Mathematics Education Reserch or Why the real numbers are not as simple as they appear (with a quiz)

Mathematics research is interesting even if largely inaccessible (in fact, often a mathematician will have trouble with research in a field of mathematics too different from her or his own). Research in mathematics education is not very interesting, but is probably more important. I’ve spent a lot of time examining how elementary college level mathematics is taught (basically, calculus and beyond), looking at the research, and looking at hundreds of textbooks all because I think that the way mathematics is taught gives students a fundamental misconception about its nature, discourages those who aren’t great with computations and deceives those who are, and fails to teach some of the most important and useful subjects (like logic, statistics, probability, etc.). Lately, however, I’ve become concerned about a much more specific problem that pervades everything from pre-college mathematics teachers’ knowledge to scientific research: the nature of the number line. Statistical hypothesis testing (or null hypothesis significance testing) is the most frequently used research method/design in numerous sciences (from medicine & climate science to psychology & sociology). Most of the time, this method depends upon the use of particular probability distributions such as the normal distribution (and more frequently the so-called t-distribution and F-distribution). These and other distributions are what are known as continuous distributions, which (put simply) means that they require possible “values” or “outcomes” to correspond to every number in the interval [0,1]. In other words, they require an understanding of the real number line. Yet continuous variables and therefore real numbers are misrepresented even in my own undergraduate statistics textbook (a popular textbook that was published in I think two more editions when I was still an undergraduate). This is not uncommon, as we find in similar textbooks that continuous variables are defined as those for which given any two values, there are infinitely many others in-between.

Continuity, in the simplest sense, can be understood by thinking of graphs of functions like f(x)=mx+b or any function with a graph that has no gaps or breaks. It would seem that if there are an infinite number of values between any two values for such a function would mean that its graph has no gaps or breaks. This is, it turns out, a major problem for college students and pre-college mathematics teachers, who use or teach real numbers and real-valued functions, but don’t really understand what it is they are working with.

The mathematics education research that investigates students’ or teachers’ understanding of rational vs. irrational (or real) numbers often does so through questionnaires. Sometimes, these can include some interesting questions that reveal how unintuitive and paradoxical the real numbers actually are.

For example, some of the below are taken from the research and some are my own:

1) Imagine that you or a computer could pick a number at random from the interval [0,1]. What is the probability that this number will turn out to be a rational number?

2) Between any two rational numbers there is another rational number. Does this mean that between any two rational numbers there are infinitely many rational numbers?

3) The number of rational numbers is infinite, but it can be proved that there are just as many natural numbers as there are rational numbers (both sets have the same cardinality). And, of course, it would seem obvious that this is so, as one infinity can’t be bigger than another, can it? To test this, determine whether the infinite set of real numbers in the interval [0,1] is the same as the set of infinite rational numbers.

4) For any natural number greater than (or perhaps equal to) 0, we can easily find the next number (just add 1). This remains true for the integers (which definitely include 0 as well as the negative numbers), so e.g., the next (greatest) integer after -3 is -2. Can you determine what the next (greatest) rational number is after 1?

5) Imagine an object like a piece of string or a brick or a concept like a time interval or an interval on the real number line. It would seem reasonable that if we chop up, divide, or break up such an object or interval, we’ll end up with all and only the pieces we would need to put together to form the original object or interval (i.e., if we chop up a piece of string, the pieces won’t add up to be less than we started with or more than we started with). Is this true of intervals of real numbers (or units of time)? Why or why not?

6) Consider the unit square:

Define a probability function P(X) where X is a set of points (or a point) in the unit square and a set of points is equal to the area they take up (i.e., the probability of P(X) where X is the set of all points in the unit square equals 1). Recall that a random pick of a card from a full deck will yield an ace with probability 4/52=1/13 and an ace of spades with probability 1/52 because there are 4 aces and 52 cards, and only one of these aces is the ace of spades. What is the probability value of the point (.5, .5)? What is the probability of any set X where X is the union of some points (x,y) in the unit square?

7) Let a probability function P(S) be defined by the unit square as above, where S is again some area or a point in the unit square. The unit square contains the point (0,0). What is the probability that any set of points in the unit square (x,y) will include this point? (Hint: If S contains the entire area of the unit square, the probability of S is by definition equal to 1).

Answers to come, but feel free to answer those you know.

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