just In Time (JIT) training. Whereas that might be good for on-the-job training, it should not be the MO for any school at any level. We want to craft and mold critical thinkers ... so we need to teach abstract skills; many of which have no immediate value. It is these skills that make American schools better than some countries which show better math scores, but clearly and consistently do not encourage creative original thinking, but instead settle for rote memorization and do extremely well at it.
No matter how dysfunctional the USA schooling system is, I always point out that most foreigners want to be in American universities because we are the best at research and ingenuity because we encourage independent thinking.
About STEM: Everyone seem s to be clamoring about it. Why. It is a great rallying cry, but why. Like you said, wher are the abundance of jobs and high paying careers in STEM. I am an electrical engineer and computer software developer/consultant for 25 years. Now I am a low paid public school math teacher. Why? I do like it, but ....
Being a Philadelphia native and always living around Philadelphia (even universities), I will be glad to show you some of the Philadelphia must-see, must-dos, including food.
From: alicet...@gmail.com
To: mathf...@googlegroups.com
Subject: RE: [Math 2.0] Is "rise over run" confusing?
Melanie
I personally found rise over run helpful. I think of it as a proportion. Say the slope is 2. That means that if I go up 2, I have to go over 1. If I go up 4, I have to go over 2. Maybe it would help to include an example from downhill skiing or snowboarding.
But... it sounds like the problem is much bigger than To Rise Over Run Or Not to Rise Over Run.
Reading your 'order of teaching', I start to feel like a hamster on an exercise wheel. In order to teach them to find a derivative, one needs to teach them such-and-so, but in order to do that, they need this-other-thing, etc. It makes me think of the Car Talk question, Does anybody screen these calls? In this case, Does anybody screen these students?
One thing that I think works pretty well in the U.S. educational system (especially in community colleges) is the college placement test. Depending on how the student does in the placement test, he may have to take some background courses before taking the classes he needs for his degree program.
As Edward mentioned, some countries sort students way before college age. I've seen this in action in Germany, and am not a fan of this approach. It's too difficult to switch tracks.
In Mexican public universities, it's sink or swim. If you fail one course, you have to repeat the whole year. You can't move up until you've passed all the required courses for the year. The most infamous example of a weeder course I can give you is anatomy, for a degree in medicine. The way you learn anatomy is by getting to know what's inside the human body, and you can only really learn it by dissecting a human cadaver. In the U.S., there are four students per group. In Mexico, there are a dozen or more. How can you pass the anatomy exam if you can't wield the scalpel yourself, can't even see the dissection because it's so crowded around the table? So you find desperate medical students digging up corpses in the middle of the night to try to learn the stuff. (Digression: I hope this will inspire people to will their body to science. My mother did it, and it was wonderful way to go out helping others.)
In the U.S. there is more mobility. The various sequences are separated out from each other as much as possible, so that getting behind in one sequence doesn't hold you back in everything.
If I assume that you're working in a system that doesn't have a placement test system as I described, then I wonder if you could do the following: have the students work on the web, preferably well before showing up for classes in their first semester. Explain, "I have designed Math1 and Math2 to help you learn the key concepts and skills you need to be successful in your Computer Science career. This diagnostic and self-taught course will help you remedy any gaps you may have before beginning Math1." Set up a peer tutoring program to help students get through this web course.
I just don't see how you can remedy the deficiencies you described AND teach them the things in the official curriculum, all in one semester. You might as well be balancing a cake on a rake while hopping on a ball.
-Melanie
LFS
Hiya all,
I don’t want to pay taxes a third time (state school system), nor do I want to be the parent who pays schooling a third time (private school system). This is what placement testing and remedial means and I say third time because we have already paid for regular school and then we pay for “tutors” just to get them through regular school and now we pay for remedial? That is ridiculous.
In my opinion, a child should be able to graph a point (and a line) when he finishes 12 years of school without any tutors or remedial.
My question is - why can’t he? Why would he think to go first vertical and then horizontal? It should be like the command “left-face” and they turn left.
I have made 3 screencasts where I graph lines (This time I used Jing so I couldn’t edit and a webcam and paper on #1 so anybody can do it for free.)
If you have 5 minutes, I would ask that you pick one of these problems (or make your own!) and make a video to show us how you would solve it in class as if it were a review for an upcoming test.
Please feel free to comment on my solutions. My point is that in each problem, they are expected to graph the line using a different technique. (Notice that n each question, they are also required to check that the algebra matches the geometry.)
Problem 1: http://www.screencast.com/t/6K9ibcsTcg Graph y=-2x+3 and check algebraically and geometrically whether (-1,1) is a point on this line.
Problem 2: http://www.screencast.com/t/njGTPjdu The solution to the system 3x+y=5 and –x-3y=1 is x=2, y=-1. Show this graphically. (First part of problem is to solve algebraically.)
Problem 3: http://www.screencast.com/t/gOun4DMtEbv For the quadratic function y=x^2-3x+2 (graphed below), find the equation of the tangent line at x=0 and graph this line.
BTW: Jing with a webcam makes ~3x bigger file.
BTW Melanie
We also have the “sort system” here as in Germany – that is, when a child is entering 9th grade he chooses his school from gymmasium (rather like a standard US hs), technical school (which is both for future engineers and for skilled machinists/…), economics school, nursing school (yes – you can be a nurse with high school), agricultural school and vocational school. (Our son went to gymnasium.) You could say that this limits their future and requires them to decide their lives when they are too young. On the other hand, they are probably going to work 45 years. Do they really want to work the same thing for 45 years? So does it matter what they choose at age 15? They should be able to change along the way.
Also, we used to have the Mexico college system you describe and kids would take 8-10 years to get their degrees. They would take exams over and over until they passed (or managed to memorize or cheat). Europe then went to the credit system, which is similar to the US system. It is still in “implementation”, but it is significantly more contiguous and better for the kids (not to mention cheaper for all).
BTW Edward
I totally agree that the US school system is still far better for creating thinking minds than anywhere else in the world. In the rest of the world, the school systems are much better at skill development. My schooling in a small town called Sparta, NJ, USA was incredible and I credit it with my success (as well as my siblings). However, in my senior year, I transferred and graduated from HS in AZ in 1972, where the school system – hmm can’t think of a nice word. When I was working on my doctorate in 1988, I was the oldest student in theoretical math. The kids from outside US had skills but couldn’t think as well as the US kids and vice-versa. My professors – many of whom were just a bit older than me - remarked that I was “old school” because I could do both. That was from NJ. I want that school system and those teachers everywhere!
Warm regards to everyone! Linda
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Melanie
Sent: Saturday, November 06, 2010 6:08 PM
To: mathf...@googlegroups.com
.ExternalClass .ExternalClass ">ecxhmmessage P {padding:0px;} .ExternalClass body.ecxhmmessage {font-size:10pt;font-family:Tahoma;} You
said: Much more info than you want :)
But
I'm so glad you shared these details. I have no personal knowledge of education
in other countries, and really appreciate hearing this kind of detail. You've
painted a vivid picture.
I've been teaching at community college level since the early 80's, and haven't
personally seen the decline you speak of. My impression is that most students
have always been as bad at math as you describe. Is it possible that a smaller
percentage of the population was attending college when you started out?
Warmly,
Sue
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kirby urner
> couldn’t edit and a webcam and paper on #1 so anybody can do it for free.)
>
Thanks for doing this. I support use of instructional video, made by
both faculty and students. Even guardians (such as parents) might
get in on the act.
> If you have 5 minutes, I would ask that you pick one of these problems (or
> make your own!) and make a video to show us how you would solve it in class
> as if it were a review for an upcoming test.
>
> Please feel free to comment on my solutions. My point is that in each
> problem, they are expected to graph the line using a different technique.
> (Notice that n each question, they are also required to check that the
> algebra matches the geometry.)
>
>
>
> Problem 1: http://www.screencast.com/t/6K9ibcsTcg Graph y=-2x+3 and check
> algebraically and geometrically whether (-1,1) is a point on this line.
>
Clear. I might encourage saying the calculation out loud yet only
writing the y answer, not showing the steps. This goes against
the grain, yet encourages mental arithmetic. They can always
rewind or try it themselves. Like those cooking shows, where you
don't see the thing baking for 30 mins, it just magically pops out
of the oven.
Also, I'm obsessing in this thread about how x is always positive
to the right by convention. Your video takes this convention so
completely for granted that no labeling is even needed. That's
quite normal of course. The culture insists the "positive is right"
meme (sounds Orwellian). I believe in flipping x and y, changing
sign and so on (might be simply a change in viewer angle i.e.
looking from the other side of the same graph).
It'd be fun to pose about 10 such problems and race through
the vids at a sped-up superhuman speed, with hands flashing
quickly, numbers appearing, answers Yes! and No! going by
lickity split, an industrial process.
How I might do it using Python:
Get some points (including where x = -1):
>>> [(x, -2*x + 3) for x in range(-5, 6)]
[(-5, 13), (-4, 11), (-3, 9), (-2, 7), (-1, 5), (0, 3), (1, 1), (2,
-1), (3, -3), (4, -5), (5, -7)]
Check if the point to be tested is in the solution set:
>>> (-1,1) in [(x, -2*x + 3) for x in range(-5, 6)]
False
Might graph it in VPython, which allows the resulting graph to be rotated
in space, in use POV-Ray like on this page?
http://www.4dsolutions.net/ocn/catenary.html
> Problem 2: http://www.screencast.com/t/njGTPjdu The solution to the system
> 3x+y=5 and –x-3y=1 is x=2, y=-1. Show this graphically. (First part of
> problem is to solve algebraically.)
>
Here I note that your "decimal equilivalents" are not precise in either case.
1/3 == 0.3 and 5/3 = 1.7 are both rounded. I'm sure they've come to expect
that but those dropping in on this video might be confused.
My question is: do students get to play with this cool interface allowing
drag and drop tables, a graph, multiple pen colors on screen? If so, that's
cool.
On the other hand, if only the teacher gets to play with these cool toys,
while students are relegated to paper and pencil, well then I think the
real teaching is that teachers are a socially privileged class and students
don't rate.
At least I'd be worried about sending that message.
> Problem 3: http://www.screencast.com/t/gOun4DMtEbv For the quadratic
> function y=x^2-3x+2 (graphed below), find the equation of the tangent line
> at x=0 and graph this line.
>
The confusion I find here (looking from a student point of view) is
endemic to calculus teaching. You say "they have to go back to
'y the function'" as you point to y = x^2 - 3x + 2.
But y is just the value that makes the equation true for a given
value of x. Saying y is really function notation and then finding
y(x) is somewhat counter-intuitive, like saying f = x^2 -3x + 2
instead of f(x): x^2 -3x + 2 with x domain = reals.
There's a difference between a function, the set of all ordered
pairs that satisfy the equation, and any given solution or
specific ordered pair.
My teacher Dr. Thurston the famous topologist used to harp
on this confusion in calculus books, when it came to confusing
function notation. He recommended Spivak's 'Calculus on
Manifolds' for being more formally correct, but that's a pretty
hard book (this was Honors Calculus at Princeton).
Of course it's not your fault that engineers use such goofy
notation. Math notation is such a junkyard!
Kirby
Phil Wagner
Maria Droujkova
Also, I'm obsessing in this thread about how x is always positive
to the right by convention. Your video takes this convention so
completely for granted that no labeling is even needed. That's
quite normal of course. The culture insists the "positive is right"
meme (sounds Orwellian). I believe in flipping x and y, changing
sign and so on (might be simply a change in viewer angle i.e.
looking from the other side of the same graph).
"In Soviet Russia" (appropriate start here http://knowyourmeme.com/memes/in-soviet-russia) they used to flip axes in math track courses (hard sciences, engineering, math) but not in general courses.
This comes up a lot, for example, if you switch coordinate systems from absolute to object-related. Which you may do in some physics problems, and in game programming. Force diagrams come to mind.
kirby urner
>
>
> On Sun, Nov 7, 2010 at 7:04 PM, kirby urner <kirby...@gmail.com> wrote:
>>
>> Also, I'm obsessing in this thread about how x is always positive
>> to the right by convention. Your video takes this convention so
>> completely for granted that no labeling is even needed. That's
>> quite normal of course. The culture insists the "positive is right"
>> meme (sounds Orwellian). I believe in flipping x and y, changing
>> sign and so on (might be simply a change in viewer angle i.e.
>> looking from the other side of the same graph).
>
> "In Soviet Russia" (appropriate start here
> http://knowyourmeme.com/memes/in-soviet-russia) they used to flip axes in
> math track courses (hard sciences, engineering, math) but not in general
> courses.
>
That's very interesting to know, I had no idea.
> This comes up a lot, for example, if you switch coordinate systems from
> absolute to object-related. Which you may do in some physics problems, and
> in game programming. Force diagrams come to mind.
>
What's on my mind in part is that no matter how you label X and Y axes,
with positive left or right, up or down, you can always find an orientation
for the viewer where the (+ +) quadrant is in the upper left, just as in
all the textbooks.
However, with that 3rd axis, Z, you have a real choice as to whether
to create a left or right handed coordinate system. You can't just change
your camera angle to turn one into the other.
Handedness (chirality, enantiomorphs) is a subtle and interesting concept,
worthy of much study. The idea of turning a glove inside out, to make a
right handed one become left handed... what textbooks deal with that?
I've been investigating space-filling tessellations of tetrahedra quite a bit.
There's this meme out there that "Aristotle was wrong" in saying tetrahedra
fill space, but that's only if we insist he meant regular ones. When it
comes to irregular tetrahedra, we have an interesting discussion, and
the notion of chirality (handedness) enters in, i.e. which space-filling
tetrahedra depend on left and right versions of themselves, versus
being all indistinguishable?
I like to bring students through Math World, the page on space-filling
polyhedra, to point out how that page lags behind the published literature.
It's silent about tetrahedra, aside from repeating the "Aristotle was wrong"
meme.
http://mathworld.wolfram.com/Space-FillingPolyhedron.html
Why I find tetrahedra especially interesting is they're topologically
minimal, i.e. no box or container of facets, nodes, edges, has
fewer than (4, 4, 6). The cube is (6, 8, 12) by contrast. In the
futuristic math courses I favor, we at least mention that making
a tetrahedron our unit of volume opens a whole new world of
whole number values for related polys. I often publish these
charts:
http://wikieducator.org/Martian_Math#Unit_of_Volume
http://wikieducator.org/Martian_Math#Unpacking_Polyhedra
(OERs, free for teacher use around the world without legal
encumbrance -- what Wikieducator contains, free and open).
Sorry, I digress, just yakking about how "handedness" is a
feature of my geometry research and classes. Also, when
it comes to XYZ coordinates, I like to point to exotic alternatives,
such as the 4-tuple-based "Quadray coordinate system"
which I participated in designing.
Here's the Wikipedia page:
http://en.wikipedia.org/wiki/Quadray_coordinates
This is a tiny niche market, certainly. My futuristic Martian
Math class was the most recent field test, at Reed College
in Portland in August of this year:
http://www.4dsolutions.net/satacad/martianmath/toc.html
Kirby
> Cheers,
> Maria Droujkova
>
> Make math your own, to make your own math.
>
Bradford Hansen-Smith
| X and Y, right and left. I agree with you Maria, this is about
proportions not labels to a floating abstraction. I have to find where
things come from to understand them. The wonder is always in the
journey. In my most recent book I approach this by going back to the first fold in the circle. Mark the two most furthest points on the circumference. Touch the points and crease. Draw a line connecting the two marked points. This reveals the center of the circle and a square relationship of two perpendicular bisectors. (You can also just fold the circle into quarters for the same results, although a limited approach, but initially may be easy for your kids.) Draw concentric circles every half inch from the center point of the two diameters. For a nine inch paper plate that will give you eight circles. Number each circle from the center out, placing a number at the four intersections of each circle. Now draw a straight line through each numbed point parallel to the diameters. You have a Cartesian grid within the circle context. It does not matter what direction the axis go or what you call them because it is about proportional relationships of concentric circles on two perpendicular lines of division. Every numbered intersection has a proportional relationship to all other numbered points of intersection locating any position in any quadrant. This is you full game board; battle ships and slops in all directions. If you want to take this into three dimensions by introducing the third axis, then simply continue to fold the circle into the regular tetrahedron and you will have raised the center point to a third axis perpendicular to the grid plane in a way you have not seen before. Kirby, if this is not Martian I don't know what is. Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ --- On Mon, 11/8/10, kirby urner <kirby...@gmail.com> wrote: |
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Maria Droujkova
On Mon, Nov 8, 2010 at 4:43 AM, Maria Droujkova <drou...@gmail.com> wrote:
What's on my mind in part is that no matter how you label X and Y axes,
> This comes up a lot, for example, if you switch coordinate systems from
> absolute to object-related. Which you may do in some physics problems, and
> in game programming. Force diagrams come to mind.
>
with positive left or right, up or down, you can always find an orientation
for the viewer where the (+ +) quadrant is in the upper left, just as in
all the textbooks.
However, with that 3rd axis, Z, you have a real choice as to whether
to create a left or right handed coordinate system. You can't just change
your camera angle to turn one into the other.
Handedness (chirality, enantiomorphs) is a subtle and interesting concept,
worthy of much study. The idea of turning a glove inside out, to make a
right handed one become left handed... what textbooks deal with that?
To the best of my recollection, elementary (undergraduate back then, probably high school now) topology, algebraic geometry, and analytical geometry courses traditionally deal with chirality. Also physics of course, because it comes up in multiple ways, like Ampere's rule.
Something like this came up in a kiddie (4-8yo) book called "Odd one out" by Tom O'Brien http://www.mathplace.net, which I've been using as a basis of several math club activities. The book had circles divided into sectors, and the odd one out wasn't identical up to rotation. Here is the screenshot of what I am talking about, from the upcoming second book. We discussed it last Saturday at the Math 2.0 event with Tom: http://mathfuture.wikispaces.com/Tom+O%27Brien
http://screencast.com/t/py67kWkBlVe
MariaD
kirby urner
If you want to take this into three dimensions by introducing the third axis, then simply continue to fold the circle into the regular tetrahedron and you will have raised the center point to a third axis perpendicular to the grid plane in a way you have not seen before. Kirby, if this is not Martian I don't know what is.
Brad
Martians begin with the tetrahedron as the most primitive object, that with an inside and outside. Spheres are actually complex as they're full of holes (this is a discrete math the Martians have, no continua, any no infinity except in the sense of a vast outwardness, a contextualizing "tuned out").
Given the tetrahedron is primitive, Martians note it's four directional. The vertexes define four arrows (as do the face centers -- that's just the dual tetrahedron (tetrahedron is self-dual)). Those four arrows cave space into four quadrants. XYZ defines eight octants in contrast. Its six spokes radiate through the vertexes of a regular octahedron, more complicated than a tetrahedron.
Whereas Earthlings are fixated on 90 degree angles (the genesis of words like "orthodox" "normal" and "foursquare"), the Martians are for more biased towards thinking of 60 degrees as "normal", with a lot more hexagonal than square tiling on surfaces. Their "holodeck" is much more like our Earthling CCP (= FCC) than a grid of cubes ala XYZ. Squares are somewhat ugly to Martians.
I think you'll be somewhat open to Martian Math because you're so into circles. Six around one make that hexagon of 7. 3-7-3 is an important pattern for us (as a cuboctahedron of volume 20), although there I'm using balls, not circles. Circles are cross-sections of balls. We never want to lose sight of volume. The Martians think 'Flatland' by Abbott is like the stupidest book ever, as there could never be "two dimensional beings" and we're pathetic for indulging in such fantasies (Earthlings are really quite retarded in their view, although we're managing to collaborate on this dam, a story problem exercise I've done some postings about, a Martian-Earthling co-venture (just have to work out differences, when it comes to unit of volume (not a cube for the Martians, never will be)).
Kirby Urner
4Dsolutions.net
(or 8D in Asia, where 4 is bad luck)
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LFS
Hi Phil and everyone else who loves GeoGebra,
The geogebramath.org website is by Joel Duffin of NLVM and eNLVM and it is a fabulous interface even though it is still in beta (the tagging and rating system are still under development.). I am simply populating it with my applet activities. For sure both visitors and contributions are welcome. Please use Mozilla FF or IE to register for an account and to add activities – Chrome is still causing problems.
Also, if you have any comments or suggestions on the interface or on any activity, feel FREE to write me or Maria Droujkova or Joel Duffin.
Phil – I would love to see any of your slope art. Meanwhile I will write Rafael Losada to see if he knows how to make his cool colors for these ( http://www.youtube.com/watch?v=af4o7qnYe1g ).
Warm regards, Linda
Here are my latest – given my frustration with my kids not being able to graph points :)
I know they are not exciting and like a game, but at least they can practice. And in GeoGebra 4, hopefully we can add sound.
Phil
Connect with me: Error!
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Contact me: Error! Filename not specified.brokenairplane
Error! Filename not specified.
Dani Novak
I am intrigued by this. Math Art is my cup of tea and I have been doing this for yours. Look at this video that one of my students did yesterday. I think it is amazing. I think that we can reach a whole new audience witht his and I envision a new kind of teacher: Math Music teacher:
Here is the link:
http://screencast.com/t/bgPMPtzEdXJ
--Dani
Math Department
Ithaca College, Ithaca NY 14850 USA
no...@ithaca.edu
http://www.ithaca.edu/dani
Phil Wagner
Phil Wagner
HTHCV
Math,Physics, and Robotics (VEX/FRC)
Education Resources Blog: www.brokenairplane.com
Contact me:
brokenairplane
Bradford Hansen-Smith
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kirby urner
Kirby,
The first fold of the circle is a tetrahedron pattern, you can not get much more primitive than that. This movement shows four points (two touching points and two end points of the folded diameter revealing six relationships, two open planes and two solid planes.) The 360 degree movement around the diameter show dual tetrahedra, one inside out of the other. It is easier than turning a glove inside out to show chirality. The circle is the compression of the sphere; it simply spreads the volume out into a circle. By cutting the sphere you immediately lose the volume. You can say circles are cross sections of balls if you want, but that is a generalization borrowed from cutting spherical fruit.
I'm staying interested in circles. I just not that accomplished at folding them. Having some Youtubes of what you're talking about would aid in parsing.
With four balls (same diameter), you just jam them together as tightly as possible, all touching one another, and there's your tetrahedron. No guess work there either.
Martian Math takes that for unit volume.
Once this unit is established, the cartoons should show pouring into commensurate vessels, such as the octahedron of volume 4 (same edge length of sphere diameter) and the rhombic dodecahedron of volume 6 (a space filler, long diagonal of each diamonds makes the octahedron just describe -- the dual cube of short diagonals, has volume 3).
Earthlings insist on worshiping the cube beyond reason, is how the Martians see it. It's not like they're unfamiliar with XYZ though, given its a pervasive apparatus and most computers expect it to be there as an API -- for example VPython, which we use in class:
http://www.4dsolutions.net/satacad/martianmath/mm20.html (large picture file included)
In folding the circle the tetrahedron comes from three diameters, without them you are either guessing or measuring to make the construction. Four is about counting; form is about formation, transformation, reformation, movement. The first fold of the circle is a right angle movement between the two touching points and the diameter that is generated. It is only a square pattern when the two furthest points are touching showing two perpendicular diameters. The square form is not structural, it is a relationship of movement that got stuck in our minds.
In the Polyhedron type of object (one of many math objects), we would typically implement (a) scaling (making bigger / smaller) (b) rotation (c) translation. There's lots to be learned ever from just these three. With scaling, we keep surface and central angles constant, meaning the shape stays invariant. However volume increases / decreases as a 3rd power of linear increase / decrease.
This is naturally true of tetrahedra as well as cubes, which is why a 3rd power sequence might just as well represent a growing tetrahedron as a growing cube. In this sense, every time an Earthling mathematician says "cubing", he thinks he's being clever, giving a shortcut. But the Martians would prefer to think in terms of tetrahedra at this juncture, so "cubing" to them doesn't sound clever, it sounds like you just don't know any Martian Math.
"The Martians think 'Flatland' by Abbott is like the stupidest book ever,..."
The Martians are right on, I could not agree with them more.
Brad
I'll let them know. Maybe you'll get lucky and be abducted (or inducted, whatever we call it). :)
Kirby
Edward Cherlin
> On Mon, Nov 8, 2010 at 4:43 AM, Maria Droujkova <drou...@gmail.com> wrote:
>>
>>
>> On Sun, Nov 7, 2010 at 7:04 PM, kirby urner <kirby...@gmail.com> wrote:
>>>
>>> Also, I'm obsessing in this thread about how x is always positive
>>> to the right by convention. Your video takes this convention so
>>> completely for granted that no labeling is even needed. That's
>>> quite normal of course. The culture insists the "positive is right"
>>> meme (sounds Orwellian). I believe in flipping x and y, changing
>>> sign and so on (might be simply a change in viewer angle i.e.
>>> looking from the other side of the same graph).
An instance where this is essential is in the so-called
backward-sloping supply curve, which is regarded as a mathematical
anomaly in economics, because it seems to represent a two-valued
function. If you swap axes, it becomes perfectly normal and ordinary.
The commonly-cited instance of backward-sloping supply is Polynesian
labor markets in which it used to be that raising wages past a certain
point reduced the supply of labor. Of course, we have instances of
this in Europe and the US, such as opera composer Rossini realizing
that he was rich, and had no need to go on composing opera, or
cartoonists Gary Larson (The Far Side) and Bill Watterson (Calvin and
Hobbes) retiring.
>> "In Soviet Russia" (appropriate start here
>> http://knowyourmeme.com/memes/in-soviet-russia) they used to flip axes in
>> math track courses (hard sciences, engineering, math) but not in general
>> courses.
>>
>
> That's very interesting to know, I had no idea.
>
>> This comes up a lot, for example, if you switch coordinate systems from
>> absolute to object-related. Which you may do in some physics problems, and
>> in game programming. Force diagrams come to mind.
There are, of course, a multitude of other coordinate transformations,
starting with polar coordinates.
> What's on my mind in part is that no matter how you label X and Y axes,
> with positive left or right, up or down, you can always find an orientation
> for the viewer where the (+ +) quadrant is in the upper left, just as in
> all the textbooks.
>
> However, with that 3rd axis, Z, you have a real choice as to whether
> to create a left or right handed coordinate system. You can't just change
> your camera angle to turn one into the other.
>
> Handedness (chirality, enantiomorphs) is a subtle and interesting concept,
> worthy of much study. The idea of turning a glove inside out, to make a
> right handed one become left handed... what textbooks deal with that?
Almost everybody thinks that mirrors reverse left and right, whereas
the physics and the geometry make it quite clear to those who can
recognize the signs that mirrors reverse front to back when you are
facing them.
> I've been investigating space-filling tessellations of tetrahedra quite a bit.
> There's this meme out there that "Aristotle was wrong" in saying tetrahedra
> fill space, but that's only if we insist he meant regular ones. When it
> comes to irregular tetrahedra, we have an interesting discussion, and
> the notion of chirality (handedness) enters in, i.e. which space-filling
> tetrahedra depend on left and right versions of themselves, versus
> being all indistinguishable?
You can dissect a cube into three tetrahedra, too, and fill space that way.
> I like to bring students through Math World, the page on space-filling
> polyhedra, to point out how that page lags behind the published literature.
> It's silent about tetrahedra, aside from repeating the "Aristotle was wrong"
> meme.
>
> http://mathworld.wolfram.com/Space-FillingPolyhedron.html
>
> Why I find tetrahedra especially interesting is they're topologically
> minimal, i.e. no box or container of facets, nodes, edges, has
> fewer than (4, 4, 6).
This is an essential point in algebraic topology, where the
tetrahedron is a 3-simplex, and all other spaces are built by joining
simplices along vertices (0-simplices), edges (1-simplices), or faces
(2-simplices) into simplicial complexes. Similarly for higher
dimensions, where N-simplices can be joined on any lower-order
simplices.
> The cube is (6, 8, 12) by contrast. In the
> futuristic math courses I favor, we at least mention that making
> a tetrahedron our unit of volume opens a whole new world of
> whole number values for related polys. I often publish these
> charts:
>
> http://wikieducator.org/Martian_Math#Unit_of_Volume
> http://wikieducator.org/Martian_Math#Unpacking_Polyhedra
>
> (OERs, free for teacher use around the world without legal
> encumbrance -- what Wikieducator contains, free and open).
>
> Sorry, I digress, just yakking about how "handedness" is a
> feature of my geometry research and classes. Also, when
> it comes to XYZ coordinates, I like to point to exotic alternatives,
> such as the 4-tuple-based "Quadray coordinate system"
> which I participated in designing.
No connection with the 4-tuple based homogeneous coordinate system for
projective space commonly used in computer graphics, I see.
> Here's the Wikipedia page:
>
> http://en.wikipedia.org/wiki/Quadray_coordinates
>
> This is a tiny niche market, certainly. My futuristic Martian
> Math class was the most recent field test, at Reed College
> in Portland in August of this year:
>
> http://www.4dsolutions.net/satacad/martianmath/toc.html
>
> Kirby
>
>> Cheers,
>> Maria Droujkova
>>
>> Make math your own, to make your own math.
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "MathFuture" group.
>> To post to this group, send email to mathf...@googlegroups.com.
>> To unsubscribe from this group, send email to
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>> For more options, visit this group at
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>>
>
> --
> You received this message because you are subscribed to the Google Groups "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
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>
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
Maria Droujkova
It's a widespread phenomenon, apparently. Here's a beautiful RSA cartoon explaining it http://www.youtube.com/watch?v=u6XAPnuFjJc
MariaD
kirby urner
Wow that was an excellent and inspiring video. I'd not watched an
RSA cartoon before. Thank you for pointing me to that needle
in a haystack.
Thanks for your comments as well Ed.
Continuing this thread regarding economics and surprising results:
I'm one of those who advocates elevating "general systems theory"
(not my invention of course) as a *competitor* for economics (the
discipline).
Economists tend to think monopolies are inefficient (at best) so
it follows that economics itself should have some competition (GST).
GST should muscle in with its own approach to the same bread
and butter issues (of resource allocation, motivation etc.).
I used to correspond with Kenneth Boulding, another Quaker, and
an economist, about such ideas. He also wrote about systems
theory quite a bit.
One of the hallmarks of GST is it credits the sun for so much
value added, making Planet Earth more like a non-profit agency
with the sun the main sponsor (donor, grant giver). That's a
different bookkeeping model than crediting "taxpayers" for
everything (for example).
Connecting to Egypt (thinking of Milo), I'm wondering if we
should embrace Pharaoh Akhenaten as one of the founding
fathers of GST? I suppose that's more a mnemonic link
(a way of wiring the circuits) than another "fringe theory"
(it's less a theory than a segue).
Note: I am aware of schools of thought within economics
that are likewise treating Earth as an open system with a
steady energy budget infusion. In fact, I suppose all of them
do that, once they stop to consider the physics. Sort of
a no-brainer I guess. Maybe there's a Youtube out there...
Kirby
Example GST page from 1990s.
http://www.grunch.net/synergetics/gst3.html
Blog post from 2006:
http://controlroom.blogspot.com/2006/05/general-systems-theory.html
LFS
Hiya Kirby,
Thank-you for watching and commenting on each video. I REALLY appreciate it. My comments on your comments are below.
>Thanks for doing this. I support use of instructional video, made by both faculty and students. Even guardians (such as parents) might get in on the act.
1. I am hoping that some day we can establish a library of “how I teach it”. (One of the things I like about Jing is the 5 minute limit …)
2. I am desperately hoping we can get kids to “talk” through their math since I don’t think we can get them to write through their math.
#1 Clear. I might encourage saying the calculation out loud yet only writing the y answer, not showing the steps. (http://www.screencast.com/t/6K9ibcsTcg)
Actually I disagree with this to some extent and that is basically for (a) the reason I made 3 videos and (b) includes the problem you mention in #3.
I make a big fuss (and take off points) about differentiating between expressions and equations and between functions and their values. I have found that if I don’t write carefully through the process, they will just start writing down “answers” – without regard for expressions and equations and often times wrongly calculated. (Of course, they expect partial credit for this since it was only a “calculation” error.)
And without the detail, they REALLY don’t get the difference in the processes involved in the 3 screencasts.
That is, 3 entirely different processes are involved in:
= graph y=mx+b by finding 2 points, (i.e. substitute 2 values for x and simplify expression for y; graph 2 points)
= graph Ax+By=C by finding 2 points (i.e. substitute 2 values for x or y and solve resulting equation; graph 2 points)
= graph y=mx+b using standard slope-intercept method (i.e. draw b on y-axis and use rise over run to find 2nd point)
In grades 5th-12th, kids learn each of these techniques separately in different grades over a period of several weeks or months. The teacher doesn’t require them to relate them to previous or future techniques. Only to solve with the technique at hand.
However, when I get them in college:
(a) they are expected to know how to do all 3 rigorously and interchangeably and
(b) if they don’t, I must teach/review them this in 2-3 classes.
Suggestion: I suggest that we throw away standard form Ax+By=C altogether (including the addition method for solving systems – if they are interested, let them learn Cramer’s method, et.al. in college) and teach them to use slope as I talked about in previous letters and screencast: http://www.screencast.com/t/vWfJdZin4L
(And yes, I am going to continue to assume that x-positive is to the right – these kids are confused enough as it is. That said, I did find the discussion quite thought provoking because I myself have always done it “the right way” – pun intended :) and never even thought about it differently except when I moved to Europe and had to do 3d with the opposite hand.)
#2 Here I note that your "decimal equilivalents" are not precise in either case. (http://www.screencast.com/t/njGTPjdu )
Done on purpose – I actually changed the problem several times until I got this. Reasons:
(a) Life is mostly bad (rounded) numbers and
(b) If you don’t allow this, then either we make our problems with nice numbers or our kids are trying to figure out whether their answer 4*sqrt(3) to “What is the area of an equilateral triangle with base 4?” is reasonable when they can easily see that Area ~7. (I make them graph quadratics with real, but ugly roots.)
#3 The confusion I find here (looking from a student point of view) is endemic to calculus teaching. You say "they have to go back to 'y the function'" as you point to y = x^2 - 3x + 2. (http://www.screencast.com/t/gOun4DMtEbv)
I totally agree. But it is not just calculus. y=3x+2 is a function in grade 8. (Not to mention my hated Ax+By=C. What is it? A relationship?)
I make a big fuss if they take the derivative and substitute a value for x in the same line, i.e. y'=2x-3 = 2*0-3 = -3.
I require y'=2x-3. (New line) y'(0)=2*0-3=-3 But many colleagues do not agree with me and think I am too fussy and you will notice that I did not obey my own strictures of writing the substituted value when graphing lines… (Sorry – don’t know Python. Here is my Ready-2-Use GeoGebra interactivity: http://tinyurl.com/36463ov (BTW: I could not get your catenary to rotate?)
>My question is: do students get to play with this cool interface allowing drag and drop tables, a graph, multiple pen colors on screen? If so, that's cool.
Unfortunately, by the time I get them they are “too embarrassed” to try. I can get maybe 2% to do it and I lend out the tablets. Interestingly, I do have better luck with my post-graduate students. They will actually make all kinds of videos and other resources. Maybe it is because they have children.
Warm regards, Linda
kirby urner
Or is it six identical Schlafli tetrahedra -- actually 3 right and 3
left handed?
http://en.wikipedia.org/wiki/File:Triangulated_cube.svg
True that the volume of a cube is 3, relative to the regular one inscribed
as face diagonals.
This web page is about space-filling tetrahedra that are truly identical
in the sense of not left and right handed:
http://demonstrations.wolfram.com/SpaceFillingTetrahedra/
Although not mentioned on the above page, Bucky Fuller also came up
with 3 of the 5 possible, which he named the Mite, Rite and Bite. The
fourth, noted by Sommerville (not mentioned by Fuller) is a 1/4 Rite.
Fuller's innovation, beyond the naming conventions, was to dissect
said Mite (which builds the Rite and Bite) into 3 components, 2 "A"
and 1 "B".
He thought this was an important discovery and in dedicating his
magnum opus to H.S.M. Coxeter, he was keen to also connect to
the latter's 'Regular Polytopes' by page number, where the Mite
is shown (not by that name).
Partly why he found the AAB dissection intriguing is it revealed
an "under the hood" handedness to the Mite, which can be assembled
from either a left or right handed B (whereas the pair of As come
as a balanced left and right set).
In other words, these two Mites are outwardly indistinguishable
yet have a different anatomy:
Mite = A+ A- B+
Mite = A+ A- B-
(kind of like how a left-side or right-side drive car looks outwardly
the same when you can't see in the windows)
Fuller wanted a place in the history books too, for an otherwise off
beat work (too off beat for most subsequent curriculum writers -- I'm
in a tiny minority in carrying on with his torch at this point).
You'll see all these funny names (A, B, Mite, Rite, Bite...) listed in
those Martian Math volumes tables. It's a way of unlocking some
esoteric literature. Fuller's writing is hard to read, especially if
you have no such key or road map. Martian Math repacks it to
make it more viewer-friendly in some ways.
>> I like to bring students through Math World, the page on space-filling
>> polyhedra, to point out how that page lags behind the published literature.
>> It's silent about tetrahedra, aside from repeating the "Aristotle was wrong"
>> meme.
>>
>> http://mathworld.wolfram.com/Space-FillingPolyhedron.html
>>
Note that the above URL fills in the missing puzzle pieces which
this space-filling polyhedron article does not supply. It would not be
hard to do an update at this point. Of course I'd like to see Fuller
mentioned but that's unlikely to happen. He's been eclipsed by
Goldberg before, when it came time to write about the micro-
architecture of the virus.
http://www.grunch.net/synergetics/virus.html
A case of deja vu then...
http://www.4dsolutions.net/synergetica/synergetica2.html
Right, no connection in this case, although I'm certainly aware of
homogeneous coordinates.
Fuller was quite the contrarian, wanted to counter some of the
most established dogmas of all time with some alternative discourse.
That's really hard to do in mathematics and I salute him for the
attempt (whether or not we judge him successful).
Quadray coordinates are useful because they help give entre
into this discourse where we say volume or space is 4D, not 3D.
You've always got those four walls and four directions of the
simplex. It's easy to fight back and uphold the traditional view.
What's harder is to allow both views to hold water i.e. to not
play either/or. We're not so used to doing that in elementary
math, which is precisely why I see this as useful exercise.
>> Here's the Wikipedia page:
>>
>> http://en.wikipedia.org/wiki/Quadray_coordinates
>>
>> This is a tiny niche market, certainly. My futuristic Martian
>> Math class was the most recent field test, at Reed College
>> in Portland in August of this year:
>>
>> http://www.4dsolutions.net/satacad/martianmath/toc.html
>>
>> Kirby
>>
>>> Cheers,
>>> Maria Droujkova
>>>
Thanks again for the link to that animation, about how people want
respect more than money, when doing stuff that's cognitively
challenging.
Kirby
>>> Make math your own, to make your own math.
> --
Maria Droujkova
Hiya Kirby,
Thank-you for watching and commenting on each video. I REALLY appreciate it. My comments on your comments are below.
>Thanks for doing this. I support use of instructional video, made by both faculty and students. Even guardians (such as parents) might get in on the act.
1. I am hoping that some day we can establish a library of “how I teach it”. (One of the things I like about Jing is the 5 minute limit …)
2. I am desperately hoping we can get kids to “talk” through their math since I don’t think we can get them to write through their math.
Cheers,
Maria Droujkova
milo gardner
|
|
|
kirby urner
On Fri, Nov 12, 2010 at 9:47 AM, milo gardner <milog...@yahoo.com> wrote:
>
> Hi Kirby,
>
> Thanks for your comments on Egyptian math and economics per:
>
> " ... Economists tend to think monopolies are inefficient (at best) so it follows that
I'm always interested in weaving in time lines.
>
> Connecting to Egypt (thinking of Milo), I'm wondering if we
> (a way of wiring the circuits) than another "fringe theory"
>
> My reply includes Dieter Mueller, "Some Remarks on
> written in 1975 that connects elite scribal thinking and
>
> http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html
>
> in which wages paid by elites were controlled 500 years before
>
> Please also note that village life after Akhenaten's life seemed
kirby urner
>>> The commonly-cited instance of backward-sloping supply is Polynesian
>>> labor markets in which it used to be that raising wages past a certain
>>> point reduced the supply of labor. Of course, we have instances of
>>> this in Europe and the US, such as opera composer Rossini realizing
>>> that he was rich, and had no need to go on composing opera, or
>>> cartoonists Gary Larson (The Far Side) and Bill Watterson (Calvin and
>>> Hobbes) retiring.
>>
>> It's a widespread phenomenon, apparently. Here's a beautiful RSA cartoon
>> explaining it http://www.youtube.com/watch?v=u6XAPnuFjJc
>>
>> MariaD
>>
>
> Wow that was an excellent and inspiring video. I'd not watched an
> RSA cartoon before. Thank you for pointing me to that needle
> in a haystack.
OK, it's only a couple of days later and I'm trying to tell myself
what I learned
from that video?
Is it that money is not a motivator when it comes to intellectual tasks,
or simply that there's no way to "force" high mental performance, if one
is trying to learn how to do something.
Pressure just adds to the stress. Here you're being offered a year's salary,
(braces for the wife and kids, new glasses) if you'll just beat some computer
at chess.
Think of doctors, often asked to prolong life, by hook or by crook, by people
with tons money. If they don't know how, they don't know how, and simply
offering 'em more dough is hardly gonna turn 'em into magicians.
Is this all they discovered, that there's no magic way to get smarter?
Surely it was a deeper insight than that. Maybe help me out here.
With mechanical labor: do it faster, do it longer, get paid more. Yeah,
that's a pretty normal reaction.
But so is getting resentful of clients who think you won't jump through
hoops just because you're holding out for a bigger reward. No, sometimes
you just can't do the job.
If you could, it'd count as "mechanical" no? Like adding a bunch of numbers
-- for some people that's no different from pushing a wheel barrow.
If that's all this was about, then I'm mystified the economists are mystified.
Maybe I should watch it again -- this was a test of my recall memory
(could be I failed the test).
Kirby
Maria Droujkova
Ed Cherlin:
OK, it's only a couple of days later and I'm trying to tell myself
>>> The commonly-cited instance of backward-sloping supply is Polynesian
>>> labor markets in which it used to be that raising wages past a certain
>>> point reduced the supply of labor. Of course, we have instances of
>>> this in Europe and the US, such as opera composer Rossini realizing
>>> that he was rich, and had no need to go on composing opera, or
>>> cartoonists Gary Larson (The Far Side) and Bill Watterson (Calvin and
>>> Hobbes) retiring.
>>
>> It's a widespread phenomenon, apparently. Here's a beautiful RSA cartoon
>> explaining it http://www.youtube.com/watch?v=u6XAPnuFjJc
>>
>> MariaD
>>
>
> Wow that was an excellent and inspiring video. I'd not watched an
> RSA cartoon before. Thank you for pointing me to that needle
> in a haystack.
what I learned
from that video?
Is it that money is not a motivator when it comes to intellectual tasks,
or simply that there's no way to "force" high mental performance, if one
is trying to learn how to do something.
I took home two things:
- That the curve of motivation/extrinsic rewards is backwards/sloping and an excellent example of why you need to turn axes around.
- That extrinsic motivation will interfere with extrinsic when there is too much or too little of it
I am teaching "immediately useful psychology" at a homeschool coop, and I am going to show this video to the kids. It's faster than reading "Punished by rewards" which I will discuss, as well.
Maria Droujkova
"Money, it seems, is a little like beer. Most people like it, but more is not necessarily better. A beer might improve your mood, but drinking 10 beers not only won’t increase your happiness tenfold, it might not increase it at all."
The un-heir of Baskin-Robbins, John Robbins, talks about his decision not to use his family fortune directly.
http://greatergood.berkeley.edu/article/item/the_economics_of_happiness/
Cheers,
MariaD
kirby urner
Kirby:
>> Is it that money is not a motivator when it comes to intellectual tasks,
>> or simply that there's no way to "force" high mental performance, if one
>> is trying to learn how to do something.
>
> I took home two things:
>
> - That the curve of motivation/extrinsic rewards is backwards/sloping and an
> excellent example of why you need to turn axes around.
I've already forgotten if there were any graphs of this nature. I loved the
fast-paced drawing though, found that highly effective.
But was the lesson that "more money can't buy higher intelligence" (or
any skill that takes real time to develop)? Indeed, when you get more
pressure (wife and kids, hoping you'll bring home a year's wages in
exchange for some feat), your tendency might be to break down.
The movie 'Slumdog Millionaire' comes to mind. Too much suspense.
> - That extrinsic motivation will interfere with extrinsic when there is too
> much or too little of it
>
The cartoon brings up the Gnu World Order thing, i.e. the rise of Linux,
fueled by any number of geeks working hard on their free time, not
getting paid. But what was the motivation in this case? If you watch
'Revolution OS' (DVD documentary) you'll get some idea.
As a computer science major, you got to work with all these cool tools,
then come graduation you're pushed out the door, and to continue
having access you'd need to work for some big company that pays
$$$ in licensing fees for the same tools. You'd have to slave for a
boss (as many were doing), could not explore your own dreams (as
this would be seen as taking away from your productivity).
As an individual computer science grad, you were out of luck, out of
work, separated from what you'd need to make a difference -- a painful
state.
So no wonder geeks all banded together to roll their own, and not
have to depend on older adults to provide them with a free operating
system. If it'd been left to the older generation, the kids would have
mentally starved to death, if not physically. A useful lesson to take
home.
However, this cartoon kind of bleeps over all these motivating factors
to suggest some new and puzzling fact of human nature that
economists don't really understand. We're left thinking it was all
mystifying romantic idealism, not the hard reality of intellectual
property laws being designed to create artificial scarcity everywhere
you turn.
Back to geeks and their world-around collaborations, it's not hard
to miss Richard Stallman's pain when he talks about working at
MIT (back to MIT again, like in the cartoon) and being told he
couldn't work on the software he and his buddies had been living
and breathing, because it was now going to be owned and controlled
by people who'd not done any of that work themselves. That's why
it's called a "revolution". This was one of the great "take backs"
of the fruits of one's labor of all time. So I have to be suspicious
of economists who come on the scene and scratch their heads
and act mystified. What is it they don't understand again? Just
asking.
> I am teaching "immediately useful psychology" at a homeschool coop, and I am
> going to show this video to the kids. It's faster than reading "Punished by
> rewards" which I will discuss, as well.
>
I need to read "Punished by Rewards" I'm thinking. Link?
Kirby
>
> Cheers,
> Maria Droujkova
>
> Make math your own, to make your own math.
>
kirby urner
> Here is another reference that just came to my mailbox from "The Daily
> Good." It expresses the backwards-sloping-ness of the money/happiness curve
> nicely:
>
> "Money, it seems, is a little like beer. Most people like it, but more is
> not necessarily better. A beer might improve your mood, but drinking 10
> beers not only won’t increase your happiness tenfold, it might not increase
> it at all."
>
Yes, money comes with the temptation of great freedoms, and we all
wish for freedoms. But then it takes lots of practice to play an instrument
(another example), to do brain surgery, to write great novels. Just having
a lot of money does not come with a guarantee of being self-disciplined
enough to actually develop one's skills. On the contrary, when you have
money, it looks like you don't need any skills, as you can always pay
others to do everything for you.
The Polynesians maybe saw the dangers of this system, when the
economists flew in from MIT, scratching their heads about the free
software movement, or students rebelling instead of taking the cash.
Why wasn't money the only necessary reward?
The Polynesians reminded the MIT economists about that story
about Faust, who was tempted to trade away his soul for a lot of
power to get what he wanted. But it turned out that, without a
soul, there's no way to get enjoyment out of what one has, no
matter how much. As the movie 'Over the Hedge' puts it (a spoof
on having too much): 'enough is never enough'.
> The un-heir of Baskin-Robbins, John Robbins, talks about his decision not to
> use his family fortune directly.
> http://greatergood.berkeley.edu/article/item/the_economics_of_happiness/
>
>
> Cheers,
> MariaD
I've been looking to Ben & Jerry's for some economics of happiness.
People talk about Bhutan a lot, and indeed, I used to go there, as my
dad was employed there, in pay of some Swiss. Mom too. She
wrote curriculum. I got to teach some computer science and do
volunteer work. I great privilege. Indeed it felt like a happier place.
Yet the GNP is not all that high, when measured in terms of money.
Average Americans would tend to look down on these people, for
not having much "net worth" (a very peculiar way of talking that
sounds pathological to outsiders).
Kirby
Algot Runeman
before they are ten. Gradually, however, our system appears to squash
the joy out of the process. It has seemed to me that we reward
"completion" of tasks, not mastery. Our curriculum is built to "cover" a
set amount of content. Our reward system is grades, and the children who
fail to achieve high grades are gradually convinced to stop making the
effort. The children who easily get through the assignments get high
grades. But those high grades are meaningless, too. The success came too
easily. The grade is "nice" but the accomplishment is insignificant
because it was too easy.
Does that mean the ones in the middle are challenged and rewarded
effectively. I doubt it. You would need to convince me that
middle-of-the-road students are getting the rewards of enjoyment,
mastery and contributing to anything mentioned in the presentation.
Teachers are usually not provided the leeway to create assignments that
engage and reward their students creative effort. They are, partly
because of standardized curricula and standardized high stakes testing,
drones working hard to provide good classroom management as they "cover
the curriculum."
--Algot
--
-------------------------
Algot Runeman
47 Walnut Street, Natick MA 01760
508-655-8399
algot....@verizon.net
Web Site: http://www.runeman.org
Twitter: http://twitter.com/algotruneman/
Open Source Blog: http://mosssig.wordpress.com
MOSS SIG Mailing List: http://groups.google.com/group/mosssig2
Bradford Hansen-Smith
| If a person has passion and is curious and excited about what they are doing, then rewards have little to do with staying with it or not. If one gets rewarded, directly or indirectly from their passion it can be either a help or hinderence depending on personal focus, dedication, and courage. If there is no passion then rewards of various kinds are always a determining factor and motivation to go further or to quit. There is no way to account for what we call passion by plotting a slope of human activity, most human activity is without passion, it is beyond some arbitrary norm and is flat lined. There is no over run unless there is a predetermined limitation that has been set; a big educational problem. Algot, I have to agree with you, cultural education is the suppression of natural individual curiosity and discourages passionate interest that goes any deeper than the educational and cultural surface. When individuals feel their passion is unacceptable, they either withdraw or rebel. We see a lot of both these days. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Sun, 11/14/10, Algot Runeman <algot....@verizon.net> wrote: |
|
|
|
Phil Wagner
Punished by rewards by alfie kohn is the name of the book but the homework myth or most anything else by him will give you the idea.
On Nov 14, 2010 4:21 PM, "Algot Runeman" <algot....@verizon.net> wrote:On 11/14/2010 04:38 PM, kirby urner wrote:
>
> On Sun, Nov 14, 2010 at 3:36 AM, Maria Droujkova<drou...
Education is fun for most students early. Children often "play school" before they are ten. Gradually, however, our system appears to squash the joy out of the process. It has seemed to me that we reward "completion" of tasks, not mastery. Our curriculum is built to "cover" a set amount of content. Our reward system is grades, and the children who fail to achieve high grades are gradually convinced to stop making the effort. The children who easily get through the assignments get high grades. But those high grades are meaningless, too. The success came too easily. The grade is "nice" but the accomplishment is insignificant because it was too easy.
Does that mean the ones in the middle are challenged and rewarded effectively. I doubt it. You would need to convince me that middle-of-the-road students are getting the rewards of enjoyment, mastery and contributing to anything mentioned in the presentation.
Teachers are usually not provided the leeway to create assignments that engage and reward their students creative effort. They are, partly because of standardized curricula and standardized high stakes testing, drones working hard to provide good classroom management as they "cover the curriculum."
--Algot
--
-------------------------
Algot Runeman
47 Walnut Street, Natick MA 01760
508-655-8399
algot....@verizon.net
Web Site: http://www.runeman.org
Twitter: http://twitter.com/algotruneman/
Open Source Blog: http://mosssig.wordpress.com
MOSS SIG Mailing List: http://groups.google.com/group/mosssig2
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milo gardner
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kirby urner
If a person has passion and is curious and excited about what they are doing, then rewards have little to do with staying with it or not. If one gets rewarded, directly or indirectly from their passion it can be either a help or hinderence depending on personal focus, dedication, and courage. If there is no passion then rewards of various kinds are always a determining factor and motivation to go further or to quit. There is no way to account for what we call passion by plotting a slope of human activity, most human activity is without passion, it is beyond some arbitrary norm and is flat lined. There is no over run unless there is a predetermined limitation that has been set; a big educational problem.
Algot, I have to agree with you, cultural education is the suppression of natural individual curiosity and discourages passionate interest that goes any deeper than the educational and cultural surface. When individuals feel their passion is unacceptable, they either withdraw or rebel. We see a lot of both these days.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Bradford Hansen-Smith
| Milo, It seems to me that in all this exploration into the psychology of education and intelligence there is nothing being said about meaning or progressive value. Maybe that is what you are referring to that is "incomplete and important considerations" in the work you sited. With education in general and math specifically there seems no concern for moral or ethical guidance, like they are not part of the human equation; something to be addressed by some other department or field of study. As an isolated, self-referencing, abstract construction, mathematics as a language does not address higher purpose or progressive meaning, particularly as it is taught. It falls short of becoming meaningful on deeper levels of personal human development. I hear what you are saying about your passion, particulary about stuttering, but I question the meaning of the word in that case because I do not see passion associated with avoidance. Passion is more about embracing life in what ever way individually makes sense. It has to do with the intense desire to gain meaningful experience in some way from what appears to be unknown. Possibly this is the direction of your passion about your speech. Brad |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Mon, 11/15/10, milo gardner <milog...@yahoo.com> wrote: |
kirby
On Nov 8, 7:22 pm, kirby urner <kirby.ur...@gmail.com> wrote:
> On Mon, Nov 8, 2010 at 4:43 AM, Maria Droujkova <droujk...@gmail.com> wrote:
> > On Sun, Nov 7, 2010 at 7:04 PM, kirby urner <kirby.ur...@gmail.com> wrote:
>
> >> Also, I'm obsessing in this thread about how x is alwayspositive
> >> to the right by convention. Your video takes this convention so
> >> completely for granted that no labeling is even needed. That's
> >> sign and so on (might be simply a change in viewer angle i.e.
> >> looking from the other side of the same graph).
>
> >http://knowyourmeme.com/memes/in-soviet-russia) they used to flip axes in
> > math track courses (hard sciences, engineering, math) but not in general
> > courses.
>
> That's very interesting to know, I had no idea.
>
> > This comes up a lot, for example, if you switch coordinate systems from
> > absolute to object-related. Which you may do in some physics problems, and
> > in game programming. Force diagrams come to mind.
>
> all the textbooks.
>
(+ +) is upper right in the vast majority of cases (not upper left).
Looking from the point of view of the stage (or page), out to the
audience (or reader), it would be to "my left" where I'm facing
forward and standing vertically in the positive Y direction.
Bows,
Kirby