[Confab] Mind Reading & Math

Scott Hills:

I give out 5-6 sets of three dice. I have the students roll them and then add up all the numbers which cannot be seen (bottom, middles and middles). Once they have the sum, they sit back with the dice still stacked and I “read their minds” to get the sum.

Kate Nowak:

So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24. They were astonished!

These moments seem infinitely preferable to just leaping into an explanation of the sums of arithmetic sequences.

Our friends who are concerned with cognitive load should be happy here because students are only accessing long-term memory when we ask them to roll dice, write down some numbers, and add them. It’s easy.

Our friends who are concerned that much of math seems needless are happy here also. With The Necessity Principle, Harel and his colleagues described five needs that drive much of our learning about mathematics. Kate and Scott are exploiting one of those needs in particular:

The need for causality is the need to explain – to determine a cause of a phenomenon, to understand what makes a phenomenon the way it is.

[..]

The need for causality does not refer to physical causality in some real-world situation being mathematically modeled, but to logical causality (explanation, mechanism) within the mathematics itself.

Here are three more examples where the teacher appears to be a mind-reader, provoking that need for causality. Then I invite you to submit other examples in the comments so we can create a resource here.

Rotational Symmetry

Here is a problem from Michael Serra’s Discovering Geometry. No need for causality yet:

141029_1lo

But at CMC in Palm Springs last weekend, Serra created that need by asking four people to come to the front of the room and hold up enlargements of those playing cards. Then he turned his back and asked someone else to turn one of the cards 180°. Then he played the mind-reader and figured out which card had been turned by exploiting the properties of rotational symmetry.

Number Theory

The Flash Mind Reader exploits a numerical relationship to predict which symbol students are thinking about. Prove the relationship.

141029_2lo

Jehu Peters:

Here is a little trick I like to call calculator magic. You will need a calculator, a 7-digit phone number and an unwitting bystander. Here goes:

Key in the first three digits of your phone number
Multiply by 80
Add 1
Multiply by 250
Add the last 4 digits of your phone number
Add the last 4 digits of your phone number again
Subtract 250
Divide the number by 2
Surprise! It is your phone number!

Sander Claassen:

A nice trick is this one with dice. A lot of dice. Let’s say 50 or so. You lay them on the ground like a long chain. The upward facing numbers should be completely random. Then you go from the one end to the other following the following rule. Look at the number of the die where you’re at. Take that many steps along the chain, towards the other end. Repeat. If you’re lucky, you already end up exactly at the last die. You’ll be a magician immediately! But usually, that isn’t the case. What you usually have to do, is take away all those dice which you jumped over during the last step. Tell them that that is “the rule during the first round”. Now the actual magic begins. You tell the audience that they can do whatever they want with the first half of the chain. They may turn around dice. Swap dice. Take dice away. Whatever. As long as they don’t do anything with the second half of the chain. [If you like risks, let them mess up a larger part of the chain.] What you’ll see, is that each and every time, they will end up exactly at the end of the chain!

Isabel Wiggins:

A few years ago, I found this “trick” on a “maths” site, not sure which, but it was UK. You need 5 index cards. Number them 1, 2, 3, 4, 5 in red ink on the front. On the reverse side, number them 6, 7, 8, 9, 10 in blue ink. Be sure that 1 and 6 are on opposite sides of the same card…same with 2 and 7, etc. Turn your back to the group of students. Have one of the students drop the 5 cards on the floor and tell you how many cards landed with the blue number face up (they don’t tell you the number, just “3 cards are written in blue”). Tell them the total of the numbers showing is 30. The key is that each blue number is 5 more than its respective red number. Red numbers total 15. Each blue number raises the total by 5. So 3 blue numbers make it 15 (the basic sum) + 15 (3 times 5). Let them figure out how you are using the number of blue numbers to find the total of the exposed numbers.

Expressions & Equations

I ran an activity with students I called “number tricks.” (Okay. Settle down. Give me a second.) I’d ask the students to pick a number at random and then perform certain operations on it. The class would wind up with the same result in spite of choosing different initial numbers. Constructing the expression and simplifying it would help us see the math behind the magic. (Handout and slides.)

141028_1lo

Kate Nerdypoo:

I do something called calendar magic where I show a calendar of the month we’re in, ask the students to select a day and add it with the day after it, the day directly under it (so a week later), and the day diagonally to the right under it, effectively forming a box. Then I ask them to give me the sum and I tell them their day.

Always a bunch of students figure out the trick, but the hardest part is writing the equation. Every year I have students totally stumped writing x+y+a+b. It’s really a reframing for them to think about the relationship between the numbers and express that algebraically.

Finally I ask them to write a rule for three consecutive numbers, but I don’t say which number you should find and inevitably someone has a rule for finding the first number and someone has one for finding the middle number. I love that!

Different Bases

Andy Zsiga suggests this card trick involving base 2.

Call for Submissions

Where else have you seen mind-reading lead to math-learning? Are there certain areas of math where this technique cannot apply?

2014 Oct 30. Megan Schmidt points us to all the NRich tasks that are labeled “Card Trick.”

2014 Oct 30. Michael Paul Goldenberg links up the book Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks.

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

25 Comments

  1. Google “The Magic Gopher” and have fun – especially with middle school students, who initially think your mind-reading gift is amazing… then show they are fully capable of figuring out the mathematics behind the magic!

  2. Dunno if this counts, but somewhere around when we learn the pythagorean formulas, I usually let the kids tell me the square of a two digit number with a calculator, and wow them with a nearly instantaneous revelation of their original number.

    Ex: They say 1369, and I say 37.

  3. I used [a variation of] the Three Swap with my students, eventually moving to an Excel spreadsheet (recreated here in Google Drive) to try really big or really small or negative numbers. For the visual students, a desmos graph shows how the end result doesn’t change, even when their number does. Then I had them design their own to show their families. A few came back the next day, proud of their “magic”.

    The Magical Triangle Theorem served some of this purpose, though that was more for the sake of drama (which earned me some points that day).

  4. I like most of these.

    Consider what information we are conveying with these types of actions:

    We convey an implication exists: there is some fact presented that implies the answer.

    And we can avoid being overly wordy by *showing* that the implication exists, instead of leaping towards the polished precise sometimes arcane symbolic expression.

    So in that respect, we could look for these opportunities whenever we have a concept that boils down to If P then Q, and Q might be just beyond the initial obviousness. Or P and Q are overly wordy.

    But we also convey something a little dangerous: that the math expert (the teacher) has a trick, and knowing the trick solves problems. Does this lead the student to construe math as the magic spellbook that must be memorized?

    I think these examples do not necessarily lead to that spellbook idea in the student’s mind, but lets examine how we might avoid it:

    (1) have students perform “the trick” for other students. This could be tricky as your exemplars must learn it also… we don’t want to have to do direct instruction for those few. A best case could be if we notice a student who gets a concept… then we pounce on it and have them do the performance.

    In Kate Nowak’s example: we know we’re going to do an arithmetic sums activity. Can we prime certain students to notice/develop Gauss’ method, then during work time, wait for opportunities to pounce? “Catch them being smart” and then draw out their knowledge in the “Mind Reading” style performance?

    (2) find places where “the trick” is worthwhile but that confirms students intuitions. A trick that produces results that students would immediately see, and even defend, as accurate.

    With the arithmetic sums, I think students are close to this: they may rely upon calculators to check the answers, but they may also see the value that Ms. Nowak has done it quicker and more efficiently.

    I remember something Kate posted years ago: students were bouncing a ball off the wall, and were quick to make conjectures about “where it would bounce” based upon the incident angle. Someone experienced can “call their shot” — they’re noticing the implication involved in the situation — and it can be immediately confirmed by the real world action.

    I like the idea… I’ll be thinking of some more examples…

  5. I use an ESP activity on the day I plan to introduce composite functions; it’s similar to your “number trick” example above. Blog post with video explanation – http://mathcoachblog.com/2012/10/18/composite-functions-and-e-s-p/

    Also, this card trick I did recently in class hooked many kids, and leaving them wanted to know how it worked. There a fun discussion of transformations to be had – that two cards dealt face down is equivalent to two cards swapped in my hand and then dealt: http://mathcoachblog.com/2014/09/24/class-opener-day-17-time-for-a-card-trick/

  6. Modular arithmetic trick: Give me a perfect cube between 0 and a million, and I’ll immediately tell you its cube root.

    If you’re much better at listening you can actually go up to a billion, but it’s pretty hard. Art Benjamin has the details: https://www.math.hmc.edu/~benjamin/papers/Cubing.pdf

    For more combinatorial classes I love the Fitch Cheney card trick, which requires two performers who know what’s going on. Audience member hands me five cards. I lay four of them out on a table in some order, then hand one card back to the audience member (who olds onto it). Assistant enters the room, looks at the four cards, and says what the fifth one was.

    How is this possible? There are only 24 ways to arrange four cards (and I’m not using any upside-down trickery), but there are 48 other cards in the deck that could be the answer, so it seems like there’s not enough information. Good writeup here: http://www5.spelman.edu/~colm/fitch.pdf

  7. Not so much a magic trick, but I feel it relates. When we have learned a particular method in a topic that isn’t particularly efficient, I race them to complete a problem. When students see that I can complete it in a fraction of the time using some more efficient method, they get curious as to how I’m doing it. When 1 student figures it out, I let them race to prove it’s not just because I’m a math teacher. For (a weak) example: while student’s use a table to graph and equation, I identify slope and y-int and graph it before they can even draw the table. It makes them hungry to know the new, faster, cooler method.

  8. Art Benjamin has a lot of cool tricks. In the mathemagician ted talk he squares large numbers, with some practice anybody can do any two digit numbers using some algebra of perfect squares.
    (a+b)^2= a^2+2ab+b^2

    Example: 53^2 is (50+3)^2=50^2 + 2(50)(3) + 3^2
    The quick way to do that in your head is square the first digit 25 put a 0, if necessary, then square the second digit 9, 2509 then double one of the digits and multiply it by the other and add it to your middle two digits 5*2=10 ->*3=30 2509+300=2809 (or multiply and double, your choice)

    Try 84^2

    (64)(16)+(6)(4)=7056

    I’ve done this one with my algebra students leading into the perfect square trinomial.

  9. I’ve used an activity in the past that has 5 cards, 16 numbers on each card, ranging from 1 to 31. One chooses a number and says which cards that number is on, and I can tell them their number. In addition to the mathematics behind it, there are lots of great patterns to consider with the cards too. One could scale down or extend the problem too.

    I couldn’t find my original source, but found this: http://www.mathmaniacs.org/lessons/01-binary/Magic_Trick/
    Unfortunately the title of the page gives too much away, but it’s worth trying to figure out what’s going on before reading the “Why does it work?”

  10. I do something called calendar magic where I show a calendar of the month we’re in, ask the students to select a day and add it with the day after it, the day directly under it (so a week later), and the day diagonally to the right under it, effectively forming a box. Then I ask them to give me the sum and I tell them their day.

    Always a bunch of students figure out the trick, but the hardest part is writing the equation. Every year I have students totally stumped writing x+y+a+b. It’s really a reframing for them to think about the *relationship* between the numbers and express that algebraically.

    Finally I ask them to write a rule for three consecutive numbers, but I don’t say which number you should find and inevitably someone has a rule for finding the first number and someone has one for finding the middle number. I love that!

  11. A nice trick is this one with dice. A lot of dice. Let’s say 50 or so. You lay them on the ground like a long chain. The upward facing numbers should be completely random. Then you go from the one end to the other following the following rule. Look at the number of the die where you’re at. Take that many steps along the chain, towards the other end. Repeat. If you’re lucky, you already end up exactly at the last die. You’ll be a magician immediately! But usually, that isn’t the case. What you usually have to do, is take away all those dice which you jumped over during the last step. Tell them that that is “the rule during the first round”. Now the actual magic begins. You tell the audience that they can do whatever they want with the first half of the chain. They may turn around dice. Swap dice. Take dice away. Whatever. As long as they don’t do anything with the second half of the chain. [If you like risks, let them mess up a larger part of the chain.] What you’ll see, is that each and every time, they will end up exactly at the end of the chain!

    I made a video of this, but I did it in Dutch, so the bonus is that you learn how to count in Dutch: een, twee, drie, vier, vijf, zes, zeven, acht, negen, tien ;-)

  12. Virginia Carmona

    October 30, 2014 - 9:47 am -

    I love the tricks from nrich that Megan has shared…

    Based on “The Best Card Trick?” from this site (http://nrich.maths.org/1479), I created a Geogebra file to be projected on the Smart Board, and I tell my students to take five cards. Then I drag for cards in the proper order on the board, and computer can guess the fifth one reading my mind….

    http://ggbtu.be/m63806

  13. One of my favorite games to play with students is “31”. You start by letting students know that we will alternate turns, each subtracting a number ranging from 1-X. The person who is left with 1 is the looser.

    I typically play this game to begin with for something like a “free homework pass.” As you can guess, this is modular arithmetic where you need to get to a multiple of (X+1) +1. When you first start out, you can randomly subtract things as a teacher to try to stimulate different student thinking. I typically pick the first interval to be 1-5 so “key numbers” of the game are 1,7,13,19,25 and 31. I always ask the student if they want to go first of if they want me to.

    It typically takes a few times playing the game before students really catch on, but it is great to see them create ideas and want to test their theory against the teacher. It is also great to see them realize their theory is wrong, and try to create a new one while playing the game. I typically get numerous requests to play the game during the year, and I will let their thinking cool down for a month or two before accepting the challenge again- but I normally throw in the twist that I pick the student to play against. It keeps them all on their toes and accountable for remembering the mathematics behind the game.

  14. Thanks, everybody. I’ve added a bunch to the main post.

    Scott Farrar:

    But we also convey something a little dangerous: that the math expert (the teacher) has a trick, and knowing the trick solves problems. Does this lead the student to construe math as the magic spellbook that must be memorized?

    I think we’re doing the opposite here.

  15. Isabel Wiggins

    October 30, 2014 - 5:26 pm -

    A few years ago, I found this “trick” on a “maths” site, not sure which, but it was UK. You need 5 index cards. Number them 1, 2, 3, 4, 5 in red ink on the front. On the reverse side, number them 6, 7, 8, 9, 10 in blue ink. Be sure that 1 and 6 are on opposite sides of the same card…same with 2 and 7, etc. Turn your back to the group of students. Have one of the students drop the 5 cards on the floor and tell you how many cards landed with the blue number face up (they don’t tell you the number, just “3 cards are written in blue”). Tell them the total of the numbers showing is 30. The key is that each blue number is 5 more than its respective red number. Red numbers total 15. Each blue number raises the total by 5. So 3 blue numbers make it 15 (the basic sum) + 15 (3 times 5). Let them figure out how you are using the number of blue numbers to find the total of the exposed numbers.