Take, for example, expanded notation [eg., 234 = (2 x 100) + (3 x 10) + (4 x 1)]. Yes, students need to be directly taught the conventions for writing a number in expanded notation. They would have no way of knowing that we put parenthesis around each multiplication expression, nor could they discover it through exploration. But they can construct their own learning about what expanded notation is (describing the value of each digit using a multiplication expression) through exploration with base-ten blocks and careful questioning by the teacher. Be sure to check out this blog post for more on that.
Another great example is decimal equivalencies. Say, for example, you are trying to teach the concept that 0.4 and 0.40 are equivalent. That is a very abstract concept for students to grasp through direct teaching. Instead, have students use base-ten blocks to build 0.4 and 0.40. Note that when using base-ten blocks for decimal values, we typically use the flat to represent ones, the rod for tenths, and the unit for hundredths. If they build those two numbers, how could the not see the equivalence? Have them build a few more similar pairs (0.7 and 0.70, etc.) and then have them explain why the two numbers in each pair are equivalent.
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In her book What’s Math Got to Do with It?, Jo Boaler describes traditional teaching methods as passive learning. She cites years of research that underscore how ineffective the lecture-demonstrate-practice cycle of teaching is for student learning. I love this quote: “Students taught through passive approaches follow and memorize methods instead of learning to inquire, ask questions, and solve problems.” If we want deeper understanding (and don’t we?), we must move past stand and deliver teaching.
The idea of direct teaching vs inquiry is not really a debate about small group instruction vs whole group instruction. It’s more a teaching style and philosophy. Unfortunately, direct instruction happens as often in a small group setting as in whole group instruction. The shift we need is more toward facilitating learning through thoughtful questioning and away from telling and showing students what they need to learn.
Do you notice the other common thread that is a must for learning through discovery? Students must be allowed to explore concepts with concrete materials for discovery to occur!
Can’t wait to hear your thoughts on direct instruction vs discovery!