Teaching with Mathemagic
To approximate the square root of $n$, find the nearest square below $n$. Call that $k^2$. Then $\sqrt{ n } \approx k + \frac{n - k^2}{2k}$. For example, $28$ is $3$ more than $25$, so we can approximate that
$$\sqrt{ 28 } \approx 5 + \frac{3}{2 \cdot 5} = 5.3$$
The true value is roughly $5.2915$. Not bad, eh?
Mathemagic tricks
Mathemagic tricks are amazing feats of calculation. Teachers sometimes use them as gimmicks to try and engage students, but these math stunts also have serious educational value. As a math teacher and magician, I think that value has long been squandered.
The key is the curiosity that a magic trick sparks. Take, for example, this trick for approximating square roots in your head. When I was in high school, I stumbled upon a video tutorial for this stunt. The video hand-waved the underlying explanation with a line like "it works because of linear approximations, which are calculus." That was the best line of the whole video. It got me asking why the trick worked. By the time I was through exploring, I understood linear approximations better than I ever had before. Heck, I understood lines better than I ever had.
A wrong way
Teachers definitely try to use math stunts. Several of my teachers did, and my peers have too. I've seen it with divisibility rules, quadratics, logs, integrals, you name it. Usually, it goes something like this:
- The teacher does a math thing they hope will be impressive. It's presented almost like a super power, and you too can gain the power (if you will just listen to my lesson, pleaaaaase)
- Students are somewhere between impressed and confused
- The teacher says "now we're going to learn how that works"
- The students dutifully follow along, and the trick is mentioned again at the end of class now that the students are supposed to know why it works
- Optionally, the students are then expected to "perform" the trick on paper for an exit ticket
This flow pains me for all sorts of reasons. First, students find it cringey. It's a gimmick, a transparent attempt to try and get students to care about the math. Second, it wastes whatever interest might have been there in the first place by turning the stunt into another math class just like all the rest. That is, the students are correct to find the trick superficial.
A right way
Using principles of both pedagogy and magic performance, we can craft principles to make a math trick work well in the classroom. My method launches a student-driven problem-solving session. In brief:
- Surprise requires expectations
- The unknown is daunting, but puzzles are fun
- A spoiled problem isn't interesting
- True curiosity is perpetual
1. Surprise requires expectations
A magic trick is gripping only if it violates an expectation of what was possible. The audience thinks, "that hat cannot possibly produce a rabbit," but then they are (apparently) proven wrong. This conflict immediately prompts the question, "How did you do that?"
Likewise, an engaging mathemagic trick requires a strong audience expectation, which is then controverted by the performance. Too often, students don't have the foundation to even see why the stunt is impressive. For the square root trick, students will only be impressed and curious if they already believe that calculating square roots is hard. And it genuinely is hard! Have students try, without your support, to estimate some square roots by hand. Then when you do it with miraculous speed and precision, they'll know something is up. They'll want to know exactly what is up, and exactly how you did it. Let the students be the ones to ask!
2. The unknown is daunting, but puzzles are fun
Just as a magic trick typically uses many sleights, mathemagic tricks usually rely on multiple mathematical ideas. If you have lots of time and all of those ideas are accessible for students to rediscover, then great. But realistically, just seeing the trick done leaves a huge space of possible secret methods for students to consider. That big of an unknown makes the problem inaccessible, especially under practical time constraints.
Instead, students will need a narrowed scope so that their curiosity is structured as a puzzle. For many tricks, I'll let students in on a little part of the solution, leaving the rest as a better-defined question. Be careful here not to rob them of the agency to ask; the question should be easier for students to pose, not already written on a slide. For the square root trick in a calc class, I might plot out my approximations over a graph of $y = \sqrt{ x }$ to produce a diagram that suggests, but does not spoil, the method. This is a type of "focusing" rather than "funneling," wherein you provide structure without boxing students into a specific line of thought.
3. A spoiled problem isn't interesting
"How did you do that?"
"Well, it was simple really. I had the rabbit in this bag under the table, and snuck her into the hat like so."
Well, that was anticlimactic. As soon as you know how the trick is done, it loses its magic. It becomes a show of dexterity rather than a wondrous impossibility. But sometimes, I show someone a card trick and they work really hard to figure it out. Once they know how it's done, they're still excited, and they want to see it again and try it themselves -- because they figured it out. That's part of the fun for a lot of audiences.
The same is true for a math stunt. If you, from on high, present students with the answer, they are likely to be more disappointed that excited. If, however, students figure it out for themselves, they will walk away proud. Once you have their curiosity and engagement, allow their figuring out to be the fun part! I recommend that you have students work in small groups, following practices of culturally responsive pedagogy. Come together periodically for whole-class discussions so that students can hear each other's ideas. This both centers student authority and helps students get un-stuck without hints from the teacher.
I additionally want to caution against spoiling the problem through ham-fisted sequencing. Students often try to exhibit what Alan Schoenberg terms "mathematical behavior" by using the techniques they think teachers are looking for. If you teach the required techniques the day before the puzzle, solving turns into the rote application of recent material -- just like in typical imperative teaching.
4. True curiosity is perpetual
When a spectator figures out how a card trick is done, they might want to try it themselves. That's nice, but it's not the same as when a magician figures it out: magicians will take what they've learned and see where else they might apply the same methods.
Likewise, a mathematician sees the solution to one problem as an opportunity to pose the next. Once students have figured out the trick, they will hopefully feel triumphant. That's excellent, but it need not be the end of the curiosity. Encourage students to take the next step on their own with extension problems. You can pose some as examples, but it is even better if the students act as mathematicians and keep asking questions of their own.
For the square root trick, I hope to hear students asking about how the approximation would work for other functions, or what values it does the worst with, or even how to make the linear approximation curved. These natural questions connect to more advanced ideas like Taylor series and analytic functions.
Conclusion
To use mathemagic tricks in your classroom, look for effects that connect to upcoming content. I'll be posting some here as I come across them, so stay tuned! Remember, the goal is to spark authentic curiosity that the students can satisfy for themselves. Create surprise, channel that surprise into a well-formed question, and let students explore. With any luck, they'll be the ones amazing you!
