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Re: Thinking Level of students

There is one other hypothesis which has not been included in your list. The
method of teaching is inappropriate to getting students to understand the
relationships. One would presume that any successful method would have to
include the missing pieces that have been listed, as well as supplying some
motivation and require active engagement rather than just note taking. No
teacher wants to admit that the methods they are using do not work, but that
may be exactly the case.

The motivational factor is obviously important. On thing that I see is that
the thinking level appears to put a limit on what can be learned, but below
that limit students achieve gain scores all the way from zero to that limit.
Perhaps increasing the types of experiences that the students have, and
working on certain basic skills may increase the limit, but increasing the
thinking level is really a better solution.

I do not think that I have said that it is impossible for the students to
learn the astronomical geometry. Rather I have pointed out some of the
barriers, all of which are formidible. The results of the problem are
evident, but the actual problem itself are not completely defined. I would
rank the solutions as 3 fold.
1. Use active engagement techniques which force the students to think about
the material.
2. Raise their thinking levels on more general tests.
3. Attack the motivational problem. (this may be the most difficult) see
results of the Maryland MPEX studies.

John M. Clement
Houston, TX




I have continued to think about the allegation that
"most college students are intrinsically unable
to grasp relationships such as the earth/moon/sun
geometric relationship."


In my judgement, that is an unproven hypothesis, because:
1) We have seen some evidence against it.
2) Although have seen some evidence that might seem to support it, this
evidence is weaker than it first appears, because it is susceptible to
other interpretations. That is, competing hypotheses have not
been ruled out.

Remember, there is a difference between the data and the interpretation(s)
placed upon the data. To say the same thing in other words: it does not
suffice to say that your favorite hypothesis is consistent with the
data; rather, you must consider all plausible competing hypotheses and
show that they are excluded.

Here is an example of what I mean:

I asked a 13-year-old if she could explain the phases of the moon. She
immediately started to consider the geometric possibilities case by
case. She started with the case where the sun and moon are at
right angles
as seen from the earth. So far so good. At this point, however,
her train
of thought got derailed, because she could not figure out what appearance
this configuration would present to earthbound observers.

Let's consider an alternative hypothesis, namely that she was limited not
so much by the alleged "intrinsic inability to grasp geometric
relationships" but rather by lack of understanding of light
propagation. Call this hypothesis #2. She was unable (using only her own
resources) to figure out that the half of the moon that was not in direct
sunlight would be very, very dark. This is a perfectly plausible
misunderstanding, because in our ordinary terrestrial experience, objects
in shadow are generally quite visible. I'm looking into my yard right now
(2 hours after sunrise). The whole yard is in the shadow of the
trees, but
I can see everything in the yard just fine.

Evidence in support of hypothesis #2 is that when I dropped the hint that
one side of the earth was daytime and the other side was
nighttime, she was
immediately able (without further hints) to generalize that notion and
apply it to the moon. She immediately understood the appearance of the
moon in the 90-degree configuration.

======================

Some related thoughts:

1) Stuff that gets used gets remembered.
-- Typical kids in Paris will have a hard time remembering the infield
fly rule, even if it has been carefully explained to them. There
is a good
logical reason for the infield fly rule, but it's awfully far removed from
anything Parisians care about.
-- The physics of the phases of the moon is rather far removed from
anything most kids nowadays care about.
-- ISTM kids who have a reason to learn and remember such things are
perfectly capable of doing so (with isolated exceptions).

2) Negative transference is very common.
-- Too much experience with low Reynolds numbers (such as dough flowing
through a spaghetti-making machine) often leads to wrong intuition about
high Reynolds numbers (such as air flowing past a propeller).
-- Experience with ordinary backyard illumination situations can easily
lead to wrong intuition about illumination in astronomical situations.
-- Experience with ordinary sizes and quantities (such as a dozen eggs)
can lead to wrong intuition about how big numbers "ought" to be. You
generally don't go down to the store and buy an astronomical
number of eggs.

3) Most people have very little innate skill at judging angles.
-- The harvest moon on the horizon looks big, whereas the full moon
overhead looks small. Of course it's the same angle; it just "looks"
different.
-- This is not restricted to astronomy. Without specific training, the
ordinary student pilot on final approach has a very hard time perceiving
the difference between a too-shallow approach angle and a too-steep
approach angle.

4) When thinking about spatial relationships, using diagrams and/or props
is much easier than using the mind's eye alone. This becomes
more and more
true as the relationships get more complicated. That's why we invent maps
and blueprints and Feynman diagrams.

4a) Drawing rough diagrams is easy. Even rather young kids can make
diagrams. Watch 5th-graders playing football. The 13-year-old quarterback
will draw the plan for the next play, using his index finger to "draw" on
his palm. (Some quarterbacks are better than others....)

4b) Drawing accurate scale diagrams is a specialized skill. It can be
taught, but it is not widely taught.

4c) Most diagrams of the solar system are !not! drawn to scale. There are
good reasons for this. Scale diagrams are not usually appropriate when
dealing with multiple length-scales that differ by astronomically large
ratios. Constructing a scale model of the solar system is a valuable
exercise, but it requires quite a bit of thought.