Conceptual and Procedural Card Tricks to demonstrate Conceptual and procedural understanding and the implications for Teaching and Learning

Card Tricks
to demonstrate Conceptual and procedural understanding
and the implications for Teaching and Learning

Magic or Mathematical?

I invite you to view this video and see if you believe magic, slight of hand, or mathematics is being used to perform this card trick?

View the video and see what you think!

Magic? Slight of hand?
or
Mathematics?

This video along with some additonal videos of card tricks are to demonstrate the pedagogical differences of conceptual and procedural knowledge and instruction. While both kinds of knowledge and instruction are important; the research strongly suggests that it is important to first learn an idea conceptually before it is learned 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 mathematical problems.

As you continue your quest there are two things that you are going to do a lot of thinking about.
1. What kind of trick is it and how does it work (knowledge).
2. What is the idea of conceptual and procedural knowledge
and their relationship to good pedagogical practice (instruction and learning).

To illustrate differences between a conceptual and procedural orientation lets suppose a teacher wants to teach a card trick to her students. She might select one of the following two paths of instruction for learning. The path on the left Procedural or the path on the right Conceptual.

This table illustrates ideas for each orientation.

Procedural orientation

Conceptual orientation

The primary goal of this orientation is to teach or learn a procedure that will provide efficient results. The primary goal of this orientation is to teach or learn the understanding of mathematical concepts or ideas.

With the example of the card trick it would be the student's and teacher's objective, or outcome, to perform the targeted trick (3 X 7 Card Trick). With the example of the card trick it would be the student's or teacher's objective, or outcome, to understand how the targeted trick (3 X 7 Card Trick) works.

Note: Normally a teacher would want students to understand and perform the trick, but these examples are being given to illustrate differences between the two. Both are important.

The teacher starts with instruction similar to that on the 3X7 Card Trick video (see video).
The teacher would start with instruction similar to that on the video, but with a the 4x4 Card Trick instead. (see video 4x4 card trick)

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 what they learned. One such challenge might be to ask if the same 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 even one card in a million by million array of cards, that's a total of
1 000 000 000 000 cards could be identified with it two questions? But, who would want to deal that array?

Then students could be asked about rectangular arrays. Will the same procedure work for an array of cards that is four columns by 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, they should 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 they, can be challenged to create a trick for tall rectangle arrays (arrays longer than wide or have more columns than rows).

After sharing and discussing their tricks their level of understanding can be assessed by performing the 7 X 3 modified card trick (see video ) and asking 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 the 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 ongoing 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".

Robert Sweetland's Notes ©

Procedural orientation	Conceptual orientation
The primary goal of this orientation is to teach or learn a procedure that will provide efficient results.	The primary goal of this orientation is to teach or learn the understanding of mathematical concepts or ideas.
With the example of the card trick it would be the student's and teacher's objective, or outcome, to perform the targeted trick (3 X 7 Card Trick).	With the example of the card trick it would be the student's or teacher's objective, or outcome, to understand how the targeted trick (3 X 7 Card Trick) works.
Note: Normally a teacher would want students to understand and perform the trick, but these examples are being given to illustrate differences between the two. Both are important.
The teacher starts with instruction similar to that on the 3X7 Card Trick video (see video).	The teacher would start with instruction similar to that on the video, but with a the 4x4 Card Trick instead. (see video 4x4 card trick) 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 what they learned. One such challenge might be to ask if the same 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 even one card in a million by million array of cards, that's a total of 1 000 000 000 000 cards could be identified with it two questions? But, who would want to deal that array? Then students could be asked about rectangular arrays. Will the same procedure work for an array of cards that is four columns by 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, they should 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 they, can be challenged to create a trick for tall rectangle arrays (arrays longer than wide or have more columns than rows). After sharing and discussing their tricks their level of understanding can be assessed by performing the 7 X 3 modified card trick (see video ) and asking 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 the 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 ongoing 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".