Why would someone need to count blocks? The key notion is that we measure slope with a number as rise/run. Do you really have to have kids count blocks to know the rise and run? Can they not see that this is a difference? Boy, if that’s the problem, we really are in trouble as kids should be doing this stuff long before we get to a discussion of slope, no?

As for the example:

“Helpful High School Teacher asks the students to find the slope between (-2, 5) and (999998, 2000005).”

Rather than see this as some confirmation that a formula is useful, it struck me as a great opportunity to talk about using estimation to get a reasonably accurate result without a pencil and paper.

An two resistors are connected in series, one resistor is known and has a resistance value of 1 ohm. The second resistor is unknown. Using a voltmeter the voltage drop across the two resistors is measured. The voltage drop across the 1 ohm resistor is 1 V and across two unknown resistor is 100 V.

a) Calculate the resistance of the unknown resistor using ohm’s law Voltage=Current*Resistance.

b) Calculate the resistance taking into account that the voltmeter has an accuracy of 1 %.

in this case counting squares can be used at (a) and the formula in (b).

I was consulting with her during a class period on a project, and decided to offer some assistance checking student work on the side. I was using my tested and true plong multiplication method to check student’s work when the teacher I was consulting with looked at with almost a bemused and bewildered look on her face.

“You don’t use the partial products method?” was her shocked question at assessing my math skills.

I felt a bit ashamed that A.) I had never learned the partial products method, and that B.) I wasn’t doing it the way she did. Don’t get me wrong, it took about 15 seconds for that sense of shame to go away before confidently telling her as long as I got the answer, it didn’t concern me. I’d accomodate her students, but how I do things is just fine with me :)

Say what?

While I agree with the first part (about treating mathematical skills as a toolkit), as an engineering professor, it is important to me that the correct answer be produced as well. Getting the right answer by magic or cheating is useless, but getting the wrong answer (even with “understanding”) is also useless.

http://det.wa.edu.au/contents/images/news/Agreement.jpg
http://www.hebron.k12.ct.us/page.cfm?p=1819
http://www.mnvalleyfcu.com/studloans.htm
http://www.doscaminos.net/
http://www.lcgsco.org/annual-genealogy-classes
…

(found with Google images, using a search from a copy of the B&W image Dina found).

So it is almost certainly a stock image.

Looking for it with the keywords
stock image
added finds the source:
http://www.visualphotos.com/image/2×2915852/teacher_pointing_at_map_of_world

http://www.highered.nysed.gov/tcert/certificate/maintaincert-prof.html

Take from this what you will.

I like your addition of step #2 “The teacher teaches the general expression for slope, but sticking to the familiar concrete examples so that the students “trust” the new method(s).”

As for intuition in students, I feel it needs to be explicitly taught to students rather than assumed, and in my own past experiences, my best teachers have tried to do that. An excellent calculus professor I once had would constantly be saying things like “…and then when you’re finished, look at the answer. Does it look right? If you get a number that’s not between 4 and 6, there’s obviously something wrong, isn’t there?”; or “…and if you don’t believe me, or you get on the exam and your mind completely blanks, try out a few easy examples and work it out. Convince yourself it’s right!”

I think for the best math students, a lot of this advice is what they’re doing anyways whether they realize it or not, but it’s not obvious for the average students.