When I wrote last year’s unit, I frequently thought of the Algebra II students I had in 1984-85. This year I taught no Algebra classes. However, I was able to do parts of the unit with an Applied Math class, students who failed the Ninth Grade Proficiency Exam.
So this year I am thinking of what you can do with non-college-prep students. Students who take long times to get started, to believe they can do the work, and finally to complete the work.
In September, full of enthusiasm from having written the unit, I started talking about probability. The concept of a common fraction is central to the Applied Math course. We did the sample space for two dice, drawing cards, and drawing marbles from bags. The students did the work, they answered the questions.
I should have continued on into statistics. Instead I went to preparing the students to pass a parallel form of the Ninth Grade Proficiency Exam, by looking at questions modeled on the exam itself. The number of students who passed it in September was small as was the number who finally passed it in June.
So we were all frustrated by the end of the year, the students much earlier than the teacher. I wanted some different, hands on activity. I decided we would do circle graphs. First we would have to learn how to use a protractor, to make angles of a given size and to measure given angles. Some students did learn to construct and measure angles. Most of the students needed much more time to even learn the concept of what an angle is and what it means to measure one. So I was still frustrated.
The text book I was using was
by Edwin I. Stein. Chapter five was “Graphs, Statistics and Probability.” A colleague says the book is written on a college reading level. The presentation is quite sparse. There are problem sets of data to use for finding mean, median, mode and for drawing histograms. Next Stein goes on to Percentiles without any motivation or visualization technique. The book is written for someone who can read but needs to find out how to do some arithmetic task. It is not written for anyone afraid of school.
So we spent some time on histograms. The students behaved as if it were a hands on activity. We did the work in class step by step. Stein treats each topic separately. I, however, had the students draw the histogram even when the book was only asking for the mean, median and mode. Stein’s next topic was percentiles. So I wanted a way to introduce percentiles from histograms. I decided to teach cumulative frequency plots as an introduction and visualization of the median and the percentiles. So a cumulative frequency column was added to the work sheet.
Stein leaves it to the instructor to ask questions to stretch the students minds. When we were doing a histogram and finding the modes, I asked what was the probability of a particular score. The students answered the question. After seven months they knew their probability. I was pleased.
The amount of time needed to achieve proficiency was much greater than what I believed when writing the unit. We did make cumulative frequency plots, but we ran out of time before we could get to translating the frequency ordinates into percentages. The student satisfaction leads me to believe I should have started with this topic earlier.
Many of the topics of Applied Math easily tie into statistics. Changing common fractions to percents is a large part of the Applied Math course, but what does it mean, why would anyone want to do it? Well changing frequencies on the left edge of a cumulative frequency plot to percentages from zero to 100 on the right edge is concrete and visual, perhaps, even motivational.
The histograms were done as a class activity: data on board, arrange in order, make tallies, count frequencies, plot histogram, find mean, median and mode. Answers were provided step by step. I waited for the students to finish a step before I gave that answer on the board. We discussed that step before they went on to the next step and so forth. Additional problems were given for homework. Some students were motivated on their own to use color for the shading on their histograms.
While this may be arithmetic, the students seemed to enjoy the organized process. I now think finding a standard deviation could be done with such students. I need some motivation to do it, however. I do not want to do it just because the students like to do arithmetic. I would want to find the standard deviations of a number of sets of data, after making their histograms. This would give a visual meaning to the standard deviation.
With the Applied Math students the data was not grouped into intervals. The data were treated as if they were scores on tests. Usually about ten scores 20 to 50 items.
So the next time I teach Applied Math I will start with probability and statistics. I will use these subjects as occasions to use and introduce the arithmetic skills of the course.
So plotting data as histograms and cumulative frequency plots is the way to go. How do we get the data to plot? Do some experiments.