Fractal flakes

To decorate for our winter party before the semester-end break, we made paper snowflakes in art class at school.
Being the geek that I am, I made a mobile from the fractal of the Koch snowflake, which starts from a single equilateral triangle, and keeps adding triangles onto the triangles. The mobile is made from the first three iterations, cut out as nested pieces, plus the background to the largest, which is trimmed as a circle.
(The mobile’s crossbar is the metal edge that came loose from a ruler; it’s being employed in this manner to prevent misuse by unruly students.)

mobile made of three successive fractal iterations of the Koch snowflake, and the background piece of the largest
mobile made of three successive fractal iterations of the Koch snowflake, and the background piece of the largest

More on the Koch snowflake: http://en.wikipedia.org/wiki/Koch_snowflake

"SORRY, OUT OF ORDER"

A Facebook friend of mine posted this problem for folks to solve:

90 – 100 ( 6 + 3 ) = ?

Answers included 0, -90, 810 and -810. The correct answer is -810. Some of you are sniggering at the errors — quit that! 
Now, if you didn’t get -810, hang on …
Why do people have problems solving math equations? It’s not that they’re stupid, but that:

  • they get confused;
  • are anxious;
  • the teaching was boring;
  • they’ve moved around and have missed bits here and there;
  • they’ve learning difficulties;
  • the teachers are trapped following the text and the text is a mile-wide and an inch deep and not in sensible order;
  • the teaching made no sense or was based upon “just memorise how to do this process” instead of understanding why or when to use what methods;
  • … and sometimes people have problems for several reasons.

Hey folks, don’t feel badly if you got it wrong. I had trouble with the maths in school, too! I didn’t even learn all my multiplication tables until 8th grade. You know what? It’s not fatal; I slowly went through some pre-College Algebra classes at my local community (junior) college, and filled in the confused bits, gained confidence, and eventually went on to introductory Calculus.
And I still have to pause and think on some of my multiplication facts, and still have days when I’m prone to reversing numbers. But those difficulties don’t detract from the fact that I am able to learn math, and they don’t mean I’m stupid. (“Take THAT, ‘Mr. Dull’!” she says, shaking her fist at a middle-school algebra teacher.)
But now I work with students in 7th – 12th grade math, and you know what? Good news! It makes a lot more sense when you go back and review it as an adult! You can fill in the parts you missed or didn’t understand, and get a better idea of how it all fits together. Honestly.
Math no longer terrifies me, even though my brain still has that glitch that prevents me from memorising the quadratic formula. But I never use the quadratic formula in real life.
I DO use ratios in real life, for example, adjusting a recipe, figuring how much stuff to put on my garden, planning travel time… And I’ll show you how to do those really easily, without getting all tangled up in multiple steps, and you don’t need some mysterious “intuitive feel for how to set the problem up”.

.~#~.

MEANWHILE, In our problem above we use Order of Operations. I tell my students, “You use Order of Operations every day! You put your tee on before you put on your shirt, and you put on your jacket last.”
The problem above is solved like this:
Continue reading "SORRY, OUT OF ORDER"

When smart people are stupid

So I’m getting the first day of class materials organised, and looking at the online class Web application.  The instructor and students can both use it for sharing documents, so tomorrow I will have to demonstrate to the students how to access the program, and where I will put files for them. The instructor can also use it to record grades and attendance.
I look at the roster, noting that there are two guys with the same common first name,
Robert
Robert
But otherwise nothing potentially problematic until I come across an unfamiliar name.  Bulgarian, maybe?  Slovak?
Demo
I then look at the family name,
Student
Oh, duh!
_____________
* Maybe Demo is related to the statistician who came up with the Student’s t-distribution test   /joke

Maths * Chem = Ranting^2

Why are so many math books poorly written? Even many of the physical sciences books seem to have this terrible dichotomy between the text explaining the concepts, and the text explaining the calculations. I suspect it’s partly because one person is writing the conceptual text, and another person is writing the calculations text. I also suspect it is because both are written by people who are naturally good at the subject, just like most maths, chem, and physics teachers are naturally good at the subject.
Well, you do want people teaching who are good at the subject. But as many of us have noticed, being naturally good at something frequently results in people who cannot understand why others aren’t equally good at it. Once in a while those adepts become snobbish, because obviously the rest of the world just isn’t smart enough to get the stuff like they are. Many of the others simply have little patience with students who “must be stupid because they can’t figure out easy things” and can’t understand the material from having the previous explanation repeated again.
Duh! If it didn’t make sense the first time around, why would repeating the same explanation make any more sense the second or third time around? What we really need is Continue reading Maths * Chem = Ranting^2

More "Trap Bias"

Whenever I read statistics about the “increasing rates of autism”, I heave a big sigh. Those statements invariable contain a whole number of assumptions, many of them flat-out wrong, or at least unexamined. In the epidemiological data, there are diagnostic issues and census issues and statistical issues and of course, the inevitable agenda issues in the reportage of the census results and analyses. I’ve previously discussed a number of these problems, including incidence versus prevalence, and correlation versus causality in the post, “Epidemics of Bad Science vs Epidemics and Bad Science”
What I would like to address today is a related issue with diagnostics and perceived prevalence, meaning, “How do we know who has autism or AD/HD or a learning disability, and how many such people are out there?”
In entomology (and in other zoological branches) we have a concept known as “trap bias”. There are a number of ways of taking a census of an animal population, including using traps. A “trap bias” means that the kind of trap you use to census a population will limit the responders to your census, and thus create unintended biases in the results.
Now, if a few synapses in your brain just fizzled from that wordy definition, let’s try a simple example. Continue reading More "Trap Bias"

Thrown a curve

(“Thrown a curve” is a phrase from baseball, meaning when someone throws you a curve ball that is difficult to hit; it can also mean running into something unexpected.)
Halfway through the semester of Gen Chem I, we had just gotten another exam back, and things were grim. On the first day of class, the prof had told us that, “Half of you are going to drop out or flunk,” and he hadn’t been kidding; as we neared the last day to Withdraw from class, the students were dropping like flies. Those of us still remaining were struggling mightily. The students were bitching about the teacher, and in turn the teacher was complaining about “the kind of students nowadays” (and this was back in the early 1980’s).
Of the several dozen who hadn’t given up and were slumped through the lecture hall staring at their exams dripping with red ink, only two had done well, meaning had correctly answered at least 70% of the questions. (Hallway discussions after lecture would yield the fact that both of them had taken chemistry in high school, so this wasn’t their first experience with the concepts.) As the instructor skimmed through and told us the correct answers to the test, the grousing turned to arguing, and then to deal-making.
“Do you grade on the curve?” pleaded one student. Everyone turned expectantly towards the prof, who as usual, looked annoyed and cross. His utter fatigue with teaching had been apparent from the first week, and had disimproved steadily with the succeeding weeks. His answer, like all other quantitative answers, began with a sigh audible all the way to the back of the lecture hall, and then he rambled on in a rush of words as to how such a calculation would work, and then why it wouldn’t change anything on today’s exam because of the two students’ A and B grades in the 90+ and 80+ percentiles. After giving them an earful of arithmetic, the energy of the protesters was worn down, and he returned to reciting the answers we should have gotten. Why we had not gotten them was not an issue he discussed.
Later on that day I was more puzzled by grading curves than by acid-base reactions. (The conceptual part of chemistry was fine, I had simply gotten tangled up in the calculations. Again.) Not yet having the awesomeness of the World Wide Web for looking things up, I flipped through some maths books at the library until I found mention of the Normal Distribution Curve in a statistics book.
I understood grading by percentile; a score greater than or equal to 90% was an A, 80% was a B, and so on. And I understood how the normal distribution curve worked as far as describing how most of the members in a set were in the middle range, and successively fewer were at the lower and higher ranges. But trying to apply that normal curve (a mound that looked like a sand dune, or slice of bologna after my dad had cooked it in the pan) to distribution of grades left my brain itchy.
Everyone knew that a C grade was “average”, and that C’s were common, and A’s and F’s were rare. That should then mean that the Normal Distribution Curve was being supported as a pedagogical concept. But something didn’t seem right. I figured that “mental itch” feeling meant there was something wrong with my understanding; after all, it was obvious that I had major problems with calculations.
In later years I studied statistics, and learned that not every data set would follow a normal distribution curve. Some of them followed asymmetric curves with their central tendencies over to one side or the other, some of them were two-humped (the Bactrian camels of statistics), and some data sets didn’t make any particular sort of curve at all. I also learned about statistical circular arguments, whereby creating a measurement algorithm that would result in survey scores with a normal distribution curve did not prove that a population set naturally fell into such a curve — the curve was simply an artifice of the algorithm.
I have since learned that the “mental itch” feeling does not necessarily mean I am being stupid; more often it means that something else is Not Right.
Weird things happen when people try to force students’ grade into the curve. It’s not that the scores cannot fall into a curve. Rather, it’s that people try to use curves when they shouldn’t.
With the standard grading scheme, a student has to achieve a certain percentage to be considered as having mastered whatever was being assessed. (Whether or not that assessment accurately reflects the learning objectives is a whole ‘nother story.) But if we instead impose the normal distribution curve to sort out the A, B, C, D and F grades, we then say that the top grades are A’s, the bottom grades are F’s, and the median (and frequently mode) grades are C’s. There are a couple of problem with this. Firstly, it requires that some students get bad grades. Secondly, the distribution of letter grades from the curve does not guarantee that the students are succeeding in meeting the required competencies.
In addition to the problems that can be created by imposing curves, we have an essential problem in assuming that grades should even result in a normal distribution curve. There’s that algorithmic artifice issue, where exams can be created that will (when given to a large number of students) result in a grade distribution that creates a normal curve. This is the rationale for the argument for using grade curves. But it’s a circular argument, because not all assessment methods will yield such score scatters, and they should not have the normal distribution curve imposed upon them.
Furthermore, we have to ask ourselves if demanding a normal distribution curve really reflects our educational goals. Do we really want to have certain percentages of students getting bad or mediocre grades? When we ask individual teachers what they want for their students, none of them say that they want lots of average students, a few really good ones, and a few really poor ones. When we read the mission statements for school districts, we find that every district has Lake Wobegon dreams, where they want all their students to be “above average”.
Another concern people have is with “grade inflation”. Because of the pedagogical bias or expectation that grades “should” fall into that fabulous normal distribution curve, when we get lots of students getting B’s and A’s (and hardly, if any, getting D’s and F’s), then people start fretting that something is terribly wrong. Why, there must be grade inflation going on. Obviously, if so many students are getting good grades, then that must mean that the work is too easy!
On the other hand, if most of our students are not only passing tests and courses, but are even doing very well, maybe that just means that the teachers and students are both succeeding in their educational goals. Don’t we want all of our students to pass subjects and succeed? Education is not a zero-sum game, where every winner must be accompanied by a loser. Likewise, if most of the students are doing very poorly, it does not necessarily mean the students are just lazy or stupid.

Geek holiday alert!

Don’t forget — this Friday is Pi Day and Albert Einstein’s birthday! (March 14 = 3/14 in American-style date marking.)
In addition to eating your favorite kind of pie, you can also enjoy the non-repeating music of pi in the key of your choice.
Hmn … gooseberry? Blueberry? Pumpkin? … mmm …
(buzy with jobs — back to normal blogging again soon)

burning questions about phonics versus … pig ovaries

Yes indeed, it’s another exciting episode of your favourite irregularly-scheduled posting, Weird Search Terms. “Teh interwebs” is a strange and wondrous place, and some of it lands here! So without further ado (cue drummer):
More queries for the Interwebs Oracle:

  • i have to tell you something important
  • sleep recording surgery in rat brain
  • burning questions about phonics versus
  • pig ovaries
  • do i have fluid in my ear
  • can’t hear the fairy music
  • oxymoron – i need the number with no dig
  • how many bottons do air-planes have

bottoms? buttons?
In answer to your question: No. No. No. No. (Wash, rinse, repeat.)

  • chronic sleep deprivation causes autism
  • challenge test for heavy metals?
  • testimonials as evidence in science
  • egg white cures mercury poisoning
  • tinnitus green tea
  • sympathy and pity help the person to adj
  • vaccinations causing learning disabiliti
  • does finger flicking pages mean autism
  • asperger self hatred
  • auditory processing disorder stupid

Things get even weirder, leaving me blinking and repeating, “G’blrrg?” (the non-word I say when I am utterly baffled): Continue reading burning questions about phonics versus … pig ovaries

The Blue People are gaining!

Here is another edition of the Weird Search Terms, because I know you folks just live for these.
Trend alert! Blue people is gaining on Cat drawing for frequency. I have no idea why folks are looking for blue people, unless they’re looking for the band Blue Man Group?* Then again, a lot of these queries don’t make sense:

Ooh shiny cat disabled autism dust

A fresh batch of Weird Search Terms, and boy, are there some whoppers in here!
With an increase in traffic comes an increase in the number of search terms that lead people to my blog — and an increase in the number of peculiar search terms. Since I started my work day at 7:30 am and finished my last class at 9:20 pm, I have not had any time for writing today. So this seems like a good day to post the latest entertaining slushpile. My favourites are at the end, of course.
The most common are still in the “how to draw a cat” category, go figure:

  • cat drawing
  • drawing cat
  • draw cats
  • how to draw cat
  • cat+draw
  • Cats drawing
  • line drawing cat

Hmn, that last one sounds like it ought to be a children’s story. “Macvicar was a line-drawing cat; he drew lines on everything: the walls, the furniture, the stairs, the rugs, even pieces of mail …”
Er, what’s this about?