Still very much in the rough-draft stage...

...but I'm starting on a video series targeted at students who are trying to study for the SAT but are too far behind to get much out of the standard prep materials. You can check out the early efforts here.

A step-up SAT/GRE problem -- Circles (Rhombus edition)

In an earlier post, we talked about "step-back problems." The idea is that, wherever possible, each problem should be associated with at least one problem that uses similar format and relies on similar concepts but which "steps up" (is more difficult) or "steps down" (is easier).

In that previous post we talked about problems where you had to find the shaded area of a circle. This problem covers similar territory but takes things up a notch.

Circle 1 and Circle 2 both have radius 2. Each passes through the center of the other. Find the area of the rhombus formed by the two points of intersection (A and B) and the centers of each circle (C1 and C2).

Solution after the break.

A note on creating geometric graphics

As you can see in the previous post, I've been working on a project involving the kind of problems you might see on the SAT or GRE. I've been focusing on geometry to start with which means I have to create lots of geometric drawings.

I recently started using Inkscape and I would highly recommend it for the following reasons:

It's a free, open-source program;

It works with  Scalable Vector Graphics (SVG). Scalability is a great feature;

It's easy.

Quick tip: you can draw circles and squares from a center point using the shift key.

A step-back SAT/GRE problem -- Circles

I've been thinking a lot about video learning, particularly video-assisted self-study. One of the ideas I've come up with is the step-back problem.

Sometimes, when you're trying to teach yourself a subject or study for a test like the SAT or GRE, you will run into a problem that still confuses you after you've gone through the explanation. If you have access to a live instructor,  you can always ask for a more detailed explanation but what do you do if you are trying to learn the subject from YouTube videos?

My thought is to pair up problems of medium to high difficulty with problems that use some of the same concepts but are much simpler.

Here's an example. The following is very similar to some problems you are likely to see on the SAT or GRE. Try it on your own then check below the fold for the answer. If you didn't get the right answer and still don't understand what you got wrong even after reading the explanation, try the second problem.

Circle 1

The radius of circle 1 is 5. Both line segments pass through the center of the circle. Find the area of the shaded region.

Puzzling your way through the GRE -- logic edition

I was looking through a GRE sample test the other day and I came across a question that looked oddly familiar. I am not going to repeat it verbatim (not quite sure of the copyright status) but the gist was that the question gave some names and some properties associated with those names and some conditions.

After a moment, I realized that this problem reminded me of an old-fashioned logic grid puzzle.

Here's an example from a popular site:

You can eliminate pairs you know aren't true with an X, and pencil in pairs you know are related with an O. If you know, for example, that Lauren wasn't born in 1961, you can add an X in the box where the Lauren column and 1961 row meet. Similarly, if you know that Bryant was born in 1971, you can add an O in the appropriate box. Furthermore, since every option can only be used once in any given puzzle, you can eliminate the four other options for Bryant in that category (1937, 1946, 1961, 1975) and the four other options for 1971 (Anahi, Jayden, Lauren and Nikolas).

You probably won't see a grid like this on the GRE or the SAT but you will run across the underlying concepts so if you're preparing for one of these tests, it might be worth your while to spend a little of your study time as play time with these puzzles.

Resources for SAT self-study -- The Khan Academy

I don't want to be too negative here. This is a decent resource and for most students there will be some value in going through these problems. If that sounds like faint praise, that's because it is. What we have here, while nice, simply isn't that much to get excited about. The Academy, it should be noted, is promising a much more complete and sophisticated set of tools next year, but for now, we pretty much have to limit ourselves to the "better than nothing" standard.

I would recommend that you go to the Khan Academy SAT site and do some exploring on your own.


In the meantime, I have included a few examples to get the discussion started.







So what are my concerns here?

I The Medium is the Message

Ideally, we'd see videos that made better use of the medium (more on that later) but the bigger, more immediate concern is is that they have simply chosen the wrong message here. What we have in these two clips is basically a standard problem session with most of the good and important parts left out.

This type of instruction, done properly, is contextualized and highly interactive. Putting problems in context is essential because, even more than drudgery, the thing that drags kids down in math is not knowing why they're doing what they're doing. Working a problem for a class should be a conversation that both frames the question and gives the students a chance to participate in the solution. (For example, ask them "If this length is x, what is this other length in terms of x?" and let them come up with the 2x.) Without the conversation, there's not much point in presenting this information in a video because...


II You've already got the message in a better medium

These clips are basically an instructor reading out loud the kind of solved examples that can easily be found in books or online. The videos add little to the printed version and they arguably lose quite a bit. This will strike many as blasphemous, but for certain tasks, print actually is a superior medium. It's good at conveying precise information and it gives the reader a level of control that is difficult to achieve with pausing and rewinding a video. It offers tremendous variety and availability. It is easily searched and database-friendly. What's more, the ability to pick up a book or go to a website and teach oneself is a valuable skill for everyone and an absolutely essential ability for anyone in STEM.

Is the spoken word ever better than print? Sure, much of the time. If you're going for a conversational voice; if the gist of the message is not dependent on any single word or phrase; if you want a particular emotional tone... Unfortunately, these clips not only do things that print would do better; they leave out the things that their medium would do better than print could.


III Rush, rush, rush

There's a choppiness here both in tone and structure, combined with a hurried quality. The approach is very much "Here's a problem. Do this. Do this. Do this. Here's the answer." No time spent orienting the student. Worse yet, no time spent checking the answer. In terms of  Pólya's How to Solve It, they tend to jump directly to "carrying out the plan" and stop there. That's very dangerous because, even if the students do manage to follow what you're doing, they won't have a firm grasp on why you were doing it and that's going to be a problem when you're not there to hold their hands.






IV Lots of trees, very little forest

If you want students to understand what they're doing and, just as important, if you want them not to feel confused and anxious, you have to give them a sense of the overall problem.

The carton question above is a perfect example of how to screw that up. The instructor jumps directly to the exact variables and equations needed, skipping a number of steps that are not at all obvious to the students. In fact, it is those skipped steps that trip up most of the students who miss these problems.

If you tried this with actual students, they would say "How did you know to do that?" and that's an extraordinarily good question. The answer is that we start with translation from English to mathematics (a faithful but not necessarily idiomatic translation, though I probably wouldn't mention that to the students). "[H]ow many of the $50 cartons did she buy?" suggests number of $50 cartons should be a variable (let's be original and call it x) and it makes sense that the number of $30 cartons would be another (y). "Sarah bought 20 cartons" becomes x + y = 20 and so on.

If students understand the why behind the what, they will be happier and learn better.




I don't want to be too harsh here. This isn't exactly a bad resource and I very much believe that the Khan Academy's heart is in the right place, but when it comes to the SAT, they still have a long ways to go.

Note: I made a few minor wording changes after first posting this.

Alexandria Puzzles for SAT vocabulary building

The following puzzles contain words from SAT vocabulary lists.

SAT Words Group 6
abdomen  abject  aborigines  abscond  abstruse  accelerate accompany accurate  Achillean  acquiesce  acrimony  adduce  adjuration  adulterant  advocate  affirmative  aggrandize  agrarian  alchemy  alienate  allot  alteration  altercation  amateur  amenable amphitheater analyze  anew  animate  anode  antemeridian Antichrist  antiquate  aperture  apotheosis  apposition  arbitrary  archangel  aristocrat  arrival  ascent  assailant  assignee  athwart  audacious  auricular  autocracy  aver  aye  baleful  bass  beau  belate  benefactor  benignity  bestride  bibliophile  biograph  blazon  bole  borough  braggart  brethren  Britannia  bronchitis  bulrush  burgher  cabinet  callosity  Canaanite  capacious  caret  castigate  caucus  centimeter  chaos  check  chronology . cite  clearance . coddle  cohesive  collector  comely  commodity  competitor  complicity  compression  concession  condensation  confederate  conflagration  congest  connoisseur  consecutive  consolidate  constituency  contaminate  contingency  contraposition  contumacious  convergent  convulse  corporate  corroboration  cosmology  countervail  cower  credible  crusade  culvert  cycloid  dauntless  decagram  deceitful  declamatory  defalcate  deficiency  degeneracy  dejection  delusion  demonstrate  denominator  deponent  depreciation  derive  desist  despotism  detriment  diacritical  didactic  digraph  dimly  disapprove  disciple  discord  disenfranchise  disinterested  dispel  disrobe  dissentious  dissonance  distillation  divagation  divisible  doleful  domination  drachma  dubious  dutiable  eclipse effect efficiency  egoist  electrotype  elude  emblazon  emigrant  emporium  encroach  energetic  enkindle  ensnare  entreaty  epidemic  epoch  equivocate  essence  euphonious	evert  excavate  exclamation  execration  exhaustible  exonerate  expectancy  explicit  extant  extinct  extravagance  facet fallacy  faulty  felonious  festive  fiducial  finite  flection  fluctuate  folio  folly forcible  forego  foresail  formidable  fragile  freethinker  frivolous  fulminate  furlough  gaiety  gamester  gendarme  geniality  gesture  glimmer  gosling  grantee  gravity  grotesque  gusto  habitude  harbinger  heathenish  herbaceous  hesitant  hibernal  hoarse  hosiery  hussar  hydrous  hypotenuse  idealize  illiberal  illusive  immaterial  immiscible  impassable  imperceptible  impervious  imply  impromptu  impunity  inane  incapacitate  inchoative  incoercible  incompressible  indefinitely  indigestible  indivisible  inefficient  inexpedient  infernal  inflammation  infusion  inherent  inkling  inquire  insentient  insolence  instructive  intangible  interact  interminable  interrogatory  intolerable  intrigue  intrusion  invasion  invidious  iota  irradiate  irresistible  irritant  itinerary  jovial  juicy  juvenile  kind-hearted  labyrinth  landlord  later  lave  legacy  leisure  lewd  liege  likelihood  linguistics  literature  liturgy  lodgment  lough  luminescence  luxuriance  madden  magnificent  malcontent  maltreat  manlike  maritime  masterpiece  maudlin  medallion  mellifluous  mentor  metamorphosis  metronome  microscopic  militarism  ministry  misbehavior  mismanage  mistrust  moccasin  mollify  monitory  monotone  moratorium  mountainous  municipality  myth  nasal  navigable  nectar  negligent  neology  nevertheless  noisy  non-combatant  novellette  numeration  obelisk  oblivion  obsolete  obviate  octogenarian  offhand  omnipotence  opalescence  opportunity  opulent  ordnance  orthopedist  out-and-out  out-of-the-way  outstrip	overpower  overture  palette  panacea  Pantheon  paralysis  Parisian  parse  paternal  paucity  pectoral  pediatrics  pendant  pension  penury  percussion  perigee  pernicious  persistence  perspire  pervasion  peter  philanthropist  phonetic  physiocracy  pillory  pitiless  playful  plenitude poetaster  polyarchy  polytheism pomposity  posit  postdate  practicable  precession  predecessor  preemption  preferential  premature  preponderate  presentient  prevalence  primitive  probate  prodigal  profiteer  proletarian  promontory  propitious  prospector  protomartyr  proverb  prudential  pudgy  pupilage  pyrotechnic  quarantine  quibble  racy  rancor  ravage  reassure  receptive  recline  reconstruct  recuperate  reducible  reflection refringent  regimen  reiterate  relinquish  remodel  renunciation  repetition  repressible  repugnant  resemblance  resonance  resurrection  retrace  revelation  rhapsody  rigmarole rotate  ruth  salient  sanguinary  sardonic  scholarly  scruple  secession  sedentary  seize  senile  sentient  sequent  severely  sibilance  simile  sinus  slothful  soldier  somnolence  soprano  species  spherometer  stagnate  stationary  stifle  strait  stupor  subliminal  subsequent  subtle  succumb  superadd  superintendent  supplicate  surmount  sustenance  symmetry  synonym  tangency  taxidermy  telltale  tenet  terminus  theocrasy  thermoelectric  tincture  tolerant trait  transcendent  transfuse  translucence  transverse  tremor  trident  trite  tutelage  tyro  unanimous  unconscious  underhanded  undulate  unisonant  untoward  upheave  usurious  vacuum  vapid  vegetarian  vendor  veracity  vernacular  vicarious  vindicative  virtual  vitalize  vociferous  votary  warlike  whereabouts  winsome  working-man  wry