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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 ----
woo hoo!
5 comments:
I think discovery learning is just a cover for avoiding the "sage on the stage", and 21st century skills are just a cover for lower expectations and accountability. It's misdirection. Let's talk about how the brain works so nobody will ask why kids are getting into fifth grade still struggling with 6 times 7.
The funny thing is that if they really paid attention to what we know about how the brain works (Daniel Willingham, for example), they would quickly find that most of their education theories would disintegrate into nothingness. If they really cared about what kind of instruction makes a difference consistently and in almost all cases, they would quickly have to eliminate most of what they're doing in the classroom. This is educational dogma, however, and the coverts to reason and rational thinking are few and far between. They will continue to do what they do so long as they have the power to do so.
well-chosen examples and clear assured style...
boy, this guy's good. and an end-of-career move
to teaching? don't get me started. i could almost
begin to believe ed schools could be made worthwhile...
Thanks, Vlorbik. I'm getting nice comments from math teachers who find the good examples of discovery quite useful--and they also agree that the examples of bad discovery approaches are indeed bad.
Yes, ed schools could be made worthwhile if they hired enough people who believed that students don't learn what they ain't been taught.
Another fine article. I'd love to hear more about your experiences in ed school. I attended when I was in my late 30's and felt it wasn't particularly helpful. Now struggling with whether to get a Master's or not. Hate to waste the money for just a piece of paper. My ed school math professor was ALL about discovery learning. (Yeah, I only had 1 math class in ed school.)
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