# Conceptual & procedural knowledge: Understanding them & the implications for teaching & learning with card tricks

## Introduction

This article uses card tricks to demonstrate the difference between conceptual and procedural mathematical knowledge and how teaching conceptually is different than teaching procedurally.

## Magic or Mathematical?

View the video and decide if it is magic or it performance can be explained by slight of hand, or with mathematical knowledge!

### Introductory card trick

What do you think?

• Magic?
• Slight of hand?
• Mathematical knowledge?

The performance of the trick can be explained in two general ways; conceptually and procedurally.

As you continue your quest to understand the trick and its relationship to teaching, there are two things on which to reflect:

1. How does the trick work? (content - knowledge).
2. What is the relationship of conceptual and procedural knowledge for pedagogical practice? (teaching & learning).

The table below includes two sets of directions, which could be used for teaching about the card trick.

While both kinds of knowledge are important; the research strongly suggests it is important learners are first experience an idea conceptually and then develop procedures to use the information.

The research further suggests that once learners 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 instructional orientation, lets suppose a teacher wants to teach this card trick.

Their instruction might be more closely related to one of the following two paths of instruction. The path on the left procedural or the path on the right conceptual.

Instructional focuses
Procedural orientation Conceptual orientation

The primary goal of this orientation is to teach or learn a procedure so the learner can perform the trick efficiently and accurately.

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, explain, and perform some kinds of card tricks.

Outcome or objective:
Perform the
3 X 7 Card Trick.

Outcome or objective:
Design, explain, and perform card tricks similar to the 3 X 7 Card Trick.

Note:

Outstanding teachers want students to have both kinds of knowledge.
The two examples are to illustrate differences between conceptual and procedural knowledge.
Again both are important.

Procedural orientation Conceptual orientation

The teacher starts instruction by demonstrating the card trick. s

Similar to the procedure in the video:

Then ask if they would like to learn how do do the trick. Or tell them they are going to learn how to do the trick today.

Then walk through the trick like in the video:

After presenting the demonstrations ask some key questions to check for understanding, like:

• How many cards are dealt? 21
• How many columns? 3
• How many cards in each column? 7
• How many times are the cards picked up and dealt? 3
• How are the cards picked up? like a sandwich with the selected row as the cheese.

Then have students pair up and practice. Alternating two times for each person.

The teacher would start by asking students if they would like to know how to design and perform card tricks.

Then, ask them to watch as she performs a card trick, and asks them to to see if they can tell how she knows the card they select.

I like to start with one where there is usually a person who can explain how it's done. Like ...

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, sometimes there are objections as it was too easy to see how it was done.

That is great as it provides an opportunity to explain magicians will then use distraction to hide the obvious to get that wonder or amazement from their audience.

The challenge.

How to extend their understanding to use what they know and make what is happening less obvious.

Suggest they explore different dimensions of the matrix or array of cards.

For example: ask if their ideas and processes could be used on a 5 x 5 array card trick, or 6 x 6, or on any size square array of cards. With the idea, more cards might make it less obvious.

However, 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 it would be possible that a card trick could be done so that one card, surreptitiously 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 that array?

So making the array or matrix larger might not be the best way to distract.

Will the same procedure work for an array of cards that is four columns with three rows?

What about 7 columns and 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.

Then share the 7X3 card trick and have them explain how it works.

Following these activities, have students summarize what they know the how tricks with different arrays work:

• arrays that have equal columns and rows (square arrays), work with two questions.
• Arrays that have more columns than rows (short rectangle arrays), work with two questions.
• arrays that have more rows than columns (tall rectangle arrays), don't work with two questions.

However, some will also realize that to identify or eliminate all cards in a tall rectangle array, they could re deal and ask additional questions for rectangles with more rows than columns (tall rectangle).

When they understand that idea, challenge them with a tall rectangle array (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. By not moving the cards to the top, but adding an additional level of distraction, by moving the known cards to the middle.

Like the introductory 3X7 card trick video

After viewing the trick, they may need a small hint that instead of putting the cards at the top of the rows, putting them in the middle hides what the magician is doing to distract the audience.

When they are confident in their knowledge of how the 3X7 card trick works, have them create a procedure so they can do the trick quickly and efficiently.

Their procedures could be something like in the 3X7 card trick video.

Assessment Assessment

The teacher might use multiple choice questions or fill in the blank similar to the following:

1. How many columns of cards are dealt? (3)
2. How many rows of cards are dealt? (7)
3. How many cards in all are dealt? (21)
4. How many times were the cards dealt? (3)
5. How many times was the person that selected the card asked what column the selected card was in? (3)

Then use a performance based assessment and have each student perform the trick.

Learners could go home and perform the trick 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 isn't the outcome.

The learners wouldn't be asked to create their own procedures or trick, because that would be viewed as less efficient and, therefore, not important.

The teacher has isolated the trick and taught what could be considered a traditional efficient procedure, but it is unlikely students will understand why it works or connect it with other mathematical ideas.

Assessment here is also 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.

Assessment information 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".

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