Note: This is not the most recent version of this paper. It is the only one in HTML. My plan is to get back to trying some of these things on other classes and adding more useful magical material. When I finish that, I'll update the HTML document.
Teaching is far more than injecting knowledge into others. A student isn't simply a sponge ready to absorb knowledge. A student must want to learn in order for the teacher's presentation of material to be grasped. The current trend in teaching all subjects has been toward an inquiry environment, where the teacher's role is as a facilitator in the finding of answers. In an ideal inquiry environment, students raise and then answer their own questions with the aid of a teacher that works to also rediscover the subject matter. This approach to teaching plays on people's inquisitive nature. Unfortunately, we can't always count on students to just want to learn and to just ask the right questions. What we need is a way to encourage the formation of those questions in student's minds.
A good teacher does this by entertaining students. It may be that the teacher entertains with interesting material that the students don't want to put down. Or it may be that the teacher acts as a performer in an attempt to make any material interesting. If we can mix elements of both, we can accomplish our goal of keeping students interested in, and thinking about, their work. A performance to start things off can act as a catalyst to get questions flowing, but what is performed should contain the material to be learned.
Many people in our society seem to think of math as a black art containing deep, dark mysteries that can't be understood. What better way to dispell this belief than to actually present math as magic? Or at least to present some mathematical property disguised as magic? Then follow up by answering their queries of ``How's it done?'' with an honest reply: ``I don't have any special powers. I just know the secret.'' The students I tried this on were more than willing to try to find the secret on their own. In this way, my performance grabbed them, and the ``secret'' hidden in the magic was what I wanted them thinking about.
The presentation I give is unique and keeps the interest and attention of the class. The purpose of magic has always been to leave audiences wondering what happened. A good magician doesn't usually leave enough clues for the spectator to determine what happened. That doesn't mean that a good magical presentation will prevent the viewers from constructing some explanations for what might have happened. Quite the contrary. A good magical presentation will cause the audience to find many possible explanations. The explanations don't necessarily lead to practical solutions. That is precisely what happened in the cases I've included here. The class took off with the material presented and found their own explanations, thought through and discussed what they saw, and in some cases came up with their own modifications to the effects presented, all while practicing basic arithmetic skills, and problem solving strategies. Best of all, they did it without being asked. They did it because they were interested.
Anyone daring enough to stand in front of a class can be a performer. Anyone willing to suspend reality momentarily can present magic. What I have written here are two of the effects used on a fourth grade class along with some of the responses I received from the students. Several other effects have been used over the course of several months. The two selected for inclusion here are simple, require no props, and demonstrate a fair amount about what can be drawn from students.
Having done magic for a number of years, I had to give serious thought to the idea of exposing secrets to a class of 33 students. ``A magician never reveals his secrets,'' has become cliche. What I realized though is that I wasn't revealing anything. I was performing the magic and letting their imaginations run with what I showed them. what I wanted them to learn was how to think critically about what they saw. It didn't really matter to me if they found the right answer or not.
They did find explanations for everything I showed them. In addition, they got so involved with the magic that they went and performed it for others, keeping the secrets to themselves. In fact, now having worked out the problems themselves, they know the working of the effects well enough that I feel confident they'll be able to continue doing these things for some time.
It's of little use to discuss what happened in the classroom if I don't first present the material that the class saw. The effects can be presented striaght, as nothing more than a puzzle for the students to ponder. However, as I already discussed, a more grand magical presentation lends itself to a grander sense of wonder, and I believe better questions. What follows are my presentations for a couple of effects.
Both of these are old mathematical tricks that I have witnessed performed by many magicians. I am unaware of the origin of either. They both fall into the category of mental magic. This is a realm where the magic appears to happen without the use of props, entirely under the control of the mentalists mind powers.
Under normal circumstances these can have as powerful an effect when done with one person as with a room full of people. Each person would be following the steps carefully to end up with the same result. However, since a group of fourth graders practicing arithmetic can't be counted on to find the same results, we worked as a class. One student followed the steps at the blackboard, while I faced the back of the room, unable to see what was being written. The class worked well together to check each other's answers. As an extra precaution, the cooperating teacher in the room kept an eye on the process to ensure that everything was done correctly.
``As everyone walked into class today I was trying to sense patterns in brainwaves. I feel confident enough in my ability to sense the way all of you think that I wrote down a prediction and sealed it inside this envelope.''
I hand out the envelope for one person to hold, but not open.
``I would like one person to stand at the blackboard and follow some basic directions.''
I pick from a vast selection of anxious volunteers.
``Write any 3 digit number on the board. Directly underneath it, write the number with the digits reversed. If you started with 123, the new number you wrote is 321. Now find difference between the number you started with and the new number. Once again, I'd like you to reverse the digits of this latest number. Add these last two numbers together.''
``What's left on the board is probably a big number. This number indicates a page and the place on a book in this classroom.''
I then picked someone in the class to grab the dictionary on a nearby table.
``How many digits are in the number on the board?'' When they answer, ``four,'' I tell them to use the first three digits as the page number and to use the last digit as the number of words to count down on the page. I turn to the board and instruct, ``turn to page 108, and read the ninth word on the page.''
When the word is read, I ask the person holding the prediction to open the envelope and read the contents to the class.
This effect is based on the 9's compliment property of base 10 arithmetic. We can start with any three digit number, provided that the number is different when it is reversed. When we find the difference between the two numbers we will always end up with a number for which the outer digits add to 9 and the middle digit is 9. Therefore, the second step will always yield 1089.
The principle here can be explored closely with the students. When they perform the sequence on their own several times with different numbers they will discover patterns. It can also be explored with 2 and 4 digit numbers to see what results the students find.
Turning this into a prediction makes it all seem magical since I somehow knew what word we would find in the dictionary. Having the end result be a word in some book in the back of the room adds another level of mystery until this problem is explored. Once everyone realizes that we always end up with the same number, they can conclude that I looked in the dictionary ahead of time.
It is probably of the greatest value for a teacher doing this to work out the explanation of this principle, however a brief explanation is provided.
To find determine why adding the outer digits always results in 9, let's look first at a simplified, but generic case with only a two digit number, ab, where a and b each represent a single digit. We write the number, reverse it, and find the difference:
ab -ba ___ cd
We would write the equation such that ab > ba. Then we know that d =10 +(b -a)= 10- (a- b), and c= (a-1) -b =1+ (a- b).
If this isn't obvious, remember that to subtract a from b we need to borrow from the ten's place. We end up subtracting a from 10 +b. That also explains the 1 in the calculation of c.
I suggested above that adding the outer digits of the solution, in this case c and d, would produce 9, as we can see here:
10-(a -b) - 1+(a -b) __________ 9
The idea is the same for a three digit number. The only limitation on this is that we use a number that is different when it is reversed. This opens up another question that may be fun to answer. If we wanted to test this exhaustively for all 3 digit numbers to make sure it worked, how many tests would we have to do?
``I am going to plant a thought in your minds. This is a large class, so I'm not sure I can get it exactly right. One or two of you may get a slightly different picture than the others. Work with me and concentrate so my thoughts can be planted in your mind.''
``Pick a number. Any number will do, but a small one may be easiest. Write that number on the blackboard. That's not actually the number we're going to use though. We're going to use that to find another number that's different from what all of us were thinking of to begin with.''
``Double the number you've selected. Add the number 8. Divide the new number by two. Subtract the number you started with from the number you have now. Find the letter in the alphabet that corresponds to that number. That is, if you have a 1, the letter you should be thinking of is `A'. If the number you have now is 2, use the letter `B', 3 is `C', 4 is `D', 5 is `E'. Continue through the alphabet until you reach your number.''
``Think of a country whose name starts with the letter you are at. Take the second letter in the country name, and find an animal that lives in that country. Think of the color of that animal.''
``Now, if everyone has been concentrating, you should all be thinking of the same animal as me. But actually, I think something went wrong. I'm sensing that you're all thinking of the same animal, but what I'm sensing doesn't make much sense. There are no gray elephants in Denmark.''
For a bit of variation we may want to reach the same result by a slightly different method.
``Pick a number. Multiply by 9. Add the digits together. If the result of adding the digits contains more than one digit, add the digits again. Continue doing this until you have a single digit number. In other words, if the answer you ended up with was 98 when you multiplied, you would add 9 and 8 to get 17. You would then add 1 and 7 to get 8. Now subtract 5 from the number remaining.''
As above they would be instructed to find the letter in the alphabet corresponding to their number and continue as earlier to end up with gray elephants in Denmark.
The steps of method 1 can be written out algebraicly. Let's use n for the number selected. Then, written more concisely, the result is obtained with
2n+ 8 _____ 2 -n.That simplifies to
n +4- n.The result is clearly 4.
When done quickly there isn't much time to think about a country. In fact, if you give it some thought you'll realize that there aren't very many choices. ``Denmark'' is chosen most often. The same goes for animals starting with ``E''. That doesn't mean others won't arise. Sometimes they do. and sometimes mistakes are made. But you can usually cover for the error by indicating just how close you came. In fact, often the audience is more impressed when you miss. Near misses indicate that your method isn't fool proof, but that you do know what you're doing.
The second method presented is based on 9's compliment. it's a simple rule of the base 10 number system that when you multiply a number by 9, the result of the added digits is also 9. You can now subtract any number you like in order to have the audience using any letter from ``A'' through ``I''.
When I presented this in class, I used both methods, on after the other. I wanted the students to conclude that I had used a mathematical trick, but by changing methods, I made the path to finding an explanation a bit more challenging. They didn't realize at first that I used different methods. I also didn't finish with elephants in Denmark, but with viewing Neptune from England. Once the basics are understood, you can end with any number you like.
These effects were presented as lessons on fourth graders in Jefferson Road Elementary School. The school houses approximately 600 students in grades K-5. It is one of five elementary schools in the Pittsford Central School district. The school curriculum is traditional. Math classes in this school are fairly structured and much of the content of the regular math lessons is based on a math textbook. As a result, curriculum-driven, teacher-centered instruction is the norm.
Most of the students are familiar with non-traditional, student-centered, inquiry-based instruction in math due to the presence of a ``math lab'' in the school. The math lab is available to all students on a voluntary basis during their lunch period. In addition to this, the teacher of the class discussed here likes to teach a Fun Friday lesson at the end of each week. This lesson is intended to be more enjoyable and less structured than their typical lessons. The lessons discussed here were used to bring inquiry into the classroom during these Fun Friday periods.
Having an arsenal of magic to throw at a class is wonderful, but what does it accomplish? Most notably, it had students concentrating on mathematical skills in a way that interested them. In these examples, they weren't doing arithmetic because they had a list of problems to get done for a grade. They were doing it because it was posed as an interesting problem. A problem that they formulated themselves from the presentation they saw. Paper and pencil calculations, as well as calculator use was acceptable. This provided a means for them to check each other's answers. They worked with other students around them to discuss solutions, practiced basic arithmetic skills, learned more about the base ten number system, and drew their own conclusions about the relationships between numbers. Most importantly, they established their own goal. They saw something that was ``magic'' and they wanted to know how it worked. That question was on their minds before we ever started discussing a solution.
I started with the dictionary prediction. Not surprisingly, the students didn't know how to approach their initial question of how the effect works. As far as I am aware they had no previous experience debunking psychic demonstrations. Rather than providing scaffolding right from the beginning, I suggested we try to repeat this. We picked another number, followed the same steps, and came up with the same answer. The class was surprised by this, and several suggested we try different numbers until we get a different answer. We tried again. After a few attempts, one student proudly declared that the trick works because we always get the same answer when we follow those steps. Heads shook in agreement. Many seemed satisfied, but some called out questions along the lines of, ``but why do we always get the same answer?'' I was quite happy to see that after they answered their question, they chose on their own to probe deeper.
Looking around the room, I could see that many of the students were still scratching at their papers trying new numbers. After a few minutes went by, one student announced that it didn't always work. He found a number that just gave him 0. He told me his number and I worked it out with the class at the blackboard. I did not immediately write in my notes what the number was, but at the completion of the lesson I did recall that it was a palindrome with a middle digit of 0. The students wanted to know if the 0 in the middle was the reason the trick failed. In answer to that question, we tried a number of my choosing with 0 in the middle. It worked for me. I soon realized that my choosing a number wasn't a good idea. Students suggested that since it was my trick I picked a number that I knew would work. I did know it would work, but not because I memorized a set of numbers that we could use. Up until this point, things were going well without my active participation. There seemed little reason to jump in now, as long as they were still progressing. With that realization, I stepped back and continued to write only what they asked me to write. It wasn't long before they decided that the numbers that weren't working were ones with the same first and last digit. Interestingly, it took a bit longer to conclude that the problem was connected with the number being the same when it was reversed. I had to ask prompting questions like, ``Why does it matter if the first and last digit are the same?''
We discussed this for a while, and I asked more questions than I answered. It was necessary for me to suggest that they look at the numbers at each step, rather than do the whole series of steps each time. I started the sequence again with one of the numbers we had before, and stopped after doing the subtraction. I immediately started on another three digit number without completing the first. ``does anyone see a pattern?'' This question started them comparing the numbers. Almost immediately, one student pointed out that in every problem we did, the middle digit was nine. Others apparently looked for more nines and found that the outer digits added to nine.
At each step of the way, as we progressed, the students found it necessary to test their theories. They checked numbers by hand and with calculators. Despite the fact that we had success in explaining things so far, the students preferred to prove this rule empirically. It was actually difficult to stop them from trying it endlessly. At last I asked a question that I thought would put an end to their search for a number that would break the rule: ``How many numbers would we have to test if we wanted to try all of the three digit numbers that are different when we reverse them?'' Various answers came from them. ``1000.'' ``900.'' ``999.'' ``10.'' Each answer called out could be explained by the students that suggested them. We were out of time and I gave them that question to answer for homework. I didn't expect a complete or accurate answer from any of them. I did expect some creative answers. I received many interesting answers, along with pages full of numbers that they tried. They wanted to prove to me that this really works on everything that they could try. Some even apologized for running out of time and not testing more numbers. Checking numbers like that isn't what I intended for them, but seeing the group this enthusiastic over math was exciting.
The best part of this is that I know they took what they learned out of class with them. Most of them tried it on friends and family members and reported back how successful they were. The only significant difficulty is that I concentrated on the mathematical aspect of this and forgot to tell the students that not all dictionaries would have the same words on page 108.
Far more could be explored just from this problem. I decided however, to abandon this particular item, and pick up a different trick the following week. Had I pursued this, I would have spent more time discussing the property of nines that allows this to work. I would have worked with the students to develop ways of using this property as a shortcut for doing arithmetic. We spoke some about the difficulty of testing every number and the reason why it's better to prove this is true without trying it. However, to satisfy their curiosity, it would not have been difficult to test all numbers on a computer. As a further exercise, I would have liked to try this with two and four digit numbers to see if the students could develop more general explanations for what they saw.
The second effect was received as well as the first. In fact, for a week I had been answering the question, ``When will we do more magic?'' I presented the effect using both methods described here, we discussed it, and then we repeated it. Having already been through this experience they already knew what was expected and they started offering explanations. The simple algebraic representation that this problem has is of no value to students that haven't yet been exposed to algebra. This proved to be a more difficult problem for the students to crack than the first. As before, they approached it from an empirical standpoint. In this case however, I pointed out that I would let them use any number they wanted. Therefore, no matter how long they tried testing numbers, they couldn't test everything. Many still expressed a belief that eventually they would find a number that didn't work.
I provided one hint for them. I suggested that I could make them come up with any letter of the alphabet. That was the key to cracking this puzzle. ``What happens if you use a different number?'' asked one student. We proceeded to try different numbers in place of the 8 in the initial instructions. Over time, other things came out, like we were multiplying by two and dividing by two. Eventually it was concluded that the number we would end with would be half of the number we add. The entire process, as with the dictionary prediction, fit within one 45 minute period. We did not have an opportunity to try adding odd numbers, or multiplying and dividing by numbers other than two. Instead, I let this lesson also rest at the end of the period, allowing them to explore further on their own and to try it on friends and parents. As I expected from the previous experience, the students enjoyed the exercise enough that they were happy to report back how their attempts and being magicians worked out.
The most important thing to be learned from this is that students will pick up and explore material that interests them. I went into these lessons with ideas of things I wanted to explore. In some cases I got to cover the material that interested me. In other cases, I had to completely change my way of thinking to match the students and to look at what they wanted to look at. Any time I chose to ask my own questions, rather than follow theirs, I felt that I was losing their attention.
Magic is a wonderful way of making math come alive for students. It allows for a true inquiry environment for learning. The questions are built into the material, and students are happy to explore them on their own.