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:

Read the rest of this entry »

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 Read the rest of this entry »

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. Read the rest of this entry »

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): Read the rest of this entry »

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?

“Searching, please hold …”

NOTE: BEVERAGE-SPEW WARNING — funny stuff ahead

  • request the pleasure of your company

Why, thank you!

Every day I like to look at the search terms that brought people to my blog. And every day there are several requests for “cat drawing” or some variation thereof, which lands people on my page about prosopagnosia, as does “faceblindness”. Of course, every Web site gets its share of weird search hits. Here are some of mine, including my all-time favourites, which are at the end of the list. (As you might guess from some of these truncated references, the WordPress search-term lister has a character limit.)

Meanwhile, I have:

  • prove you are not a robot

That would be related to the devilish annoyance of CAPTCHA. I hope! Sometimes I think people view search engines as modern versions of crystal balls — a lot of queries are scripted in ways that suggest someone is asking questions of an oracle: Read the rest of this entry »

“Innumerancy Taxes”

I once saw a bumper sticker that claimed lotteries were “a tax on the innumerate”, meaning that most of the people who gamble on such do so because they don’t really understand the mathematics of basic probability (chance). It does seem to be alarmingly true that a great number of people don’t have a good understanding of odds. Sure, some people simply gamble for the gaming aspect, but casinos aren’t getting rich off folks like my grandma who got together with friends at each other’s homes once a month to chat and play penny-ante poker — they’re in business to make money off those who keep thinking that they’ve figured out some kind of “system” or that they’ve some kind of special “luck” or who are addicted to gambling.

There are some really odd ways the human brain works against reality, especially when it comes to understanding probabilities. The brain likes to find patterns, even when they aren’t there. Read the rest of this entry »

Model5

(For the less geeky, the post title is “Models” — a play on Numb3rs)

For someone who deals with statistics only when I absolutely have to (the formulae make my head swimmy), I still have a fondness for doing comparative measurements. Most of the online personality-type tests are an absolute waste of time (I’d much rather work out a Sudoku), but once in a great while one will catch my attention long enough for me to actually complete it, such as the nerd test. Okay, so at a 93% I’m not as nerdy as Bug Girl, who earned a “Nerd God” score of 99!

On the other hand, last time I took the AQ Test Read the rest of this entry »

Math and Science, Bass-Ackwards

In one of my jobs, I’m a paraprofessional in a high school science classroom. Last week in Biology we were in the unit on carbohydrates, lipids and proteins as polymers. Of course, the students have been struggling because we’re touching on biochemistry concepts, and they’ve not had chemistry.

So I asked the teacher, “If biology depends upon chemistry, and chemistry depends upon physics, then WHY don’t we start with physics, and then go to chemistry and then biology?”

(I mean like, duh!)

And he replied, Read the rest of this entry »

Which Is Better?

When people ask, “Which is better?” for most anything, my response is, “Better for what?”

The same is true for any kind of debate about different teaching approaches, whether the subject is language, mathematics, or how we design classroom environments.

Take for example the whole debate about phonics versus whole-word approaches to reading. Each method is useful in different ways, and to different people. Phonics does give you tools to decode a great many words. But because English is not a strictly phonetic language, phonics can break down in the pronunciation ability, and especially in the spelling ability. One can usually come up a number of phonetically rational ways to spell a word, but only one or two will be correct (e.g. the British kerb and the American curb). So, let’s spell a word (I bet you can come up with even more ways than I’ve listed here!): Read the rest of this entry »

Rainbow Cracking

The other week after my blogging about dyspraxia and such, hubby found an article in wired blogs (“Hacking My Child’s Brain”) and a recent article in the New York Times, “The Disorder Is Sensory; the Diagnosis, Elusive”. Although sensory integration remains a vaguely-defined albeit real disorder, treatments are highly varied and disputed. Some treatment approaches lack rigorous testing for efficacy, creating difficulties for insurance coverage.

One approach mentioned in the former article is from the Sensory Learning Center in Boulder, Colorado (US), and is described as suitable for a long list of issues: autism, Asperger’s Syndrome, acquired brain injury, developmental delays, birth trauma, behaviour problems, ADHD, and for “learning enhancement”. Their Web site is rife with testimonials from clients and practitioners.

Well, testimonials don’t sway me, Read the rest of this entry »

Slices (Episode 1)

The best definition of “poetry” I’ve ever encountered is, “Poetry is life condensed”. In a similar way, cartoons condense a slice of life into just a few panels.

All four of these reflect different aspects of dealing with the social world, from blocking off unwanted interaction to the absurdity of Read the rest of this entry »

Running With the Red Queen

Everyone in life has to compensate in some manner or another, because no one excels at everything. If you are not mechanically inclined, you take your car to a shop to get the oil changed, and you call a plumber to fix leaks or replace worn faucets. If you’re not comfortable with arithmetic calculations, you have a a tax specialist do your annual return, and you arrange for automatic payroll deposits and bill payments with your bank. These are ways that ordinary people deal with ordinary difficulties, and no one thinks any less of them. In fact, the economy depends upon people’s interdependency — earning your living doing things for others is important to the Gross National Product, is important to a town’s sense of community, and is important to a person’s self-worth from feeling useful.

It is curious that people who have others do everyday things for them because they are rich are envied, whereas people who have others do everyday things for them because they are unable to do them are looked down upon. People with ability sets that are different than the “average” person’s run into problems because they are being “inappropriately incompetant”. Some of those “should be able to” things are related to sex-rôle stereotypes: a man should be able to fix a leaky faucet, a woman should be able to sew her own shirts. Among more traditional or conservative populations, a person is not faulted if they are incompetant at a skill that is reserved for the other gender. However, when someone cannot do something that is expected of everyone, or cannot do it well, or cannot do it consistently, they are then open to derision.

The Austrian psychologist Alfred Adler noted how people compensated and even over-compensated as ways of dealing with perceived incompentancies and avoiding feelings of inferiority. Not all “incompentancies” really are gross difficulties — they may merely be assigned as such by others around us.

I’ve mentioned before that my life is a mass of compensatory strategies. I compensate for auditory processing problems, and the tinnitus that increases the background noise problem. I compensate for prosopagnosia (difficulties recognising people from their faces). I compensate for all those organisational, time-sense, and executive-functioning issues related to ADHD and Asperger’s (planning, executing tasks including the getting-past-the-inertia stages, self-monitoring). I compensate for the hyperacusis, and my general clumsiness, tics and stuttering, and migraines. Generally speaking I compensate fairly well. So much so that most people don’t realise that I am working much harder to achieve nearly as well. I “pass for normal” most days, so people can’t understand why I’m having problems when I’m ill or stressed or simply trying to compensate for too many things simultaneously.

Adler would probably say that I over-compensate.

I had to go through Driver’s Education class twice to acquire the necessary motor skills. I did eventually learn to drive stick shift (manual transmission) and have even driven in both the UK and US. The day that I parallel-parked in front of my high school to request a transcript to be sent to a college was indeed a threshold moment in my life. (Even the transcript part was a highlight, as assaying higher education was uncertain due to my previous academic difficulties.) My husband once asked me, “What, can’t you drive and talk at the same time?” and I did not feel that it was unreasonable to answer, “No, I can’t.” I cannot drive a stick shift vehicle through city traffic, trying to find a business I had never been to, and talk on a cell phone. (I have Auditory Processing Disorder and he has a severe hearing loss — talking on the phone can be inherently confusing in its own right.)

There are classes when I struggle to keep my attention focused on the instructor, and also to understand what they are saying, especially if the classroom is mechanically noisy, or if the instructor mumbles or talks while facing the whiteboard or doesn’t present information in a clearly-defined format or use supplementary visuals. Because I am very good at being able to distinguish the important material in an educational presentation and record those details in sensible paragraphs, I have been a note-taker for dysgraphic or hearing-impaired students. But I have only been able to do that in those subjects where I was already familiar with most of the information — I could not be a note-taker for others if I was still learning all the vocabulary and concepts myself.

Mathematics presents special difficulties for me because of problems with sequencing, slow working speed, and occasional transpositions. It took me four years to memorise my multiplication tables, and I have flunked a number of tests over the years, and nearly had to take a class over. In university I dropped a course that I was getting D or F grades, to try it again later on to get C, B or A grades, and did that with more than one course. It was slow, difficult work slogging through college algebra, trigonometry, calculus, statistics, physics, and four semesters of chemistry. One of my current jobs is working as a special education paraprofessional. I help in the science classroom, but my main assignment is in the math classroom. The extremely ironic thing is that not only am I helping students with mathematics, but also that I am doing so in the very same school I attended years ago, in the same classrooms where I had once sat flunking math tests. (My first work week was not only difficult from the prosopagnosia-aggravated new-job disorientation, but also from “post-traumatic school disorder” as I had ongoing flashbacks.)

I actually did flunk a semester of secondary English and had to re-take that portion of the course. I have also written a book and hundreds of articles (on a variety of subjects) for magazines and newspapers. I tutor college students in composition classes.

Given these examples, it might sound as though my difficulties were all in the past, and have been made up for by my recent successes. That isn’t quite true. What I have done is learned how to work around some kinds of difficulties. With others I simply have to work harder to puzzle through consciously to figure out those things that most people do easily and without conscious effort. Some days I feel like Alice Through the Looking Glass, running as fast as I can just to stay in place.

The problem with over-compensation is that although I have at times felt that I had vanquished my personal demons of incompetancy by having overcome various failures with landmark achievements, those successes do not mean that I cannot or will not have future problems! What helped more than those moments of personal glory (exhilerating though they were, despite lacking exciting soundtrack music), has been finding out why I have problems, how those problems manifest in my daily life, and how to work with them. Self-understanding improves self-image because it gives me tools for those ongoing and future difficulties. Self-understanding means that the next time I fail something (not “if” but “when”, because everyone does fail periodically), I will have the necessary cognitive and emotional tools to handle the disappointment. I will be able to handle defeat graciously, because it is a failure of task-specific achievement, not moral failure. Furthermore, I can extend that same grace to others, because we all have such problems, even though the details differ.

Out in our various communities, we need to be able to not only acknowledge that Yes, not everyone can do the same thing, but also destigmatise that fact. One of the tragedies with the current paradigms in the helping professions is the disdain and depersonalisation from “care-givers” to that people who need personal attendant services or other forms of assistance. We can’t all do the same things. Needing someone to change your diaper should be no more stigmatising than needing someone to change the oil in your car. There’s really something sick about people who feel superior those whom they serve — there’s an element of self-loathing transferred from one’s self to one’s job to the client. It is overcompensation of the soul-eating malicious sort. Service to others is about sharing strengths, not about bolstering one’s damaged self-worth at the expense of others’.

We should not have to overwork ourselves to over-compensate just to earn other’s acceptance.

Doing Things the Wrong Way

I was in my teens when my mother announced in a fit of supreme annoyance, “You know Andrea, all children rebel, but you’re doing it all wrong!”

This comment required some thinking on my part. Indeed, it rolled around in my head for hours as I tried in vain to make sense of it. Granted, I was continuing to have academic difficulties, but those did not stem from rebelliousness. What was I doing wrong? I didn’t date (so no sex), didn’t drink, didn’t do drugs, didn’t even have my driver’s license to be engaging in reckless behavior, didn’t ditch school (wasn’t truant), and wasn’t grossly disrespectful. If someone had created a list of the Six Dreadful D’s that a teen could engage in, I would have been clear of the whole list.

The “doing something all wrong” part of itself wasn’t the difficulty; that was a sadly familiar refrain. It was attaching “all children rebel” to it. The words implied that there was a “right” way to rebel that I was failing to accomplish. But parents never wanted their children to rebel … what a double-bind! Oh, it made my head hurt. Finally by the next day I decided that her comment simply did not make sense. That would later prove to be the turning point of my tediously slow process of untangling an alarming number of double-binds that had for years tied my head up in knots.

Part of the reason that I had trouble understanding the nonsensical nature of that remark was that my mother was not the only person from whom I’d heard this refrain about “doing things the wrong way”.

I had inexplicably run into problems in art class (of all places surprisingly – this subject was normally a source of outstanding marks) because I wasn’t following the directions for figure drawing. We were supposed to be drawing the person perched on a high stool by creating a series of connected ovoids for the torso, limbs, and appendages, and then connecting those ovals and smoothing them to create the figure. That didn’t make much sense to me; it seemed like a lot of unnecessary work. I simply started at the top of the head and proceeded to draw the silhouette. Sometimes I would erase a small section to refine the line, but otherwise I would work my way around to the beginning point, and then filled in the interior details.

My art teacher however, was a stickler for “Process, process, process!” She had managed to get everyone successfully through single and double vanishing-point perspective by careful adherence to procedure, and she was determined to have all her students complete satisfactory still-life drawings of bottles, cow skulls, and humans by careful adherence to procedure. Initially we’d started our still-life work with the typical assemblages of fruits-as-Platonic-solids, but this class was right before lunch and the props kept disappearing. The bottles proved to be adequate subjects for learning techniques, but the cow skulls proved daunting. The system of Platonic solids and ovoids proved to be no match for the murderous complexity created by the mandible and orbital cavities. I was able to draw a respectable cow skull only by virtue of the fact that I could visualize it as a two-dimensional image and then transfer that mental image to my paper, fait accompli. I have no idea if her distrust of my personal process was related to the fact that I wasn’t complying with the given directions (and thus had succeeded in completing the assignment but left her with little to calculate in her grading rubric), or whether it was related to the fact that she had no idea how I could draw by finished silhouette. Even the token artistic genius of the class had to sketch and re-sketch lines repeatedly, for all her finished product was the most refined.

Trouble was constantly simmering over in my maths class, and boiled over every nine weeks as progress reports were sent home. Whereas beginning algebra had been a minefield of flunked exams, geometry was taking a much different turn, and not always for the best. It wasn’t that I didn’t understand geometry with all its angles and parallel lines and intersections of compass-drawn circles. Indeed, it was the first time I had excelled in understanding anything mathematic. I could consistently answer the homework and exam questions correctly. I just couldn’t consistently show the steps or name the proofs that described how I’d reached those answers. As far as I was concerned, the exam requirements of List the proofs and Show your work were the bane of my life. Generally there weren’t any steps to be had! The answers were obvious. So much so that I spent most of the class lecture time just doodling on the margins of my notepaper, creating recursive labyrinths, spiraling pursuit curves, or re-inventing Voronoi tessellations by marking the areas of influence around random blemishes in the paper.

When my maths instructor had taken me aside one day after class to find out just how I was getting my answers (there were suspicions of cheating), I then stupefied him by announcing answers by glancing sideways at the problems. He was totally flummoxed when he found that I figured sums of several numbers by initially clumping complementary pairs of digits in each column into sets of ten before adding them up, rather than starting at the top of the column and consecutively adding each digit. I couldn’t understand why my approach wasn’t natural to everyone, because after all, we were using a base ten system. At least he was satisfied that I was producing the correct answers on my own, no matter what obscure method I used to produce them.

When I sat and contemplated my place in the grand scheme of things, I found myself wondering just how it was that I could be “doing things the wrong way” and yet still be producing the right results. Were the processes really as important as the results? Apparently so, for I was increasingly finding that style was as important as substance when I found myself in social situations. You weren’t supposed to lie, you weren’t supposed to sit there and not participate, and yet you weren’t supposed to say what was really going on. Amazing how often one could be deemed rude for merely sharing facts or for being specific. I repeatedly found myself doing things the wrong way and thus going against what people were telling me to do. Maybe I was rebelling after all.

It’s just … that wasn’t my intent at all.

Testing, 1, 2, 3 …

The other week I was typing math tests, generally a task as dull as dusting door lintels. But this time I was enthused because I was re-typing the tests in order to make them more accessible.

You see, the old tests were done in a small 10-point font, with the arithmetic problems set up in the traditional manner of stacking them in long columns and aligned rows. Many of our students have a variety of learning disabilities, and I suspected the very layout of the tests was aggravating some of the visual and/or graphomotor difficulties.

Firstly I increased the numerals to a 14-point font. This is much closer to natural handwriting size, so it’s easier for the students to write their own numbers under the columns of existing digits. For dysgraphic students, anything that gives them more room to write is beneficial. Therefore I also increased the amount of space between the problems, both within the rows and between them. This way there would be sufficient room for working out the calculations, especially the long division problems.

Another reason for giving extra room between the rows was that I wanted to avoid making the students squeeze their answers around smudged calculations. Nor did I want to have them transfer their answers to a separate page, which could incur errors involving number transpositions, correspondence between the problem and its specifically numbered answer blank, or some of the answers not even getting transferred over.

Next I put the problem numbers (enumeration) on different lines than the problems, so there would be less confusion about which was which. In contrast, the operations signs (plus, minus, multiply or divide) were moved closer to the problems to reduce any confusion about what the student was to do.

Another important step was to arrange the individual problems so they were not stacked directly above and below each other. This reduces some of the spatially-related difficulties some students have, and prevents confusion about which number is involved in a given problem. It’s too easy to pick up the wrong number or even skip a problem when all the digits are piled up in long wriggling stacks. Offsetting the problems helps isolate each one in a larger area of white-space.

The combination of offset problems plus using a larger font resulted in using two rows for five problems, rather than just one row. In turn, the tests usually grew longer by a page. I don’t consider that to be a problem; there’s a time for “saving trees” (conserving paper) and a time when that is a false economy because it creates other problems. When photocopying the tests, I did not copy on both front and back. It’s too easy to miss a chunk of problems on a test when they are “hidden” on the back. Plus, having blank page backs automatically gives blank space for any additional little calculations that the students need to do.

These mathematics tests don’t have much in the way of worded questions, although for those that were included, I doubled the length of the answer blanks so they would be roomy enough for handwritten responses.

When laying out tests with worded questions, there are some other techniques that can make test-taking less difficult on the practical end. Many things are good common sense, but we have to be aware of them to be sure of including them. These include methods such as:

• In matching questions, have the descriptions in column one and terms in column two on the same page (no run-ons to another page);
• Use numbers for one column in the matching and letters for the other column;
• Spell out the words True – False to be circled (rather than the student writing T or F or t or f and letting the grader guess which was written down);
• Avoid the use of double-negatives in true-false or multiple-choice questions;
• Use capitals in matching or multiple choice (A, B, C, D, E) instead of lower case (a, b, c, d, e) that can be confusing to the student or to the grader (a – d, b – d, or c – e can look similar), and be sure to give a blank to write the answer upon.

(As you might guess, this particular grader has her own difficulties reading small font sizes, visually tracking numbers, or sometimes distinguishing certain letters.)

The benefit to all these various techniques is that they help all the students, not only those who have particular disabilities that have been diagnosed and for whom accommodations have been established. Other students who have undiagnosed problems, marginal problems, those who are simply tired or sick, and even those in top form will all benefit from having tests that are easier to read. (Ditto the teaching staff!)

This is the joy of universal design for learning: make as much of the material as accommodating as possible for a wide group of students, and you will have fewer specific changes to make for individual students, plus everyone will be able to use the material more easily.

After all, our end goal is to assess the students’ acquisition of knowledge, not their ability to decipher tests..

Dividing We Stand

So there I was mulling over how to approach long division with one of my seventh-grade students. There are several difficulties involved in his learning of the process, and I’ve only identified a few of them. One thing I do know for sure is that he has a low frustration threshold, and that mathematics is neither an easy nor an interesting subject for him. (Last week he had a meltdown after just a few problems and wouldn’t do any work for the rest of the hour.)

I thought back to yesterday’s class. A large part of the problem is that he doesn’t have his multiplication facts memorized. This could be from problems with rote memorization, and it could be also from problems with retrieval of information he already knows. Either or both gives the same result behaviorally. I have to be able to sift through what I observe and what he says, to determine what’s happening. After watching him remember most of what we went over the previous day, and watching him have to stop and calculate 22 minus 18, I suspected that it’s probably more of a rote memorization issue than a recall problem.

He also needed a more efficient method of calculating. I showed him that instead of going through the whole rote process of subtracting 8 from 12 by borrowing the 1 from the tens column, he could count from 18 to 22, and (looking at his fingers) see that there’s a difference of four. That sped up his working pace and reduced the cognitive load. It also helped him see that subtracting is finding the distance between two amounts, rather than just cranking through stacks of numerals.

He can calculate his multiplication facts (every single time he needs one) because he understands them as adding by multiples, and he figures the product by adding, “4 …. 8 …. 12 … 16” with each group of four on one finger, then look at his fingers and know that 4 times 4 is 16. Doing all this arithmetic with every step (such as figuring out how many 6’s are close to 37 for the first value of the quotient) places heavy demands upon working memory, and thus reduces his ability to learn and recall the larger process. All that work makes it hard to keep the data in short-term memory, and without that, it never makes it to long-term memory.

So a couple of days ago I brought out a multiplication table, but he didn’t understand it. Time to backtrack and get a better grip on multiplication! I got some manipulative cubes, and we built up a partial table, setting up 2 sets of 3. He counted each cube, “That makes six,” and then 3 sets of 2, “That also makes six,” and he wrote 6’s in both squares. We went over 3×4 and 4×3, and he wrote 12’s in both squares. Ah-ha! The order you multiply doesn’t matter. “That’s called the Commutative Law,” I explained. “It doesn’t matter if you multiply 4×5 or 5×4, whatever order you multiply them, you still get 20. That means you only have to learn HALF of the multiplication table!” Then we went over 3×3 and 4×4 and 5×5 and learned why a number multiplied times itself is called a square – the blocks stacked up into squares. Finally he understood how the multiplication table is built and what he can do with it, so we decided to use it in his long division problems.

Sequencing is definitely a difficulty; he’s having problems remembering when he’s dividing and when he’s subtracting. He’s also getting confused on whether to put a number down as part of the quotient or as a product. That could also be a spatial processing issue. Some of our students have problems with their columns of numbers wandering about, which plays extra havoc when they get to decimals (I liken it to “getting decimated”), so I have them turn lined paper sideways and write their numbers in the columns between the lines.

One thing I had noticed yesterday was that he could describe the process to me verbally more easily than he could write the problem. He might very well be an auditory learner. This might also be a fine motor coördination issue (he writes his numeral 4 with three separate strokes) so we’ve been doing the problems with a whiteboard and marker. This makes it easier to write the problems down and also erase errors, in contrast to doing them with pencil and paper. The marker glides more easily, and the numerals are naturally larger. It’s also easier for me to see what he’s doing without breathing down his neck, which is more comfortable for both of us.

Today I started him out by asking him what his favorite sport is. Yes, that’s an odd way to start math class, and there was a few seconds of delay before he answered, “Baseball.” Tying the subject to his special interest makes it more interesting and relevant, and thus be more likely to “stick”. Starting with something that he likes also helps reduce his aversion to the subject. In this case, we needed to learn the names of the different parts of the division equation. Previously he’d been telling me to, “Put the 2 over the 3,” but knowing how to do one problem by rote process doesn’t always help when you get to a different kind of problem. He also needs to be able to understand that all the problems are built of the same types of pieces. So I explained that just as baseball teams all had the same kinds of positions (catchers, pitchers, basemen, outfielders and so on), so did division problems (divisor, dividend, quotient, product and remainder). Just as each team has different people playing those positions, different problems had different numbers playing different positions. Well, that made sense.

With this base of understanding, we began reviewing the process he’d mastered yesterday. Because of his low frustration tolerance, I wanted to be especially sure of emphasizing his achievement. Then we did four problems together, with me correcting errors and also doing the scribing. Having refreshed the process, for the second quartet of problems I had him tell me what do write, and he was nearly soloing. For the third quartet I had him tell me what he’s doing, and he did the writing as well. Then after all that achievement, we looked at the two different ways of writing the same problem, with the bracket or the dotted sign.

Of course, the big questions are whether or not today’s understanding made it into his long-term memory (if he can retrieve that process after a day or a weekend or a month), and if he understood what it is actually about.

Tomorrow we’ll go over again what a division problem means. 295 divided by 36 describes, “How many sets of 36 can we make from 295? Do we have any left over, or does it come out even?” I’ll also have him describe to me the overall process of long division, which I will type up for him to keep. Having the student explain something in their own words requires a higher taxonomic level of learning than just shuffling around a bunch of numbers. Using verbal description also ties the learning to another part of the memory.

The problem with learning rote processes without conceptual understanding is that the students will then stumble in pre-algebra. They will need to use abstract reasoning to evaluate which method to use when. Part of that abstract reasoning simply comes from the maturation of the brain, and part of it comes from creating that deeper understanding of different methods.

I can sympathize with our students’ mathematical difficulties. It took me four years to learn my multiplication tables, and even in statistics and calculus I still have pauses in recall. (Calculus concepts are a breeze, but I can’t memorize a formula to save my life.) I had also flunked a number of math tests when attending this very same school building, and now here I was teaching it to students. (The irony!) But I take that understanding of the frustration with me every day, and express it as patience. I apply everything I have learned (and continue to learn) about cognition and learning, and everything I have learned about observing people, and put them together in my work as a paraprofessional and as a college tutor.

All told, this student successfully completed 13 long division problems today. The whole process is making much more sense, and he persevered with the work through most of the class period. I told him that since he’d stuck with it so well (even when he got a bit frustrated) he could take a break for the last ten minutes of class. He commented that it was kind of fun. “Yup,” I agreed, “Math is like games or puzzles once you understand the process!” This is a good sign. It may not last – one good day after unknown months of difficulties isn’t enough to turn around a student, but it is part of a good start with a new teaching relationship.

Techniques & Tips from a “Professional Student”

It’s too easy for blogging to end up as nothing more than a series of rants, so here’s something positive.

It’s that time of year when millions of people (leastwise, those in the northern hemisphere) are starting new school years. As someone who tutors (other) students with ADD and learning disabilities, I thought I’d share a bevy of helpful ideas I’ve scraped together over the years.

GETTING READY TO READ

Put the material into the Big Picture. Before starting a chapter/ module/ unit, review your syllabus to see how the content of this one fits within the logical flow of the previous unit, and how it might be important to the next unit. This helps the material make more sense and seem less like a giant pile of loose facts.

Read the textbook backwards.
Start with the Summary in the back of the chapter; this is the “TV Guide” version to what the chapter is about, so you know what you’re heading into before you dive into all the excruciating details. Read over the new terms in the Glossary, so when you encounter them in the text you won’t have those unintelligible speed-bumps that interrupt your understanding of the reading.

This is helpful if this subject is entirely new to you and you have little or no background in the concepts and terminology of this particular field of study.

From the first day of class, create a personal glossary of new terms and their definitions. This is imperative if you are starting a new field of study because you will soon find yourself in possession of a swarm of new words for which you are responsible. Trying to look up a word for its definition by flipping through masses of notes, handouts and textbooks only slows you down and makes you frustrated. Staring into space, pacing, rocking or banging your head do not aid in remembering new terms, so having that personal glossary will give you a ready list to access. Don’t forget to add helpful tips to your definitions, such as cautions about similar-sounding words that you might confuse, or terms with complementary or opposite meanings.

This is especially helpful if you are slow at recalling words, or have difficulties with spelling.

Block off distracting printed material with a mask. Use a half sheet of thin cardboard, a 3/4 sheet cut into an “L” shape, or two blank index cards to mask off distracting graphics, or simply to block off everything but the single question, objective, or paragraph you need to focus upon.

This is helpful if you are someone who is easily distracted by fascinating pictures, or if you have reading difficulties.

TAME THE PAPER TIGER

Assign a particular color to each class. I like to have the binder match the textbook color, so when I’m getting things together for class I only have to grab “two red things”. After the test, keep the notes and handouts in the colored binder or manilla folder. Use that color of ink to mark due dates for assignments and test dates on your calendar. Use that color of manilla or pocket folder to keep all the stray bits of useful stuff you are collecting for a report/project – having that special “parking place” will help organize and reduce the “file by pile” mess on your desk, floor, table, window ledge and other random surfaces…

Buy a hole punch with a trap. The trap collects all the “dots” so they don’t litter the floor. A 3- or 4-hole punch (depending on whether you use 8.5″ x 11″ or A4 paper) is vastly easier than a single-hole punch, as it not only reduces the number of clenches you have to perform, but also because it makes hole spacing that is perfectly even for the binder. Hole-punch all of your handouts and put them into the binder with your notes, so the two can live in wedded bliss.

Buy several packages of index dividers so you can separate the different chapters/units in your binder and more quickly flip through them for studying.

Make liberal use of colored sticky-notes. These are the greatest invention since the microwave oven! They will save tremendous amounts of time from having to endlessly flip through textbook, lab manual, handout, and note pages to track down important information.

Use colored sticky-notes to mark where important graphs, lists, charts, and diagrams are located in the textbook – write a key word on the external, flagging end of the sticky.

Use different colors of sticky-note for different chapters/modules/units, to make studying easier when you have tests that come after you have begun the next chapter/module/unit.

Use sticky-notes to mark chapter sections for those classes that skip around a lot within a textbook. If you are only using section 3.2 of a chapter, then you may begin by reading the summary for just section 3.2 of that chapter, but it might also be helpful to briefly review what the rest of the chapter summary has to say, to understand how the ideas in this section are connected to other ideas.

NOTABLE TIPS FOR NOTE-TAKING & STUDYING

Always take notes in black ink. There is nothing more horrifying during Midterm or Final Exams than discovering that a semester’s worth of pencil-written lecture notes has turned into a smeary, unreadable mess. Oh, the horror… Also, some kinds of blue ink are close to “non-photo / non-repro” blue, a color that’s nearly invisible to many photocopiers; this is usually not a problem unless you need to photocopy those notes for any reason.

Always date and/or number your note pages. Of course, if you live a charmed life and never have sudden “gravity fluctuations” in your part of the planet that cause you to drop or spill note papers, or you never own binders that lose their “bite”, then don’t bother. Otherwise, dating the pages lets you keep track of what was lectured on at a particular time (handy if someone asks to borrow your notes from last Tuesday). If you take more than one page of notes per day (which is nearly always) then numbering the pages instead of or in addition to dating them makes it even easier to put spilled pages back to rights.

Title each page.
Even if it’s just by abbreviation, describe the page of notes by the lecture topic, the unit or chapter title. This not only makes it easier to find the right notes when studying for tests, but it also helps you remember what the overall pattern of ideas is during the course of the class across the semester.
Example:
MITOSIS WED 2 FEB p.1

Take notes in two columns: the left side for listing the main idea titles, important names, terms, dates or formulae, and the right side for all the regular details and sentences. If there is a page in your textbook, lab manual or whatever that has a particular graph, chart or listing, write down that page number on the left side as well, as well as a word or two to title why that page number is important. This speeds up your test studying because you can glance through pages of notes to find the one that has the specific information you’re looking for.

Use the Objectives listed in the chapter/unit/module as your study guide for the test, and write out a full answer to each one as though it were a question. Pay attention to key verbs such as Describe, Compare, List, Define or Identify – these can give you an idea of what kind of test question could be asked. Writing these out does two things: it not only helps you self-test your own understanding before you get to the class test, but it also changes your answers from something you have to invent during the test (which is time-consuming) into something you just have to recall during the test (which is much quicker and easier).

Writing out answers to the objectives in full sentences is especially helpful if English is not your first language, and/or if you are slow at remembering words,and/or otherwise have difficulty expressing the knowledge that’s stuck in your head.

DECIMATED BY NUMBERS

Turn lined paper sideways to have ready-made columns for keeping your place-values straight in big arithmetic calculations. Another option is to use green “engineer’s paper” that has graph squares on one side and is blank on the other side, but the graph grid is still somewhat visible on the blank side, and the green tint is more restful on the eyes.

This is especially helpful if your handwriting tends to wander around or slope down a page, and will keep your numbers and decimals in order.

If you are doing mathematical equations or other things that are processes, write out your own set of numbered directions describing how to do the process. For instance, it may not be as obvious to you as it was to the author of the formula that you need to determine the value of “C” before you put the other values into the formula. So in your own directions, you should note “Find the value for “C” by ~ ~ ~” as one of the earlier steps.
Whenever you solve an equation or do a statistical analysis, write out in a complete sentence what the answer to the calculations MEANS in regards to the original problem/story/question given.

These are especially helpful if you are more of a “words” person than a “numbers” person.

If you have several different formulae , make yourself a flow-chart (meaning, a series of decisions) that helps you figure out which one you use for different kinds of circumstances. When you are studying a chapter or doing that day’s homework, it’s obvious which one you need to use – it’s the one you’re learning that day! But come test time, you will need to be able to understand which one you use for each kind of situation.

This is especially helpful if you are one of those people for whom “all the formulae look the same”.

Use name and address labels on everything, and add your phone number or email as well. Put them on your textbooks, lab manuals, various notebooks, calculator, data CD, flash/keychain drive, assorted binders, notepads, calendar-organizer, each piece of art & drafting equipment plus the carrying case, and all the other things that you need to survive as a student, to help guarantee that the person who finds them can help get them back home to you.

This is especially helpful if you are forgetful, distractible, prone to leaving things in various places, and/or are juggling a variety of classes and jobs. (You can imagine why I know this.)

Bibliomeme

Mum-is-thinking tagged me to answer a book survey. My answers are a motley collection, and I think that motley collections are always the most interesting. I’m guessing that people like to read these kinds of meme-tag surveys because they either want to hear how others have loved the same books they have, or else want to hear about books they had not yet (or possibly would not have) encountered, but would also enjoy.

One book that changed my life
I’ll have to take this is “one of many” rather than as “the one with the greatest impact” because surely different books have had done this at different stages in my life. There are a lot of contenders for books that were the first (if not always the best) to open up my knowledge-base to completely new fields of understanding, such as those on AD/HD or autism. Those are valuable in that regard, but more important are the books that give a different kind of insight, looking behind social paradigms to critically analyse the how and why of human interaction.

For the way that humans interact with their environments, Donald A. Norman’s The Design of Everyday Things looks at the problems that bad design causes people, and how people assume that their difficulties are considered to be their fault, rather than bad design. He touches but lightly on the issues of handicap accessibility, and I don’t think he mentions Universal Design at all, but the central message is still the same. My inner geek adores good, useful, imaginative and æsthetic design, and it drives me nutz when tools, machines or environments are badly designed.

For the way that humans interact with medical & emotional health care providers, Paula Kamen’s All In My Head: An Epic Quest to Cure an Unrelenting, Totally Unreasonable, And Only Slightly Enlightening Headache that describes some of the problems with the medical models of psychology, such as being a problem patient rather than a person with a problem, or the need to find “cures” for everything when instead one can be helped and be healed without being cured.

Strong messages from both of these books.

One book that you’ve read more than once
Who doesn’t have a comfily-tattered set of J.R.R. Tolkien’s four-volume Middle Earth trilogy? (Yes, trilogy means three books, but The Hobbit is part of the Lord of the Rings, and science fiction & fantasy is rife with trilogies composed of more than three volumes.) For my favorite re-read when stuck abed with a nasty virus, I really enjoy Anne McCaffrey & S.M. Stirling’s The City Who Fought. It’s a fun piece of adult science fiction with the well-drawn characters and nitty-gritty techy details and swashbuckling action that make for a engaging read.

One book you’d want on a desert island
Most people like to pack either something really long, or else an extensive practical reference book. But I don’t think that I’d want to be stuck with some interminably long piece of fiction, no matter how well-written, and I’ve probably read enough references over the years that I could eventually solve any manner of functional issues. What I want would be a huge book of blank pages, so I could keep a journal of thoughts about various things. It’s often difficult for me to work out mental explorations without a written medium. I’ll remember or figure out the right knots for lashing together poles, but being able to compose my thoughts is integral to my equalibrium.

One book that made you laugh
Terry Pratchett’s Mort was the first Discworld novel I ever read, and Death is still my favorite character, possibly because he’s so practical and the human world doesn’t always make sense to him. Plus, he talks in ALL CAPS. Soul Music is damn funny, too. I love the puns and unexpected turns in Pratchett’s books.

One book that made you cry
Ebbing & Gammon’s General Chemistry (sixth edition). The authors of this uninspired, heavy tome had an interminable number of equations to solve. I made it through four semesters of chemistry and sweated through this volume for half of them.

One book you wish you had written
Actually, I’m still compiling thoughts for my next book. I don’t tend to dwell on wish-I-had’s.

One book you’re currently reading
I never read just one book at a time, which explains why it takes me so long to finish anything! I just finished Joseph P. Shapiro’s No Pity. I’m furthest into Majia Nadesan’s most interesting Constructing Autism, which I will finish as soon as I remember where the hell I left the book laying about.

Currently my bedside pile contains: Thomas Skrtic’s Behind Special Education, Alfie Kohn’s What Does It Mean to Be Educated?, Kegan & Lahey’s How the Way We Talk Can Change the Way We Work, Marshall B. Rosenberg’s Nonviolent Communications, Fisher & Shapiro’s beyond reason, and Walter Kauffmann’s translation of Basic Writings of Nietzsche (maybe after finishing the book I’ll be able to spell N’s name without looking it up every time). I had just started on Richard Dawkin’s The Selfish Gene and then my daughter took it back with her to college; bad girl. By default I’m also reading Hardman, Drew & Egan’s Human Exceptionality: School, Community and Family because it’s my current textbook.

One book you’ve been meaning to read
The future pile-by-my-bed: Daniel C. Dennett’s freedom evolves, John H. Holland’s Hidden Order: How Adaptation Builds Complexity, Douglas R. Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid (I think that one may take a study-buddy to gain the most benefit), the Routledge Critical Thinker’s series editions about Gilles Deleuze, Jaques Derrida, and Michel Foucault, Eli Maor’s e: the Story of a Number, and David Darling’s Universal Book of Mathematics. Doubtless there’s more, but that’s what’s on that section of my bookcase.

Tag five other book lovers
Anna, Catana, David, Liam, and Whomever wishes they’d been tagged but felt like they needed some kind of “official” sanction to simply write and post a list!

Walking the Mine Field: Misadventures in Mathematics

Doing a complex calculation is not the simple matter than many people perceive it to be. “It’s simple,” I’ve heard repeatedly, “Just memorize the equation. Then it’s just ‘plug-and-chug’.”

In truth, performing calculations requires a great many steps, any one of which can be mistaken (or miss-taken), leading to disaster. It doesn’t matter if we’re doing calculus or statistics; either way, there’s a formula to choose and data to crunch through.

Identify the data from the problem. Not as straight-forward as you’d think; often there is extraneous data running around in there that’s not needed. We also run into charming instances of professors and books using different terms to mean the same thing, or handwriting that makes the same term look like a completely different character. “Is that a sigma or a delta?” Pro mea lingua Graeca est. For added entertainment value, let’s have a hand-written test.

Be able to correctly transform the data into the necessary forms, pre-flight. Are the numbers in the correct units? Do we first need to find means and standard deviations of the 30 values listed in the table? Do we need to flip through 15 pages of lecture notes and a chapter of textbook pages to find or verify the transformational technique? Yes, I have a page of formulae that I’m building. As we slog through the class, this is how I know what to put on the page. Let us hope that I don’t have a transcription error on my formula page!

Be able to correctly transcribe the data, without transpositions, morphing of numerals, or loss of data. This is often where I get into trouble – there’s a sense of spatial meandering as the numbers seem to wander around like ants. Pages with several problems on them make this worse, cluttering the search image. Sometimes I cover over parts of the page with index cards to reduce the visual clutter, but then I have to turn the pages… Double-check the numbers you’ve entered into the calculator before punching Enter. Oops, dropped a digit; re-enter all the numbers again.

Identify which procedure is being called for. Often indirectly stated, especially on tests. Once you know what you need to do, then you need to select which of that mass of massive, messy equations does that trick. It sure would be nice if the professors would spend less time explaining how someone derived the formula – for all I care, it could have arrived fully-formed, like Athena from the forehead of Zeus. Copying down all the steps someone used to transform one formula into another nifty new formula is not helpful to me – it just gives me pages of notes of half-cooked formulae that I need to puzzle through while trying to track down the one I really need. I’d rather they spent more time taking us through a flow-chart of how we determine which formula we need.

Transcribe the formula onto the homework page, without error. Then be able to correctly transcribe the data into the formula format, without transpositions, morphing of numerals, or loss of data. Stop and compare the numbers here with those in the problem. All systems “GO”? Clearance from the control tower?

Be able to remember where in the sequence of functions you are in the procedure. What was a bit awkward in “borrowing” during subtraction, became confusing in long division, and is downright maddening in regression analyses where each problem is a series of computational subsets. (I sure hope this problem doesn’t take more than one side of the page.) Sometimes I put labels alongside the subsets so I know what the pieces are, but sometimes writing ∑xy or s2 on the page only adds more ants. The page is already messy looking from several erasures. Flick rubber crumbs off the table.

Double-check the numbers you’ve entered into the calculator before punching Enter. Got “decimated” – transposed a zero and the decimal point. Re-enter all the numbers again.

Be able to correctly transcribe the correct data, in its proper transformation, without transpositions, morphing of numerals, or loss of data. Yes, we’re stuck in a loop of trying to keep track of a swarm of answers, some of which are raw, some of which are cooked, and it’s not impossible for one to roll off the counter and end up forgotten on the floor.

Next step of the procedure: double-check the numbers you’ve entered into the calculator before punching Enter. So now what do we plug this answer into?

Be able to interpret the significance of the numeric result. So what does “17.2” mean? (Do I care?) Re-read the problem again. Did I use the right formula? Oh, yeah. Write out the answer verbally, because by tomorrow in class this home-work page will have reverted to an unintelligible ant-farm of digits. I really do NOT recall what I did on a math problem from one day to the next.

Congratulations. You have finished the first homework problem. Only fifteen more to go. Um, are we doing problem 56 or 65? Did I get the right answer, or am I practicing doing the problem incorrectly?

In the Final analysis. Of course, in a homework assignment, you know what formula(e) you’re supposed to be using; it’s the one related to that section of the book. Now let’s go to a test, where we’re doing several different kinds of problems.

The test questions written by the professor state the problems differently than the book did, and require using the formulae in different ways than in the homework, to asses our understanding of the concepts. Naturally, this means that the problems on the tests don’t look at all familiar, because they aren’t set up the same way that the problems were on the homework. Before tackling the brute calculations, we have to decipher just what is in front of us. (Where are we going, and what am I doing in this hand-basket full of eraser crumbs and ants?)

“It’s simple,” they tell me, “Just memorize the equation. Then it’s just ‘plug-and-chug’.”

::sigh::

Cognitive Bias, Patterns & Pseudoscience

(It’s been a long, long day. So here’s an only-slightly-used, gently-recycled essay, but with an Brand New! hyperlink for your enjoyment. Bon appétit!)

“It has been said that man is a rational animal. All my life I have been searching for evidence which could support this.”
~ Bertrand Russell

Here’s our new word for the day: pareidolia. It comes from the Greek, para = almost and eidos = form. The word itself originates in psychology, and refers to that cognitive process that results in people seeing images (often faces) that aren’t really there: the man or rabbit in the moon, canals or face on Mars, faces of holy people in tortillas or stains in plaster … It also sometimes refers to hearing things that aren’t really there in random background noise (Electronic Voice Phenomena: EVP). Pareidolia is what makes Rorschach inkblot tests possible (attribution errors are what make Rorschach tests fairly unreliable).

The human brain is “wired” to see patterns, especially those of faces. Creating and perceiving patterns is what allows all animals to operate more efficiently in their environments. You need to be able to quickly find your food sources, your mates, your offspring, and the predators in the busy matrices of sensory inputs. Camouflage relies upon being able to become part of a pattern, and therefore less recognizable. Aposematic warning coloration, such as black and yellow wasps, does the reverse, by creating a specific kind of pattern that stands out.

Sometimes people subconsciously assign patterns and meanings to things, even though they don’t intend to do so. This is why we have double-blind studies, so the people who are collecting the data don’t unconsciously assign results to the treatment replications by increasing or suppressing or noticing effects in some trial subjects. Prometheus has a lovely blogpost about this: The Seven Most Common Thinking Errors of Highly Amusing Quacks and Pseudoscientists (Part 3). (This series of his just gets better and better!)

Seeing patterns can lead to weird cognitive biases and fallacies, like the clustering illusion, where meanings are falsely assigned to chunks of information. The fact is that clusters or strings or short repeats of things will naturally happen in random spatial or temporal collections of objects or events. A lot of people think that “random” means these won’t happen (which makes assigning correct answers for multiple choice tests an interesting process; students get suspicious if they notice too much of a pattern and then start out-guessing their correct answers to either fit or break the perceived pattern).

Sometimes the reverse can happen, where instead of seeing patterns in data, people put some of the data into patterns. This is known as the Texas Sharpshooter Fallacy: a cowboy randomly riddles the side of a barn with bullets, and then draws a target where there is a cluster of bullet holes. People will perceive a pattern of events, and then assume that there is a common causal factor to those, because of the perceived pattern. This is why statistics was invented – to suss out if there is a pattern, and how likely it is. Mathematics takes the cognitive kinks out of the data so the analysis is objective, rather than subjective.

Statistics also gives research rules about how best to proceed in experiments, to avoid various errors. One of those is deciding what kinds of analyses will be used for the type of data set that is produced by the experimental design. Note that this is decided beforehand! The reason for that is because people want to see patterns, and (even unconsciously) researchers want to see results. The purpose of testing for a null hypothesis is to try to disprove the given hypothesis, to avoid these kinds of issues.

It doesn’t matter how noble your intentions are – wrong results are still wrong results, no matter how they are achieved, or to what purpose.

To look at the data and then start picking through it for patterns, (“massaging the data” or “datamining”) is inappropriate for these very reasons. The greatest problem with doing analyses retroactively is that one can end up fitting the data to their pet theory, rather than testing the theory with the data. Mark Chu-Carroll’s post on the Geiers’ crappy and self-serving data “analysis” is an elegant dissection of how this kind of gross error is done. (Note that is MCC’s old blog address; his current blog is here at ScienceBlogs.)

Doing this intentionally is not only bad statistics, it’s bad science as well. The results come from anecdotes or data sets that are incomplete or obtained inaccurately. Correlations that may or may not exist are seen as having a common causality that also may or may not exist. It’s pick-and-choose and drawing erroneous, unsupported conclusions. People want to see patterns, and do. Even worse, they create patterns and results.

The seriously bad thing is that con artists and purveyors of various kinds of pseudoscience do this a lot. The intent is to deceive or mislead in order to sell something (ideas or objects or methods).

The people who then buy into these things then think they are seeing treatment results because they want to see them. Take this secret herbal cold medication, and your cold will be cured in just seven days! (Amazingly, one will get over a cold in a week anyway.) Give your child this treatment and they will be able to learn and develop normally! (Amazingly, children will learn and develop as they get older, for all not everyone follows the same timelines – developmental charts are population averages.)

Meanwhile, the well-intended but scientifically ignorant people who buy into these things are being duped by charlatans, sometimes with loss of life as well as with great monetary expense.

Economists will tell you that the cost of something is also what you did/could not buy, and when time and money is spent on false promises, it deprives everyone involved of the opportunity to pursue truly beneficial treatments.

Then the problem is propagated because those well-intended but scientifically ignorant people become meme agents, earnestly spreading the false gospel …

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