to demonstrate Conceptual and procedural understanding
and the implications for Teaching and Learning
Magic or Mathematical?
View the video and decide if it is magic or can explained by slight of hand or with mathematical knowledge to explain how to perform it.
What you think!
Magic? Slight of hand?
The performance of the trick can be explained in two general ways; conceptual and procedural.
As you continue your quest to understand the trick and its relationship to teaching, there are two things to focus on:
- How does the trick work? (content - knowledge).
- What is the relationship of conceptual and procedural knowledge to good pedagogical practice? (instruction and learning).
The table below includes two sets of directions for teaching how to perform this card trick.
While both kinds of knowledge are important; the research strongly suggests it is important to first learn an idea conceptually and then procedurally. The research further suggests that once students understand an idea conceptually that they are able to invent their own procedures and have a better understanding and flexibility in using those procedures for solving problems.
To illustrate differences between a conceptual and procedural orientation lets suppose a teacher wants to teach this card trick. Her instruction will be closely related to one of the two paths of instruction: The path on the left procedural or the path on the right conceptual.
Procedural orientation Conceptual orientation The primary goal of this orientation is to teach or learn a procedure that give efficient and accurate results all of the time. The primary goal of this orientation is to teach or learn how card tricks are designed and performed. How mathematical concepts can be used to design and explain some kinds of card tricks. For the card trick an outcome or objective would be to perform the card trick (3 X 7 Card Trick). For the card trick an outcome or objective would be to design and explain how the card trick workds (3 X 7 Card Trick) . Note: An outstanding teacher would want her students to be able to both design, explain, and and perform the trick, but these two examples are to illustrate differences between conceptual and procedural knowledge. Again both are important. The teacher starts with instruction similar to the procedure on the 3X7 Card Trick video.
The teacher would start by asking student if they wouldlike to know how to design and perform card tricks. Then ask them to watch as she performed a 4x4 Card Trick to see if they can tell how she knows the selected card. 4x4 card trick video.
After students discuss their ideas and conclude that a process of elimination or identification can be used by the magician to know what card was selected, the teacher would challenge the students to extend that understanding to other dimensions of card displays. For example: ask if their ideas and processes could be used on 5 x 5 array card trick, or 6 x 6, or on any size square array of cards.
After students are confident that any card in a square array of cards can be identified by the same process, and they have communicated conjectures and support for the confidence of their answer, the teacher can ask if a card trick could be done so the one card, serepticiously selected, in a million by million array, that's a total of 1 000 000 000 000 cards could be identified by asking only two questions? Yes, but, who would want to deal the array?
Then students could be asked about rectangular arrays.
Will the same procedure work for an array of cards that is four columns with three rows?
After students have discussed their conjectures for the possible outcomes they can demonstrate and discuss their conjectures for the 4 x 3 array (short rectangle) card trick.
Following these activities, students should know the procedure works for arrays that have equal columns and rows (square arrays) and for arrays that have more columns than rows (short rectangle arrays), but not for rectangles that have more rows than columns (tall rectangle arrays).
However, some will also realize that to identify or eliminate all cards in a tall rectangle array, they could redeal and ask additional questions for rectangles with more rows than columns (tall rectangle). When they understand that idea, challenge them with a tall rectangle arrays (arrays longer than wide or have more columns than rows) card trick.
After sharing and discussing their tricks their level of understanding can be assessed by performing a modified tall rectangle array trick. Example see this 3 X 7 modified card trick similar to the procedure in this video to do and ask students to explain how they could do the trick.
After completion of these activities students can be shown the 3 x 7 card trick at the top of this page and asked to explain how it works. The trick could be done by the teacher or maybe a student would have already extended these ideas and created this card trick or a similar one that he or she could demonstrate to the class and challenge them to discover how it works.
The teacher might assess the students with multiple choice questions or fill in the blank similar to the following:
- How many columns of cards are dealt? (3)
- How many rows of cards are dealt? (7)
- How many cards in all are dealt? (21)
- How many times were the cards dealt? (3)
- How many times was the person that selected the card asked what column the selected card was in? (3)
The teacher might also choose to use performance based assessment and ask the students to perform the trick.
Students could go home and perform the test for their families and their families would be impressed with their accomplishments and the teacher's ability to teach.
The teacher wouldn't ask how or why the trick works, because that wouldn't of prime importance.
The students wouldn't be asked to create their own procedures, because they wouldn't be viewed as less efficient and therefore, not important.
The teacher has isolated the trick and taught the traditonal efficient procedue, but it is unlikely students will understand why it works or connect it with other mathematical ideas.
Assessment is on going authentic assessment that supports students' self assessment which is continually necessary for the student and teacher to know what questions to ask or ideas to think about and try to facilitate understanding. This assessment information must be more than right or wrong answers. It must be sufficient to provide adequate information for the teacher and student to make appropriate decisions to learn. This elevates the importance of the assessment process as well as the conceptual understanding of the mathematical ideas. With out that understanding the student and teacher must retrace their steps to where they last understood and try to continue from there. That is a realistic mathematical environment created for the purpose of understanding and justifying explanations to problems that people have a desire to solve and share with each other.
I think Dolk and Fosnot would call it "mathematizing".
Dr. Robert Sweetland's Notes ©