kohn

Quantitative Reasoning Project for Math 1.95

 Marvin J. Kohn

Department of Mathematics

            Math 1.95 is Brooklyn College’s “math for elementary ed majors” course.  It is required of all elementary education majors.  The syllabus includes the mathematical topics that are represented in the National Council of Teachers of Mathematics K-6 standards, http://www.nctm.org.  We try to teach these topics from an advanced mathematical standpoint.  We also try to emphasize “problem solving” throughout the course, as advocated by the NCTM.

            The students who take Math 1.95 have widely different levels of mathematical ability.  Some are very bright and intellectually adventurous.  Others are very weak.  Many of the students have very poor basic mathematical skills.  Most of the students are quite anxious about mathematics.  Almost all of the students search for specific formulas or algorithms to memorize.  This is much safer for them than open-ended problem solving.

 

Statistics in Math 1.95

 

            One of the syllabus topics of Math 1.95 is an introduction to descriptive statistics.  We teach the students to organize data into tables and graphs of various types.  We compute measures of central tendency (mean, median, and mode).  We try to emphasize that one of these measures might be appropriate in one situation but misleading in a different situation.  We also briefly study measures of dispersion, such as standard deviation.

            When I teach the statistics topic in this course, I usually begin the lecture by putting on the blackboard a list of numerical exam scores, representing the scores on a recent Math 1.95 exam.  I show the students how to compute the mean, median, and mode, and the class discusses the relevancy and appropriateness of each of these measures.  I then organize the list into a grouped frequency distribution.  For this specific data, I use class intervals 90-99, 80-89, 70-79, 60-69, and below 60, because these are the intervals that correspond to A, B, C, D, and F at Brooklyn College.  From the grouped frequency table, we construct bar graphs, histograms, and frequency polygons.

            The statistics chapter is generally very easy for the students to master.  Most of the students have no difficulty with homework problems that mimic my class lectures.  However, many of them have difficulty choosing the most appropriate statistics to use in applied problems.

The Student Project

 

            In conjunction with the Transitions Quantitative Reasoning project, I decided to assign my students a statistics assignment with a twist.  I told the students to pretend that they were fourth grade teachers, and I gave them a list of numbers that represented the scores of their fourth graders on a standardized test.  They were to numerically and graphically organize this data.  However, they were not required to use the most appropriate measures and parameters.  Instead, they were to use those numerical measures and parameters that made their class look as good as possible.  They could not omit or change data, but they should organize the data so that it would be seen in the best possible light.  The student assignment is reproduced in Attachment 1 at the end of this paper.

            Here is the list of scores that I gave the students:

89, 91, 21, 67, 89, 69, 74, 28, 89, 66, 95, 59, 84, 92, 12, 79, 69.

This data was not randomly chosen.  I selected particular numbers so that the students would have to cope with several issues.  First, I included a few very low scores.  These were unrepresentative of the class and distorted the class average.  Next, I included three scores of 89.  This made the mode of the data 89, but the mode is clearly an inappropriate statistic here.  Finally, I included several other scores that ended in ‘9’, i.e. 59, 69, 79.  If a student were to construct a frequency distribution exactly as I had done in my class lecture, with intervals of 90-99, 80-89, etc, then most of the class intervals would be over-represented by numbers at the top of the range.  This is shown in Table 1 below:

 

 

Interval

Scores

 

 

90-99

91, 95, 92

80-89

89, 89, 89, 84

70-79

74, 79

60-69

67, 69, 66, 69

Below 60

21, 28, 59, 12

 

 

 

 

Table 1

A histogram drawn from this table would not be a particularly flattering or accurate representation of the data.

            I had hoped that some students would recognize that the class intervals 90-99, 80-89, etc, were not sacrosanct.  (We are dealing with a standardized fourth grade test, not a Brooklyn College course).  For example, if one were to instead use class intervals 89-98, 79-88, etc, then the resulting frequency distribution would be much more gratifying, as shown in Table 2.

Interval

Scores

89-98

89, 89, 89 91, 95, 92

79-88

79, 84

69-78

69, 69, 74

59-68

59, 67, 66

49-58

 

39-48

 

29-38

 

19-28

21, 28

  9-18

12

 

 

Table 2

Student Results

 

            Most of my students took the assignment seriously and devoted substantial effort to it.  Several students made serious errors in grouping data or computing averages, but most students were able to correctly execute these tasks.  One student threw out the low scores, saying that they were unrepresentative, and re-worked the assignment using only the higher scores.  This clearly was against the rules.

            In this data set, the mean is 69, the median is 74, and the mode is 89.  To my thinking, it is pretty clear that the median represents the best measure of central tendency for this problem.  The mean is too low because of the few very low scores.  The mode, even though it is the largest number, is just a fluke and signifies nothing.

            Many students agreed with my thinking on central tendency.  However, some argued that the mean was the only true measure, because it is the only measure that takes all students into account.  A few students contended that the mode was the best measure, not because it was highest, but because it was the most suitable – it was the score that the largest number of students earned. 

I was proud to read several other student papers in which the students agonized over using the mode.  They realized that its use was deceptive here; however since it was the highest measure, these students verbally wrestled with the morality of using it to fulfill my instruction of viewing the data in the best light.

There was quite a bit of variation in the ways that the students graphed the data.  Many students constructed graphs exactly as I had done in the introductory lecture, using the same class intervals as in Table 1.  These graphs did nothing to make the class look good.  Other students made pie charts with only two sectors—one representing the 60-99 interval (PASS) and the other representing the below 60 scores (FAIL).  This is creative but really not appropriate, because it arbitrarily sets the grade of 60 as passing on a fourth grade standardized test, and also because it does not highlight those many students who got very high scores.  A few other students created graphs that were completely meaningless.  See Attachment 2 for an example.

At the other extreme, several students were imaginative enough to use more favorable (but still fair) intervals in the frequency distribution.  I have included a few of these graphs at the end of this report, Attachment 3.  A few students used alternative graphing methods.  One of these was a box-and-whisker plot.  (I did not teach box-and-whisker plots, but they are in the textbook.  I was happy that this student searched beyond her lecture notes).

I hope that my students learned several lessons from this exercise.  One is that they can benefit by thinking creatively (in a problem solving way), and not by just using the professor’s lecture as a template to be duplicated.  Another is that they have the power and ability to deal with numbers and statistics to their advantage.  And of course, I hope that my students learned to view the statistics of others with critical analysis and some skepticism.

 

 

 

 

           

 

 


 

 
 

 

 



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Appendix 1

 

Math 1.95                                 Fall 2001                                   Prof. Kohn

 

Quantitative Reasoning Assignment

 

 

Due date:          Wednesday, December 5

 

 

 

The scores below represent the grades of a class of fourth grade students on a standardized exam:

 

 

89, 91, 21, 67, 89, 69, 74, 28, 89, 66, 95, 59, 84, 92, 12, 79, 69

 

 

Pretend that you are the teacher, and that you need to organize this data and present it to your supervisor in numeric and graphical form.  You would like to do this in such a way that makes the class performance look as good as possible.

 

Write a one or two page paper in which you numerically and graphically analyze the data, to achieve this end.  You have the freedom to decide which numeric measures to compute (and which to omit), and to choose the graphical format and parameters that are best for this task.  You should use the statistical measures and parameters that put the class in the best light, not those that might be most appropriate or customary.  After you have presented the numeric and graphical representation of the data, you should explain why you have chosen these particular statistics and rejected others.

 

You can work individually or collaboratively, in groups of two or three.  If you choose to work collaboratively, then hand in only one copy of your paper, with the name of everyone in the group.