How low can you see that the earth is curved

At how low an altitude does the earth look like a disk, in other words, from how low does a line from the horizon to your eye trace out the surface of a cone (with angle a shown from the centre) ?

Surprisingly low. The depression of the horizon below level is noticable from an altitude as low as 218m. (Lysterfield Lake trig point 37°56'31" S 145°16'06" E, Elevation 218m)

View looking south east, the faintly visible Strezleki ranges are level with the line

View looking north west, the horizon is below the string

Looking east, the horizon is below the two tape markers

Looking west, the horizon is just below the two tape markers

The average depression was 7mm at a distance of 147cm

a = sin (a) for small a

a = 0.7/147 = 0.00476

r = 2h/a^2

r = 2 *218/(0.00476^2)

r = 19,000 km

Compared with the actual value of 6,371 km

Notes
Earth from space, image
Small angle formula

East Timor continued, video

(10 minutes)

Or see the short version (3 minutes)
http://www.youtube.com/watch?v=1C6Ru1hpDjk

Turtle calculus

Its easy to plug in various functions as a graph's gradient in Turtle Art. The resulting graph is the integral of the function used as the gradient. In the below case, the gradient is 1 and its integral is a straight line, f(x)=x



Plug in a linear gradient, -2+x/100, and make a parabola with zero crossings at x=0 and x=400



A quadratic gradient generates a cubic with zero crossings at x=0 and x=300

The integral of sin is -cos
(by the chain rule)

making dy/dx =y gives an exponential curve
but a nonzero value at x=0 is necessary, in this case forward 1 means f(0)=1
starting at 0, f(0)=0 has a solution of a straight line f(x)=0


Challenges
Graph for negative x
Graph the exponential for f(0)=-1 explain
The exponential above is for dy/dx = y/100 what exactly is the formula for the graph?
Find more interesting integrals
Graph the derivative of a function
Iteratively solve d2y/dx2 =-y
Explain the offset in the graph of -cos
A step size of 1 is used, what error does the step size introduce?

Fractions and decimals 4th grade maths

Fractions
TurtleArt project to meet outcomes 4N3 to 4N6 in the 4th grade math curriculum. Enter a single digit numerator and denominator to hit a target which is expressed as a decimal 0<1.

Open the source and paste into TurtleArt.
Could be improved with rewards and score.

Multiplication version
Source

Counting
Enter the number of balloons. If the turtle hits the top balloon, all are popped.

Source

Turtle flip and rotate, 4th grade maths

An activity in Turtle Art to meet the 4th grade curriculum item "4.G.7 Describe and apply techniques such as reflections (flips), rotations (turns), and translations (slides) for determining if two shapes are congruent"

A number of shapes are randomly generated (in this prototype just a right angle triangle and a trapezium) unrotated and unflipped on the left and randomly rotated and flipped on the right. The student flips and rotates to match the two polygons.

In boxes (variables) 11 to 19, the trapezium is stored, distance, rotation, distance, rotation ... and terminated buy -1. Similarly, the triangle is stored in variables 21 to 27.

poly is a random number, either 10 for the trapezium or 20 for the triangle. Rotate is a random number, (0-7) x 45 and flip is either -1 or 1.

The program loops, waiting either a 'F' key, 102 or a 'R' key, 114. The polygon is drawn once on the left unflipped and unrotated (netflip =12 and netrotate =0) and drawn once on the right flipped and rotated (netflip = flip x userflip and netrotate = rotate + userrotate)

drawpoly draws the polygon specified in box 11-19 or 21-27 terminating on a -1 value, the polygon has rotation and flip specified in netrotate and netflip.

The activity could be improved with a score or reward system, sideways translation, more polygons and randomly congruent and incongruent pairs

Source

Turtle Pi (another one)

see also turtle-pi

Each side is of length 500/n where there are n sides, so half the perimeter of the closed polygon is of length 500, the polygon is of perimeter 1000, the diameter is xcor

perimeter/diameter approaches pi for large n

http://wiki.sugarlabs.org/go/File:Turtle_Blocks_pi_another.ta

En español
Véase también turtle-pi

Cada lado tiene una longitud de 500 / n donde hay n lados, por lo que la mitad del perímetro del polígono cerrado tiene una longitud de 500, el polígono es el perímetro de 1000, el diámetro es coorx

perímetro / diámetro pi enfoques para n grande

Listen

Read phonetically

Turtle Sieve of Eratosthenes

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. First delete the multiples of two, then multiples of three, skip 4 because it has been deleted, delete the multiples of 5 and so on.

In this program, the primes are built up visually, 0-49 in the bottom row, 50-99 in the second ...

Main calls setup which initialises 2500 integers. It then calls deletemultiple for each prime in box1 between 2 and 1250. (Student question: how far does it actually need to go?)

Setup sets up an array of 2500 variables: box (box1 +"") = 1
The value 1 is a flag that for the moment number box1 is considered a prime.
The null string "" was necessary because of a bug in TurtleArt V84.

Deletemultiple deletes multiples of box1 up to a product of 2500.
store in box (box1*box2+"")=0
marks the number as a non prime
again the null string "" is needed because of a bug in TA V84

Finally 2 stacks to plot and print the results.



Turtle_Art_Activity_eratos.ta

Septiembre 2010
En español
La Criba de Eratóstenes es un algoritmo simple, antiguo para encontrar todos los números primos hasta un número entero especificado. En primer lugar eliminar los múltiplos de dos, luego múltiplos de tres, cuatro saltar porque ha sido eliminado, eliminar los múltiplos de 5 y así sucesivamente.

En este programa, los primos se construyen visualmente, 0-49 en la fila inferior, 50-99 en el segundo ...

Main llama a la configuración que inicializa 2.500 enteros. A continuación, llama deletemultiple para cada uno de primera en box1 entre 2 y 1250. (Pregunta Estudiante: ¿hasta qué punto lo que realmente necesita para ir)


El programa de instalación crea una matriz de 2500 las variables: caja (caja 1 + "") = 1
El valor 1 es un indicador que para el número momento box1 se considera un número primo.
La cadena vacía "" era necesario debido a un error en TurtleArt V84, probablemente no necesita.

Deletemultiple elimina los múltiplos de box1 hasta un producto de 2500.
almacenar en la caja (caja 1 * BOX2 +"")= 0
marca el número como no principales
otra vez la cadena vacía "" era necesaria debido a un error en AT V84


Por último dos pilas para trazar e imprimir los resultados.

Turtle e

Calculation of e using the Taylor series

http://en.wikipedia.org/wiki/E_(mathematical_constant)#Complex_numbers

(Turtle Art seems to now only display one decimal point at V83 so I have scaled up by 10^12 to show decimal places, Previous example of Pi showed decimal places)

Student challenge: calculate square roots with the Taylor series

http://en.wikipedia.org/wiki/Methods_of_computing_square_roots

File:Turtle_e.ta

En español
Cálculo de correo utilizando la serie de Taylor

(Turtle Art muestra limitada decimales por lo que han ampliado en un 10 ^ 12 para mostrar más cifras decimales, el ejemplo anterior de Pi mostraron cifras decimales)

Estudiantes desafío: calcular raíces cuadradas con la serie de Taylor

Maths Wars

http://commons.wikimedia.org/wiki/File:Euler%27s_formula.png

"Maths Wars" refers to the tension between instruction and construction in teaching/learning

The following interesting quote came via Artichoke

"Many of those wanting to build relational understanding with students assume that spending time on rote procedural knowledge is an important precursor for developing deeper conceptual understanding. This seems like a common sense approach – a let’s keep a foot in both camps kind of approach. However, research findings in math education suggest otherwise (Pesek and Kirshner 2000). It seems more likely that, in maths education at least, time spent building prior instrumental understanding is an interference to, not an aid to, developing relational understanding."

Interesting because Australia schools at least seem to spend a lot of time on rote procedural knowledge in mathematics.

More on Pesek and Kirshner 2000

"Students who received the procedural instruction prior to the conceptual instruction learned less than did students who received only the conceptual instruction. Somehow, exposure to routine application of formulas inhibited or interfered with students' subsequent conceptual learning."

It still seems likely to me that the best teaching will incorporate both solid procedural knowledge and problem solving in authentic and relevant contexts. This study does not destroy that belief, but it does sound a warning that you can't just mix a bit of both and always get good results.

Problem visualisation in mathematics and programming

I have been researching into problem visualisation in mathematics and programming following a discussion, originally by Mark Guzdial and continued by Bill Kerr Alan Kay and others and also the parallel discussion by Rob Costello

One paper that has me thinking is the influence of texts' mental images upon problems' resolutions by Giorgio Bagni (2000). It examines the importance of the level of detail of the mental models constructed by students. Bagni asks whether imagining a situation in all its details helps problem solving. Bagni suggests that it can be an obstacle.

Problems were given to 3 Italian school classes, (13-14 years), (14-15 years) and (15-16years).

The first version was briefly stated as an abstract mathematical problem:

The length of the basis AB of an isosceles triangle ABN is 1 000 000 m; the sum of its sides AN, BN is 1 000 001 m; find the length of the height NM.

The second version was embedded in a real world context and included a diagram:

Let us tie a row to two nails very far, say… 1000 km; let us imagine to use a row whose length is exactly 1000 km: so this row will be tight. Then let us tie to the same nails another row, whose length is 1000 km and 1 m; so this second row is a bit longer than the distance between the nails and it will not be tight: in order to stretch it, let us bring the second row in its middle point and let us “raise” such point (see the picture), in order to take it away from the first row, until the second row is completely tight.

a diagram was shown followed by more text

Well, how much must we take away the middle point of the second row? Find the distance between the middle point of the first row, M, and the middle point of the second row, N.

I am hoping that the second version has suffered in the translation from the original Italian to English because it does not read well in English.

Finally the students were shown the correct solution by Pythagoras' theorem which has the result, which may be surprising at first, that by allowing a 1 metre increase in total path length, it is possible to deviate sideways by over 700 metres.

The students in the first group, those given the abstract problem, did significantly better than the second group with the real world problem.

When shown the correct solution, many in the second group, the group that had the problem given in a real world context, had difficulty accepting the correctness of the solution because it was counter intuitive that a such a small increase in total path length would allow such a disproportionately large sideways displacement.

Bagni concludes: “that D’Amore (1997) clearly proved that the full possibility to imagine a situation does not help pupils; now we state that, sometimes, this full possibility can constitute an obstacle to the resolution (or, as in the examined case, to the acceptation of the correct resolution)”.

I hear echoes of Cognitive Load Theory here. The idea that if you are teaching Pythagoras, you should remove any distractors from Pythagoras, so as to minimise the cognitive load on the learner. The learner can then focus their limited processing powers on the material which is to be learnt.

My thought is that the cognitive conflict created by the second problem is valuable for learning and that students should be given time to experience and resolve such cognitive conflict if they are to have more than a superficial understanding of a subject.

This particular cognitive conflict relates to deeper understanding of geometry and physics. It is closely related to the concepts that:

For small x, sin(x) = x.

Over a given distance, the tangent is very close to the circumference if the circle origin is remote.

At x=0, the derivative of sin(x) is maximum but the derivative of cos(x) is zero.

Why is it increasingly hard to pull a rope span, a catenary, the closer to straight it gets? Because your mechanical advantage tends to 1/infinity. Which group of students would you rather have specifying the support structure for electricity cables crossing a busy road?

Jonassen talks of schooling to create real world problem solvers. In school you know that you are doing Pythagoras this week so any problem you are given will be solved in a similar way. Real world problems are poorly specified and multi disciplinary, they require students to have engaged in messy problems.

OLCC2007 How the Read/Write Web Challenges Traditional Practice

Todays session of OLCC2007 was not about the learning theory as such but more about the oportunities for connected learning or as one chat comment said: I don't think there is a completely distinct line between this theory and others--for myself, I see this as one more set of tools to use

We see the undeniable effect of Web2 or public authorship tools creating an engaging and relevant context for learning. Students who would never write an essay contribute to Wikipedia, Youtube, Myspace etc. Players of World of Warcraft are heavily dependant on text communication. The old school will criticise it for bad spelling and grammar but it is effective communication, suited to modern times.

I have written previously on this blog on the value of WoW, as a programming environment, an example of economics but mostly as a supportive global village where children learn management skills from a community.

Second Life was mentioned in chat. I think that the attempt to re-create traditional learning spaces in SL is misguided, it remains a creative space with much potential.

The internet has given access to a wide range of educational material, Maths Demo's, Applets and Virtual Manipulative this is really fun stuff,

Networking creates the opportunity to meet people of common interests. The edublogging community is great for teachers, but have students benefited? We need to teach students to be lifelong learners, teachers need to model this, teachers need to be transparent learners. For example, kids need to be shown how they can set up RSS feeds.

The opportunities for collaboration have increased. My experience with kids learning mathematics and programming through making computer games is that you create an environment where peer tutoring arises naturally but that experience is not universal .

The concept of “trusted sources” worries me. The advice was to assemble a network who think like us. That could lead to isolated extremist thinking.

First post

Well this is the first post. I have added another learning object at http://www.freewebs.com/schoolgamemaker/#lobject . Its a Lissajous generator. Why? Mainly because it was fun but also because I like the idea of learning objects being reprogrammable by the learner.

I was inspired by the excellent set of mathematical gif animations at http://mathworld.wolfram.com/topics/AnimatedGIFs.html and then re-visited Virtual Manipulatives at http://nlvm.usu.edu/en/nav/vlibrary.html and http://www.fi.uu.nl/wisweb/en/welcome.html

Seems that there is a continuum:
Objects for viewing
Objects for manipulating & viewing
Objects for re-programming, manipulating and viewing

Maybe if the third option was done by somebody more skilled than me, it would be the most powerful learning object. There's your challenge!