Date: Nov 15, 2012 9:48 PM
Author: kirby urner
Subject: Re: How teaching factors rather than multiplicand & multiplier<br> confuses kids!

On Thu, Nov 15, 2012 at 6:31 PM, Robert Hansen <> wrote:
> It isn't that I disagree with all of these neat discussions, but these discussions are meaningless without proper development.
> What use is it to ask "What instruments and controls might we count as continuous versus discrete?" of students that haven't even fully developed the concept yet? You seem to think that students have some innate understanding of continuity, intermediate value theorem, discrete and analog, and have been just itching to discuss it with someone. These are great discussions for students that have already tackled these concepts, but not for students that haven't.

In a group setting, when you ask these questions, you start getting
answers, because the "digital" vs. "analog" concept is easily

An "analog clock" has a sweeping second hand whereas a "digital clock"
flips through digits.

It's the Socratic method. You're teasing it out of them. When they
hear their peers offering suggestions, it's emboldening, usually.

> What you are doing here is simply thinking up good examples of continuity, but they are only good if you understand continuity. They are confusing to someone that doesn't. The kids would not know what you are talking about. Not without development.

This is what development looks like.

Same thing when you have them go around the room identifying features
that are vertex-like (corners), face-like (walls and doors),

The point is to stay with everyday experience as a foundation. Not
that every student has seen a dashboard or airplane cockpit.

One needs to be culturally sensitive and / or project more pictures.
This might have been the banter that goes with a slide show.

Continuous is like a sheet of plastic. Not-continuous is like a fish
net. Slide. Slide.

> Can you tell me how you picture the students' minds unravelling these examples into a generic theory of continuity?

I'm not necessarily looking for a "generic theory of continuity" as I
don't imagine most adults have that either.

With that graphite pencil line, I'd point out that up close, through
the microscope, you could probably find a pathway through the line
that didn't touch the graphite i.e. it's not really that continuous.

The other thing to keep in mind is in hopping to Avogadro's Number
when talking about gas tanks, and referring to internal combustion,
I'm showing what a STEM teacher is like.

In STEM teaching, it's encouraged to hop around between S, T, E and M
without putting up strict fences between them. We have removed those

If you've only talked "math" for awhile and have brought in no facts
about science, well, time to restore some balance to your STEM talk.

Don't be two "one side".

Note also that whereas I alluded to my own high school experience, my
subsequent bouncing around did not presume the average age of my

Lets say that was an audience of STEM teachers I was talking to,
average age 37, some with prior military or para-military experience
(e.g. returning vets, looking for new interesting stuff to do).