Discovery learning in math: Exercises versus problems (pdf file)
by Barry Garelick
Our fourth example offers a sharp contrast to the other three. This problem comes from the fourth grade textbook in the series called Primary Mathematics from Singapore.6 It is well posed and requires students to apply their prior knowledge.
“What is the value of the digit 8 in each of the following?
a) 72,845 b) 80,375 c) 901,982 d) 810,034 e) 9,648,000 f) 8,162,000”
Students cannot escape the lesson about place value since they cannot simply note where the 8s are, they must know what the various positions of the 8s mean. Preceding this problem in the Singapore text are other problems that introduce the concept of a number being a representation of the sum of smaller components of that number by virtue of place value; i.e. 1,269 can be expressed as 1,000 + 200 + 60 + 9.
Similarly, students are asked to express written out numbers, such as ninety thousand ninety, using numerals in the standard form (i.e., 90,090). They are also asked to write numbers in numeral form, such as 805,620, in words.
In short, students are asked no ambiguous questions, and the underlying concept of place value is indicated clearly via examples that can be applied directly to problems. By the time students reach the problem asking for the value of “8” in the various numbers, they have a working knowledge of what the numbers in various positions represent. This problem pushes them to apply that knowledge, thereby revealing any confusion they may have and also providing enough guidance for them to see that the position of the number dictates its value.†
Advocates of complex problems that get students “off the script” may think this problem is not challenging enough. After all, any discovery students make is inherent in the presentation of the problem and the solution clearly comes from work that the students have just completed. But as anyone recalls from the early days of having to learn something new, it feels a whole lot different answering questions on your own, even after having received the explanation. In fact, such experience constitutes discovery. So I have to ask, what is wrong with acquiring incremental amounts of knowledge through well-posed problems? It is, after all, much more efficient than discovery-type problems that require Herculean sense-making efforts and leave most floundering for a solution, without a clear sense of whether they are right or wrong.
herculean sense-making ----