question for the instructivist

Instructivist left this comment:

When I was teaching DI to 8th graders some of the hurdles were understanding that conversion factors like 1 ft/12 in mean one and that we are taking advantage of the identity element for multiplication. Another hurdle was to figure out which unit of the CF should be in the numerator and vice-versa. I thought I had developed crystal-clear strategies for a foolproof approach, even though the approach didn't sink in easily.

I always insisted on writing out each step and wrote the direction of the conversion on top of the problem with an arrow to minimize confusion, e.g. sec --> hours. Not all students were converts to my approach to conversion.

I don't quite get the arrow part.

I wish to heck I'd kept more notes on dimensional analysis.

Saxon teaches dimensional analysis throughout all his books, starting maybe in 7-6.

Several times I've thought I had it down cold, and then encountered a problem that stumped me.


notes:

• every single time we work on dimensional analysis I say to Christopher: "What does 1 ft/12" equal? Then I wait 'til he tells me it equals 1. Sometimes he doesn't tell me it equals 1, so I tell him. Then I say, "Why can we multiply the initial value by 1 ft/12"? That he always gets: we're multiplying by 1. Then I say, "Have we changed this initial amount? Is it a different amount after we've done all this multiplying by unit multipliers?" He gets that one, though he's slightly hesitant.... "...No..." Finally I say, "What has changed?" He may or may not say that the unit has changed, but that's only because he's not necessarily following my train of thought. As soon as I say it he gets it. I've become a huge fan of scripted instruction. I do this script every time we work on unit multipliers; when Christopher reaches the point where the script seems stupid and obvious to him I'll know he's got them conceptually as well as procedurally. (Or at least that he's got a far more solid conceptual understanding than he did when we started out.)
  

• Christopher has no trouble figuring out which unit has to go in the numerator and denominator, and neither did I including back when I first learned unit multipliers. I think there's something visual about it (and I believe visual memory is "stickier" though I have yet to review all that research).
  

• He sometimes gets confused about which number is which: for a particular problem he'll know he has to put yards in the numerator and feet in the denominator, but he'll write 3 yards/1 ft because 3-to-1 makes more sense or is more familiar (you probably know what I mean). So, although he has zero confusion about the canceling aspect of unit multipliers, the very fact that sometimes yards will be in the numerator and sometimes they'll be in the denominator can trip him up.

• I had a bit of trouble moving from "easy" unit multipliers (centimeters to yards) to rate unit multipliers (mph to meters per second), but Christopher has had no trouble at all. I'm sure that's because Christopher still writes the number 1 in the denominator of the rate: 60 miles/1 hour. I wish I'd thought of that. For quite awhile I kept thinking things like, "Wait! I have two units in the numerator! (60 mph - it's all one chunk) What do I do now!" This is one of those times where having a fresher brain is an advantage.

• I think the single hardest aspect of unit multipliers is knowing which number to put first. To this day I don't quite know whether it matters; I've gotten jumbled up in long problems before and had to unjumble myself by deciding there was one, and just one, number that could start the whole thing out. When Christopher reads a simple unit multiplier word problem I have him circle the value to be translated and underline the unit he's supposed to end up with. He's not particularly interested in doing that, but on the other hand the fact that we have done it seems to have made it fairly easy for him to figure out where to start.

• I have him do the cancellations as he goes along. I learned this the hard way. By the time you get to Saxon Algebra 2 you're doing some long-chain dimensional analysis; more than once I've lost my place and had to start over.

• having the student write two unit multipliers for each conversion (1 yd/3 feet versus 3 feet/1 yd) is a very good thing to do.

• dimensional analysis word problems are also a very good thing to do. For me it was a terrific exercise to use dimensional analysis to solve everyday word problems I had never used DI to solve before.
  

• Terrific DI problem from Saxon Math: The Adams' car has a 16-gallon gas tank. How many tanks of gas will the car use on a 2000-mile trip if the car averages 25 miles per gallon? 
(source: Saxon 8/7 Lesson 96 page 660 #3 - answer: 5 tanks)

  
  

Dimensional analysis is the simplest procedure on the planet, and yet it's strangely challenging to learn. I think this is entirely due to Wickelgren's observation about all math looking alike.

Dimensional analysis is the ultimate exemplar of the practice, practice, practice theory of knowledge.

It's easy, but it's confusing.

Practice solves that problem.