Open-Ended Questions

On the MathFuture mailing list, we have been discussing open-ended questions intended to spark discussion and exploration, and to form the basis for a deeper understanding, not just problems where the only concern is to get the Right Answer. The idea is to bring up schoolchildren to the idea that they can learn things themselves, and are not limited to only those topics presented by a teacher. Of course, pre-school children learn an entire world without teachers, but we dismiss that as play.

Here is one example of the approach to open-ended questions and discovery taken by Turing Ward winner Ken Iverson. I will extract others for this blog from time to time. This extract comes from his 1972 textbook, Algebra: An Algorithmic Treatment, now available under Creative Commons license. I am translating this book from 1970s APL into J, Iverson’s last programming language, on the Sugar Labs booki server for the Replacing Textbooks program. J is considerably more powerful and more expressive than APL, with more facilities for doing algebra. An example is the polynomial adverb p., such that 1 2 3 p. x is equivalent to 1+2x+3x².

Within this text, the term verb is used in place of function, and an adverb is a higher-order function that yields a verb as a result. Iverson noticed a significant parallel between the language of algebra and the language of grammar, and other parts of English:

• data object = noun
• variable = pronoun
• function = verb
• vector = list
• matrix = table
• operator = adverb or conjunction

This refers to operators in the sense used in quantum mechanics and other high-level mathematics, that is, to functions acting on functions to produce new functions, like the derivative operator. Other programming languages use the term operator to refer to built-in functions. Such is the confusion of language, under the Humpty-Dumpty rule: “When I use a word, it means just what I choose it to mean—neither more nor less.” But that is a topic for another day.

And now, without further ado, Ken Iverson.

The basic rule for reading a function table is very simple: to evaluate a function, find the row in which the value of the first argument occurs (in the first column, not in the body of the table) and find the column in which the second argument occurs (in the first row) and select the element at the intersection of the selected row and the selected column. However, just a there is more to reading an English sentence than pronouncing the individual words, so a table can be “read” so as to yield useful information about a function beyond that obtained by simply evaluating it for a few cases.

 Small Medium Large H 57 105 113 121 W E 58 107 115 123 E I 59 109 117 125 I G 60 112 120 128 G H 61 115 123 131 H T 62 118 126 135 T 63 122 130 139 I 64 126 134 143 I N 65 129 137 147 N 66 133 141 151 I 67 137 145 155 P N 68 141 149 158 O C 69 145 153 162 U H 70 149 157 165 N E 71 153 161 169 D S 72 157 165 173 S

For example, this weight table can be “read” so as to answer the following questions:

1. Can two women of different heights have the some normal weight?
2. For a given frame type, does normal weight always increase with increasing height?
3. For a given height, does normal weight increase with frame type?
4. How many inches of height produce (about) the same change in weight as the change from small to large frame? Does this change remain about the same throughout the table?

Arithmetic verbs are more orderly than a function such as that represented by the table above, and the patterns that can be detected in reading their function tables are more striking and interesting. Consider, for example, an attempt to read this table to answer the following questions:

1. The second column of the body (which was previously remarked to represent the “times two” verb) contains the numbers 2 4 6, etc., which are encountered in “counting by twos”. Can a similar statement be made about the other columns?
2. Is there any relation between corresponding rows and columns of the body, e.g., between the third row and the third column?
3. Can every result in the body be obtained in at least two different ways? Are there any results which can be obtained in only two ways?

Similarly, one can  construct a verb table for addition and read it to determine answers to the following questions:

1. In how many different ways can the result 6 be obtained by addition? Does the result 6 occur in the table in some pattern and if so does a similar pattern apply to other results such as 7, 8, etc.?
2. What is the relation between two successive rows of the table?

Because of the patterns they exhibit, verb tables can be very helpful in gaining an understanding of unfamiliar mathematical verbs. For this reason they will be used extensively in succeeding chapters. 