The following is a set of tasks which I think are great questions for assessing understanding of irrational numbers. These tasks were from the study of Natasa Sirotic and Rina Zazkis. The responses were analysed in terms of algorithmic, formal, and intuitive knowledge described at the end of the post.
Set A
This set of tasks assesses the formal and intuitive knowledge about about the relative sizes of two infinite sets – rationals and irrationals.
- Which set do you think is “richer”, rationals or irrationals (i.e. which do we have more of)?
- Suppose you pick a number at random from [0,1] interval (on the real number line). What is the probability of getting a rational number?
This set assesses knowledge about how the rational and irrational numbers fit together in relation to the density of both sets.
- It is always possible to find a rational number between any two irrational numbers. Determine True or False and explain your thinking.
- It is always possible to find an irrational number between any two irrational numbers. Determine True or False and explain your thinking.
- It is always possible to find an irrational number between any two rational numbers. Determine True or False and explain your thinking.
- It is always possible to find a rational number between any two rational numbers. Determine True or False and explain your thinking.
This set investigate knowledge of the effects of operations between irrational numbers
- If you add two positive irrational numbers the result is always irrational. True or false? Explain your thinking.
- If you multiply two different irrational numbers the result is always irrational. True or false? Explain your thinking.
You may want to analyse the responses using Tirosh et al.’s (1998) dimensions of knowledge:
- The algorithmic dimension is procedural in nature – it consists of the knowledge of rules and prescriptions with regard to a certain mathematical domain and it involves a learner’s capability to explain the successive steps involved in various standard operations.
- The formal dimension is represented by definitions of concepts and structures relevant to a specific content domain, as well as by theorems and their proofs; it involves a learner’s capability to recall and implement definitions and theorems in a problem solving situation.
- The intuitive dimension of knowledge (also referred to as intuitive knowledge) is composed of a learner’s intuitions, ideas and beliefs about mathematical entities, and it includes mental models used to represent number concepts and operations.
At the conclusion of the study, Sirotic and Zaskis reported this short exchange:
What do you think of the teacher’s answer?
You may want to share your responses to the questions in the comment section below.
Regarding the “teacher’s answer” it might help to address very early on the fact that the product (or sum) of an irrational and a non-zero rational is always irrational. Then after reminding the student that the irrationality of pi already implies that there are at least as many irrationals as rationals, mentioning root2 could be used to show that there are actually more, with root3 and so on extending this to infinitely many times more, and so giving some sense of the reasonableness of the facts in group A and B.
But re A2 I think the idea of “you pick a number at random from [0,1] interval” is not well defined without reference to concepts which are well beyond the elementary math level.