I’ve been substitute teaching for about a month now, which has been a roller-coaster ride (the fun kind). On a few of those days, I was left a lesson or activity to facilitate, but most days I’m not so lucky. Understandably, most teachers prefer to leave substitutes with a work period (alright, let’s call it what it is – glorified babysitting). However, I really enjoy engaging with students, especially when math is involved, and so I usually can’t resist showing the students a mathematical “magic” trick that I leave as a challenge for them to figure out during the period. I now have a small collection of tried and true “tricks” that I like to pull out at the beginning of class, but there’s one in particular that kills every. single. time, no matter the age group (although it will likely need to be adapted for grades below 6; I haven’t tried it).
I offer this simple but impressive trick to future or current substitute teachers, or even as a class starter / warm-up for classroom teachers, but I take no credit for it and am only paying it forward: I was introduced to it by Dr. Egan Chernoff during his Methods in Secondary Mathematics course, who learned it in a class with Dr. Peter Liljedahl, who in turn, came across it in an issue of Vector (journal of the British Columbia Association of Mathematics Teachers). Naturally, although I will describe it below, I will not explain how it works… I leave that to you. (I thought about it… but then I remembered Chernoff calling us “answer junkies” and thought better of it.) Afterwards, I’ll reflect a bit on when I’ve found it to be particularly successful and make some guesses as to why.
I start the trick by telling the students that I’m a time traveler. (Of course, you can devise your own story.) “Yes, it’s true. And I’ll prove it to you by adding up a whole bunch of numbers really fast. ‘Cause, see, what I do is go forward in time about a minute, see the answer, then come back and write it on the board before you’re even able to punch in the first number into your calculator. Do you believe me?” (I adapt the story for higher grades so as to avoid the sting of 30 simultaneous eye rolls.)
Have the students take out their calculators. You will also need a student volunteer. Explain that he or she can write any three 4-digit numbers on the board. You won’t look at them while they’re writing, but afterwards, you’ll turn around and write two more. Then, when you say “Go!,” everyone will start adding the numbers. Make sure you tell the students to start putting the numbers into their calculators only when you say so – you will all be starting at the same time. Try not to mess up (although this can give students a good hint of what’s going on), prepare for the ensuing amazement.
Here’s an example:
1. Student writes three 4-digit numbers on the board
2. You add two numbers of your own
3. When you say “go,” the students start adding the numbers. You, meanwhile, will be able to write the answer within 2-3 seconds:
Simple. But it never fails to amaze. (I once had a Grade 9 student stand up, slowly, and ask: “What are you?”)
Of course, I don’t show the trick to the students so that they think I’m a time lord, but because it’s a fun challenge to figure out how I was able to do it. Typically, I will do the trick at least twice (often, students will ask themselves to see it several times) to give the students a bit more to go off of. (I’ve included a couple more examples at the end of this post to help you recognize the pattern, if necessary.) After this, I’ll simply give the day’s assignment, leaving the examples on the board.
And then, I simply watch. Because without fail, there is always (always) at least one student who will stare at the board for a few minutes, then copy down the numbers and try to work it out for him or herself. Sometimes, they will leave it until I give a guiding hint in the middle of class; very often, though, they simply will not be able to leave the problem alone. (When they should be working on Section 4.3, #1-12… oops.) Sometimes, they’ll recognize interesting but non-relevant patterns in the numbers that seem to offer a clue, until I ask whether the pattern holds in all of the examples. By the end of class, someone will have figured it out, and I get them to demonstrate their newfound time-traveling powers in front of the class.
Although I love to see students in any class working on this problem, what makes me most excited about this trick is that, at least in my experience, it has been most readily picked up in “stretch” or “extended” math courses… i.e., by students who aren’t normally expected to enjoy math, or who don’t think that they’re any good at it. Just this morning I was in such a class (Grade 11), and a group of boys declared that it was now their “sole purpose in life” to figure it out. All throughout the class, they persevered. They struggled. They argued. And then they solved it. Everyone clapped at the end of the period when one of them demonstrated it on the board.
But why? Why would students who seem to have no interest in math spend a class – by their own choice – trying to figure out a silly little addition trick? Well, to start, I don’t believe that there is single a student out there who has absolutely no capacity to get interested in a mathematical problem. The first key, I think, is perplexity, which according to Dan Meyer is the goal of engagement: in his words, the task is engaging because it “induce[s] in the student a perplexed, curious state.” I encourage you to try this in your own classroom one day, just for fun, and experience seeing the shock and wonder on students’ faces. (And notice that although it’s very engaging, the problem isn’t “contextual” or “real world” – I’m not trying to get the students excited by trying to wedge Snapchat or Beyoncé into a math problem. Not that the latter is necessarily a poor strategy, but, as Nat Banting writes, “I would much rather think of classroom materials as either mind numbing or thought provoking. […] A real-world context is the cherry on top.”)
The second key, as I see it, is the task’s accessibility. Although it may not be the best example of one, I do see this trick as belonging in the class of low-floor, high-ceiling tasks (also briefly discussed by Dan Meyer, among many others) – the floor certainly is low, as the majority of even the most struggling high school students understand integer addition and can look for patterns in a set of numbers, and one can think of many extensions to the trick for students who are able to go further (I’ve suggested a few below). It’s easy to understand the problem (essentially, the question is: why did I choose the numbers I chose?), satisfying to figure it out, and even more satisfying for a student who has typically struggled in math class go up to the board and solve what would otherwise be a tedious problem in a matter of seconds, in front of his or her peers. And while I’m not so naive as to think that a student’s confidence (and interest) in math can be dramatically improved within a class period, perhaps a steady drip of these kinds of experiences can help.
I hope you give this little trick a try (and thank @MatthewMaddux – or maybe not, he’ll probably tell me off for “waxing poetically”… do it anyway). Let me know how it goes.
- Do the numbers have to have 4 digits?
- Could we alternate writing the numbers? Who has to go first?
- What if the volunteer had written 4 numbers instead of 3? 2? 5? 10? n?
- How can the trick be modified so that I know the answer before any numbers are even written on the board? (I’ve done this one too, taping the answer under a student’s chair before class . Also very impressive.)
One day, after staring at the board for a while, a student concluded (in jest) that this was why I was so good at addition: