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[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:

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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.

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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.)

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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.

PhotoMath is an app that wants to do your students’ math homework for them. Its demo video was tweeted at me a dozen times yesterday and it is a trending search in the United States App Store.

In theory, you hold your cameraphone up to the math problem you want to solve. It detects the problem, solves it, and shows you the steps, so you can write them down for your math teacher who insists you always need to show your steps.

We should be so lucky. The initial reviews seem to comprise loads of people who are thrilled the app exists (“I really wish I had something like this when I was in school.”) while those who seem to have actually downloaded the app are underwhelmed. (“Didn’t work with anything I fed it.”) A glowing Yahoo Tech review includes as evidence of PhotoMath’s awesomeness this example of PhotoMath choking dramatically on a simple problem.

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But we should wish PhotoMath abundant success – perfect character recognition and downloads on every student’s smartphone. Because the only problems PhotoMath could conceivably solve are the ones that are boring and over-represented in our math textbooks.

It’s conceivable PhotoMath could be great for problems with verbs like “compute,” “solve,” and “evaluate.” In some alternate universe where technology didn’t disappoint and PhotoMath worked perfectly, all the most fun verbs would then be left behind: “justify,” “argue,” “model,” “generalize,” “estimate,” “construct,” etc. In that alternate universe, we could quickly evaluate the value of our assignments:

“Could PhotoMath solve this? Then why are we wasting our time?”

2014 Oct 22. Glenn Waddell seizes this moment to write an open letter to his math department.

2014 Oct 22. David Petro posts a couple of pretty disastrous screenshots of PhotoMath in action.

2014 Oct 23. John Scammell puts PhotoMath to work on tests throughout grade 7-12. More disaster.

2014 Oct 24. New York Daily News interviewed me about PhotoMath.

2014 Oct 27. Jim Pai asked some teachers and students to download and use PhotoMath. Then he surveyed their thoughts.

Featured Comment

Kathy Henderson gets the app to recognize a problem but its solution is mystifying:

I find this one of the most convoluted methods to solve this problem! I may show my seventh graders some screen shots from the app tomorrow and ask them what they think of this solution – a teachable moment from a poorly written app!

M Ruppel:

I we are structuring this the right way, kids (a) won’t use the app when developing the concept, (b) have a degree of comfort with doing it themselves after developing the concept and (c) take the app out when they end up with something crazy like -16t2+400t+987=0, and factoring/solving by hand would take forever.

Sander Claassen:

The point in this case isn’t how well the character recognition is. Or how correct the solutions are. Because it’s just a matter of time before apps like these solve handwritten algebra problems perfectly in seconds, providing a clear description of all steps taken.

The point is: who provides the equation to be solved by the app? I have never seen an algebraic equation that presented itself miraculously to me in daily life.

Kenneth Tilton:

ps. Photomath is just a “stupid pet trick” they did to market their recognition engine.

This is a talk I gave awhile ago looking at why students hate word problems, posing five ways to improve them, and introducing this thing called “three-act math.”

My 2015 Speaking Schedule

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Here is my speaking calendar for 2015 in case anybody is interested in attending Dan’s Blog: The Unplugged Experience. Some of these sessions are private, others have open registration pages (see the links), and others have waiting lists. Feel free to send an e-mail to dan@mrmeyer.com with inquiries about any of them. It’d be a treat to see you at a workshop or a conference.

BTW. Delaware, Idaho, Nebraska, Rhode Island, Tennessee, West Virginia, and Wyoming will complete my United States bingo card. If you’re the sort of person who schedules these kinds of sessions for a school or district or conference in any of those states, please get in touch.

From Pearson’s Common Core Algebra 2 text (and everyone else’s Algebra 2 text for that matter):

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

The only math students who like these problems are the ones who grow up to be math teachers.

One fix here is to locate a context that is more relevant to students than this contrivance about coins, which is a flimsy hangar for the skill of “solving systems of equations” if I ever saw one. The other fix recognizes that the work is fake also, that “solving a system of equations” is dull, formal, and procedural where “setting up a system of equations” is more interesting, informal, and relational.

Here is that fix. Show this brief clip:

Ask students to write down their best estimates of a) what kinds of coins there are, b) how many total coins there are, c) what the coins are worth.

The work in the original problem is pitched at such a formal level you’ll have students raising their hands around the room asking you how to start. In our revision, which of your students will struggle to participate?

Now tell them the coins are worth $62.00. Find out who guessed closest. Now ask them to find out what could be the answer – a number of quarters and pennies that adds up to $62.00. Write all the possibilities on the board. Do we all have the same pair? No? Then we need to know more information.

Now tell them there are 1,400 coins. Find out who guessed closest. Ask them if they think there are more quarters or pennies and how they know. Ask them now to find out what could be the answer – the coins still have to add up to $62.00 and now we know there are 1,400 of them.

This will be more challenging, but the challenge will motivate your instruction. As students guess and check and guess and check, they may experience the “need for computation“. So step in then and help them develop their ability to compute the solution of a system of equations. And once students locate an answer (200 quarters and 1200 pennies) don’t be quick to confirm it’s the only possible answer. Play coy. Sow doubt. Start a fight. “Find another possibility,” you can free to tell your fast finishers, knowing full well they’ve found the only possibility. “Okay, fine,” you can say when they call you on your ruse. “Prove that’s the only possible solution. How do you know?”

Again, I’m asking us to look at the work and not just the world. When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones. The answer isn’t just that, anyway. The answer is to look first at what students are doing with the coins – just solving a system of equations – and add more interesting work – estimating, arguing about, and formulating a system of equations first, and then solving it.

This is a series about “developing the question” in math class.

Featured Tweets

I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:

2014 Oct 20. Michael Gier used this approach in class.

Featured Comments

Isaac D:

One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

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