Unless a student understands a mathematical concept, it isn’t possible to transfer the concept to a related one. Understanding mathematical concepts requires undoing, so to speak, the rote methods taught to students in earlier grades (e. g. , rote memorization of the multiplication tables). The most important tool in fostering understanding is the use of concrete examples which provide opportunities for students to test their answers against what is and is not possible.
Participation is a way for students to assess their own understanding, thus one should ask questions addressed to the class as a whole and to individuals who do not participate on their own (however, it cannot be overstressed that at the beginning of the semester, you should assure students that you will not call on anyone who privately lets you know they do not want to be called on. Doing so avoids the real possibility that at least a few students will spend the semester unable to concentrate or learn anything because their attention is on their fearing being called on.
Regarding a widespread fear of being unable to learn statistics, it would seem that the only chance of helping students to overcome this fear is to try from the beginning to introduce topics by drawing on what you believe they already understand and next moving to concepts you believe are easiest to understand. However, while beyond the scope of this paper, eventually it becomes necessary to understand abstract concepts (e. g.
, sampling distributions), requiring at least a minimal ability to reason at a formal operational level (Piaget, as cited in Shaffer, 1999), an ability as high as more than 50% of college students (depending on school) have not acquired Central Tendency and Spread (all Statistical Information from Watkins, Scheaffer, & Cobb, 2004) The most important relationship between measures of central tendency and measures of spread is that there is no relationship – the two measures are independent. Because this information is counter to the intuitions of typical students, they have difficulty in understanding the concept.
Thus, after learning what the measures are, concrete examples need to be used that are counter to their probable first intuition that higher measures of the center also have higher measures of spread (e. g. , examples such as “5, 6, 7, 8, 9,” “105, 106, 107, 108, 109,” “50, 60, 70, 80, 90,” “5, 20, 35, 50, 65,” “-100, -50, 0, 50, 100”). Measures of central tendency. Possibly because of how arithmetic might have been taught to students in elementary school, the first task is to teach how to test answers against common sense, especially necessary because it’s easy to accidentally push the wrong calculator button.
One essential rule is to understand that measures of the center can not be higher than the highest score or lower than the lowest one (and obviously if all scores are the same, that score also is the only measure of the center). For all of these measures, students also should be able to understand that an answer larger than the highest number indicates the answer is too large and one that is lower than the smallest number is too small. Regarding the mean, unlike the other two measures, it would be useful if students wrote down the numerator and denominator they used.
Then, correcting the error noted above could be accomplished by checking only the denominator and if it’s incorrect, then dividing the obtained numerator by the correct denominator (the answer may not be correct, but it should not violate common sense by being an impossible answer. More sophisticated ways of estimating eventually should be taught, and regarding tests in general, students should have learned about the value of using extra time at the end of an exam to re-check. ) For both the median and mode, example numbers should be ordered from highest to lowest or lowest to highest.
If ordered correctly, it should be impossible to be unable to find the correct mode. However, students are more likely to remember what they understand, so that in explaining the mode is the most frequent score, it would be useful to note that it is rarely used except with nominal variables and to provide a concrete example, such as in an election, where the values are names of candidates and typically the one with the highest frequency wins (unless, of course, there are rules requiring run-offs).
Before covering adjustments for tied scores and even before using examples of large data sets, it should be easy to explain how to count to the middle number when N is odd and the two middle ones when N is even (and the mean of the two is used). A good way to introduce the topic of spread is to ask students how they would decide whether to use the mean or median. Relationship between Central Tendency and Spread (based on Watkins, Scheaffer, & Cobb, 2004) One might first simply present a normally distributed data set, such as “5, 6, 7, 8, 9.
” It should be easy for students to recognize that “7” is both the mean and medium. One might then draw a concrete normal distribution, where there’s an actual lowest and highest possible score and students already probably know what used to be true of grades. Using a plot where on the horizontal axis grades are ordered F, D, C, B, and A and using percentages of students on the vertical axis, one might start by asking what grade used to be most common, and then, assuming they correctly responded C, drawing a dot corresponding with any reasonable percentage of those receiving Cs that students provide.
Eventually, students should be able to recognize a normally distributed variable where the mean and median are the same. Next, it might be best to change to a distribution where there’s a real lower limit but not a higher one, such as income. After they have suggested drawing a long right tail, they then should be asked to compare the two measures of the center and understand the mean would be higher than the median (and the reverse would result if one were able to find an actual variable with a highest but not lowest possible score).
To begin leading them to conclude the median better represents data with skewed distributions, one should use a concrete example such as income or housing costs and provide an actual graph representing the data so they can see the long right tail. The range and quartiles. An understandable way to explain the range is to reintroduce the concrete example of a normal distribution with an actual lowest and highest score. Students would then probably understand the range is the difference between the highest and lowest score.
A problem that more than a few students probably have involves examples with positive and negative scores. Thus, in ordering a small data set, it might be necessary to explain why the higher absolute value of a negative number is higher than a low one, a task that might be accomplished by drawing a line with 0 in the center and ordered positive numbers to the right and negative ones to the left. When a typical error occurs where subtraction results in a negative range (e. g.
, erroneously finding that “5-(-6) = -1! ”), it is time to explain – and in great detail – why no measure of spread can be negative – why there can’t be less than no spread at all. To begin explaining quartiles, one might go back to the normal distribution illustrating grades (noted above). If the horizontal axis on the right and left sides are divided in half, students would be able to see what a quartile is and that most people fall between the lines in the middle of each side.
By changing “F through A” to “0 through 4,” students could see that most people fall between 1 (D) and 3 (B) and thus learn about the inter-quartile range. ) References Shaffer, D. R. (1999). Developmental psychology: Childhood & adolescence. Pacific Grove, CA: Brooks/Cole. Watkins, A. E. , Scheaffer, R. L. , & Cobb, G. W. (2004). Statistics in action: Understanding a world of data. Emeryville, CA: 94608.