D. UnCertainty
By the end of the 12th grade, students should know that
-
Even when there are plentiful data, it may not be obvious what
mathematical model to use to make
predictions from them or there may be insufficient computing power
to use some models.
- When people estimate a statistic, they may also be able to say
how far off the estimate might be.
- The middle of a data distribution may be misleadingÑwhen the
data are not distributed symmetrically,
or when there are extreme high or low values, or when the
distribution is not reasonably smooth.
- The way data are displayed can make a big difference in how they
are interpreted.
- Both percentages and actual numbers have to be taken into
account in comparing different groups;
using either category by itself could be misleading.
- Considering whether two variables are correlated requires
inspecting their distributions, such as in
two-way tables or scatterplots. A believable correlation between
two variables doesn't mean that either
one causes the other; perhaps some other variable causes them both
or the correlation might be
attributable to chance alone. A true correlation means that
differences in one variable imply
differences in the other when all other things are equal.
- The larger a well-chosen sample of a population is, the
better it estimates population summary
statistics. For a well-chosen sample, the size of the sample is
much more important than the size of
the population. To avoid intentional or unintentional bias,
samples are usually selected by some
random system.
- A physical or mathematical model can be used to estimate the
probability of real-world events.
E. Reasoning
By the end of the 12th grade, students should know that
- To be convincing, an argument needs to have both true statements and
valid connections among them.
Formal logic is mostly about connections among statements, not
about whether they are true. People
sometimes use poor logic even if they begin with true statements,
and sometimes they use logic that
begins with untrue statements.
- Logic requires a clear distinction among reasons: A reason
may be sufficient to get a result, but
perhaps is not the only way to get there; or, a reason may be
necessary to get the result, but it may
not be enough by itself; some reasons may be both sufficient and
necessary.
- Wherever a general rule comes from, logic can be used in testing
how well it works. Proving a
generalization to be false (just one exception will do) is easier
than proving it to be true (for all
possible cases). Logic may be of limited help in finding solutions
to problems if one isn't sure that
general rules always hold or that particular information is
correct; most often, one has to deal with
probabilities rather than certainties.
- Once a person believes in a general rule, he or she may be more
likely to notice cases that agree with
it and to ignore cases that don't. To avoid biased observations,
scientific studies sometimes use
observers who don't know what the results are "supposed" to be.
- Very complex logical arguments can be made from a lot of small
logical steps. Computers are
particularly good at working with complex logic but not all
logical problems can be solved by
computers. High-speed computers can examine the validity of some
logical propositions for a very
large number of cases, although that may not be a perfect proof.
Benchmark 11 Common Themes
A. Systems
By the end of the 12th grade, students should know that
- A system usually has some properties that are different from those of
its parts, but appear because of
the interaction of those parts.
- Understanding how things work and designing solutions to
problems of almost any kind can be
facilitated by systems analysis. In defining a system, it is
important to specify its boundaries and
subsystems, indicate its relation to other systems, and identify
what its input and its output are
expected to be.
- The successful operation of a designed system usually involves
feedback. The feedback of output
from some parts of a system to input of other parts can be used to
encourage what is going on in a
system, discourage it, or reduce its discrepancy from some desired
value. The stability of a system
can be greater when it includes appropriate feedback mechanisms.
- Even in some very simple systems, it may not always be
possible to predict accurately the result of
changing some part or connection.