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## Procedural versus Conceptual Knowledge

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**Invert and multiply.This is ProceduralKnowledge.Knowledge**about an algorithm, that is a procedure to compute an answer.**A story about Procedural knowledge.**Told by researchers About 5th Graders in North Carolina Who had learned well how to invert and multiply**After lunch on a day that the cafeteria served pizza, the**researchers posed a question. What is one-half of a pizza divided by one-fourth of a pizza?**The students inverted and multiplied and said that the**answer was 2. “Two what?” the researchers asked. After some thought and discussion, the student responded, “2 pizzas”.**What do you mean, 2 pizzas.**Drawings were made of two whole pizzas. How can that be? We divided 1/2 of a pizza by 1/4 of a pizza. Where did all that pizza come from. After much thought the students responded, “ We don’t know where the pizza came from, but we know the answer is 2.**The students’ misunderstanding occurred because they only**had the Procedural knowledge--Invert and Multiple. They did not understand what they were doing at the conceptual level.**They have not connected the algorithm with**real/concrete/pictorial situations. They lacked the “conceptual knowledge” to understand.**Fraction Circles**1/4 1/2**1/4**1/2**Benefit of Conceptual Knowledge**You UNDERSTAND the math that you are doing. Students should always have conceptual knowledge before they learn an algorithm.**Part 2 Another Story**Sometimes I reminisce about my high school days. What comes to mind is Geometry Class and a formula I learned.**The Formula:**(N-2) * 180o = the sum of the interior angles of a polygon. An Example: (5-2) * 180o = 3 *180o = 540o**I remember the teacher showed me this formula.**I remember it was in our textbook. I remember that I remembered it for a test. I remember that I forgot it soon afterwards. I forgot it because I never understood it, I only memorized it.**How Could I Have Learned It Better?**• Explore • Investigate • Conjecture • Discover • Construct • Describe