In article <[EMAIL PROTECTED]>, Carl Lee <[EMAIL PROTECTED]> wrote:
>Using introductory statistics as an example, concepts are built in a certain
>sequence. If students get lost at a certain stage, s/he will have difficulty
>to connect the later concepts together. Therefore, it is crucial to test the
>understanding of the connection (or relationship) among related concepts. For
>example, you may be surprised that the concept of histogram is much more
>difficult for students than we thought. Try the following problem in your
>final exam, you may be surprised by the outcome:
The "concept" of histogram? I cannot see it as a concept.
The concept of measure, which includes probability and
frequency and much more, is quite simple if one does not
try to make it complicated. The oldest measure used by
mankind is that of the number of objects.
But histogram is presentation, not statistics. I am not
convinced that it should even be used in a beginning course,
because it is confusing and misleading.
>If you collect a random sample of 100 salaries of working individuals who are
>40 years or older. Ask students to describe the shape of the histogram that
>is more likely to occur, and their reason. Then, ask students to verbally
>describe the Y-axis and X-axis of this histogram.
I cannot see that this is a statistics question at all.
>I have collected data for this problem for several years. When I first asked
>this question, I was shocked that 80% of students got confused between
>scatter plot and histogram. I began to pay attention and used a variety of
>strategies to help students. We usually think people have seen histograms all
>the time, it must be simple. However, this test problem seems to indicate
>that we may have overlooked simple concepts such as this.
This is not too surprising considering that our miseducational
system is doing its best to keep students from even seeing
concepts; the teachers have themselves lost the ability.
>If we think about the construction of histogram a little more, we see that a
>histogram is a transformation of raw data into two-dimensional presentation
>for a response variable. This indeed is very different from our common
>experience of two-dimensional plot, which is usually involved with two
>response variables, a scatter plot.
The problem is that they cannot understand English, or any
other language. I also do not see this as statistics, but
as graphical "crutches" to look at data in a non-statistical
manner.
>One assessment tool I use to test student's understanding of concepts is to
>test how well they understand the relationship among related concepts, not
>just stand-alone concept. For example, the relationship among time series
>plot, box plot, histogram, outliers, mean, median, standard deviation and
>range is important for understanding variation, distribution and later the
>sampling distribution of sample mean. I have developed a series of questions
>for testing their understanding of the relationships using the project of
>investigating stock prices. There is no formula neither computation is
>required by students in answering these questions.
None of the statistical concepts make any sense without having
an understanding of the abstract concept of probability. If
you are putting all of this in a beginning course, there is no
possibility of understanding the real concepts.
The above are not concepts, but definitional constructs. The
only reason for using standard deviation to any extent is its
mathematical properties; it works. One CAN give a reason
why it is called that, but it is more formal than anything
else; it is that a two-point distribution with the same first
two moments and all of its mass at the same distance from the
mean has its mass one STANDARD deviation away from the mean.
If this gives you no insight, that is because there is none
there to be had.
As to WHY one should use any of these, mathematical results
are needed. To understand mean, one needs to understand
expectation, which is a form of integral, and integrals
were understood better by the merchants of 5000 years ago
than by the students who have taken calculus today. One
also needs to simultaneously use different sample spaces
for the same problem to understand the properties of event
and probability.
>Another assessment tool that I use is to ask students give the reasons of
>their answers verbally. Again, no formula neither computation is needed. What
>I intend to find out is how they think and how they solve the problem. This
>has helped me greatly to study how students learn a variety of statistics
>concepts and which concept students tend to get lost at the early stage of
>their learning.
On the contrary, expressing everything in symbols is what
is needed. They need to be able to use mathematics as a
language, to formulate problems, not to solve them.
>Assessment, learning and teaching are closely connected. And understanding
>how students learn is most important of the three. A first step toward
>understanding how learning take place is to conduct a good assessment,
>especially assessing the process of reasoning. Teaching strategies and
>instructional material can then be better prepared.
The process of reasoning is formal mathematics. It need
not be deep, but that is what it is.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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