WTF Math Problems

As I mentioned on Twitter earlier this week, I find a particular kind of math problem extremely exciting now. Here are five of them. I want to know what to call them. I want to know what are their essential features. I want more of them and I want to read more about them.

Here is one of the five, taken from Scott Farrand’s presentation at CMC North.

Here are some points in the plane:

(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11)
(20, 33), (7,7), (-5, -17), (10, 13)

Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?

From Henri Picciotto’s review of Farrand’s session:

Students are stunned to learn that everyone in the class gets the same slope. This sets the stage for proving that the slope between any two points on a given line is always the same, no matter what points you pick.

In an email conversation with Farrand, he proposed the term “WTF Problems” because they all, ideally, involve a moment where the student exclaims “WTF”:

Set up a surprise, such that resolution of that becomes the lesson that you intended. Anything that makes students ask the question that you plan to answer in the lesson is good, because answering questions that haven’t been asked is inherently uninteresting.

These seem like essential features:

  • These problems are all brief. They slot easily into an opener.
  • They look forward and backward. They fit right in the gap between an old concept and the new. They review the old (slope in this case) while setting up the new (collinearity).
  • Students encounter an unexpected result. The world is either more orderly (the slope example above) or less orderly (see problem #2) than they thought.

And the weirdest feature:

  • They require the teacher to be cunning, actively concealing the upcoming WTF, assuring students that, yes, this problem is as trivial as you think it is, knowing all the while that it isn’t.

When did they teach you that in your teacher training?

It’s striking to me that the history of mathematics is driven by the explanations following these WTF moments:

  • We knew how to divide numbers. We didn’t know how to divide by zero. Enter Newton & Leibniz explanation of calculus.
  • We knew how to find the square roots of positive numbers, but not negative. Enter Euler’s explanation of imaginary numbers.
  • We knew what Eucld’s geometry looked like, but what if parallel lines could meet. Enter the explanation of hyperbolic, spherical, and other non-Euclidean geometries.
  • There are lots of WTF moments that haven’t yet been explained.

In school mathematics, though, we simply give the explanations, without paying even the briefest homage to the WTFs that provoked them.

What Farrand and you and I are trying to do here is restore some of that WTF to our math curriculum, without forcing students to re-create thousands of years of intellectual struggle.

So help me out:

  • Have you seen other problems like these?
  • Who else has written about these problems? I believe we’re talking about disequilibrium here, which is Piaget’s territory, but I’m looking for writing local to mathematics.

Featured Comments

David Wees cautions us that the effect of these problems depends on a student’s background knowledge. If you don’t know how to calculate slope, the problem above won’t surprise, just confound. I agree, but the same is true of textbooks and nearly every other resource.

Michael Pershan worries that the “twist” in these problems will become overused, that students will become bored or expectant. (Clara Maxcy echoes.) I demur.

Dan Anderson offers other examples. As do Mike Lawler, Federico Chialvo, Kyle Pearce, Jeff Morrison, and Michael Serra.

Franklin Mason critiques my math history without (I think) critiquing my main point about math history.

Scott Farrand, whose presentation at CMC-North inspired this post, elaborates.

Ben Orlin summarizes the design of these problems in four useful steps.

Terri Gilbert summarizes this post in a t-shirt.

Featured Tweet

2016 Jan 24. An example from systems of equations.

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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.

36 Comments

  1. Reply

    Love this. It’s a beautiful encapsulation of the initial stage of the learning process advocated in National Academies Press’ book, How People Learn — except it’s got a much catchier title. Thanks for flagging this!

    – Elizabeth (@cheesemonkeysf)

  2. Reply

    Newton and Leibiz didn’t divide by zero. Infinitesimals are non-zero quantities.

    By definition, parallels do not meet. In the non-Euclidean geometries, we either have many parallels to a given line through a given point (hyperbolic) or we have none (elliptic). But we never have parallels that meet.

  3. Reply

    Hi Dan. This is a variation of your act 1 and 2. I think ‘the hook’ is a good phrase to use because it’s meaning is well established (and we have enough vocabulary).

    I am no longer impressed by single lessons. I (we) need a year’s worth. Make me (us) a year of lessons or work with those who have (like MVP).

  4. Reply

    Here’s one of my favorite examples of this kind of problem. Take a multidigit number, reverse the order of the digits in the number. Find the difference between the two numbers. What do you notice.

    I start with two digits typically, and let students discover whats going on with those before letting them move on to 3-digit, and then 4-digit.

    I remember seeing an article about this problem in an NCTM magazine a year ago. I wrote about it also here: https://artofmathstudio.wordpress.com/2013/09/18/subtraction-reversal-game-and-investigation/

  5. Reply

    I really credit you for zoning in on WTF moments. Some of my favorite problems are surprising in the absolutely unsettling way that I hear you describing in this post. Here are two problems that had this sort of productively disorienting impact on me:

    1. The opening problem of this set involving number theory’s sigma function. It was sort of a Stairway to Heaven for me, revealing new folds and surprises just when I thought I had it all figured out.

    2. From Shell Centre’s Blue Book: you can only press “3” “4” “x” and “-” on a calculator. Find a way to produce numbers 1-10. This made me wonder if *any* number could be produced, which surprised me and pushed me to make a connection to greatest common divisors and number theory.

    I keep on coming back to your ideas here because they’re just so interesting to me. Maybe all student engagement with math can be captured with the notion of surprise.

    Maybe. But I don’t think “surprise” completely explains all the engagement with math I see from myself and from my students. I’m thinking now about Cuoco et. al’s piece “Habits of Mind.” They describe many sort of things that mathematicians do, and it’s been my experience that people enjoy dong all these different things:

    Mathematicians explain.
    Mathematicians invent.
    Mathematicians deduce what they find via experimentation, and experiment with what they deduce.
    Mathematicians push the language.
    Mathematicians systematize.
    Mathematicians categorize.

    Using your memorable phrase, I’m worried that “students are engaged by surprises” is a limited theory of engagement. We’ll miss out on lots of good stuff if we let WTF moments swallow all of math.

    This comment is not, though, an argument that (a) there shouldn’t be more WTF moments in math class or (b) that WTF moments aren’t particularly engaging. I just want to put pressure on some of the more expansive versions of your claim.

  6. Reply

    An example that comes to mind is an introductory activity to graphing linear relations using slope y-intercept form.

    Students are offered a Desmos file with y = mx + b and corresponding sliders. I then challenge them to do their best to graph the first letter of their name, using the sliders as a guide.

    Most students end up with something looking kind of like a letter, but some students are not satisfied. They want to “cut off” the lines so that it doesn’t go on forever. Then, we talk about domain and range (not in our curriculum at this point, but it serves a purpose here). Great.

    After some time, I hope for a student to ask how to make a line vertical. Question comes out something like this:

    “Sir, how do I make a line go straight up and down? I know it has something to do with slope, but I can’t seem to get it.”

    I redirect the rest of the class to get on the case. Most will start with m = 10, some will go to m = 20, while others might go m = 1000. As students pick higher numbers, they seem content that they have made a vertical line.

    At this point, I have a student share their iPad screen via Apple TV and I ask them to keep zooming in. Eventually, Desmos reveals that their lines are not vertical.

    “WTF!”

    And I’m sure you know where we go from here…

    Great visual discussing the difference between “WTF then explanation” rather than the most common “explanation then explanation.”

  7. Reply

    I really like this idea of striving for disequilibrium but it’s difficult to nail down and own as a teacher. Being able to anticipate a student reaction is key here, but it all stems from the strategic and purposeful use of problems such as these. Knowing when to use them and how to foster the WTF moment in students is hard work!
    Not sure I agree that we should use these problems sparingly. In doing so, we deprive our students the opportunity to engage in SMP 7&8. We want students to internally ask questions like “What do the numbers tell me? How are these numbers related? Is there a pattern and can I find more numbers that fit the pattern?” and that’s what these problems do. Problems and opportunities that encourage this type of metacognition are limited in the elementary grades. This goes back to what Dave said relating to prior knowledge…I wish there were more.
    We’ll engage 3rd and 4th grade students in a problem such as this when we introduce them equivalent fractions. Until students encounter fractions they focus solely on digits to order numbers. This really screws up their fractional reasoning and understanding (which we use to our advantage in creating the WTF moment). Knowing that students have no previous knowledge with equivalent fractions we give each student a fraction card; two-thirds, four-sixths, six-ninths, eight-twelfths, etc… and a blank open square. Each student represents their fraction in their square. After each student has represented their fraction they’re asked to order them from least to greatest. Once students begin to order them they realize that they’re all the equal (WTF). The numbers aren’t the same but the shaded parts are equal…but why? The conversations and explanations that follow are gold.

  8. Reply

    Here’s another way to characterize these questions, which I think applies to all your examples so far:

    1. They start with a basic task that can be completed using methods currently known to the students.

    2. The solution to that task is surprising.

    3. That emotion of surprise invites pattern-spotting, hypothesizing, and inductive reasoning.

    4. This lets you ride a wave of student ideas and enthusiasm into the coming lesson, rather than trying to start the car cold.

    But I wonder if I’m describing a narrower set of tasks than you’re aiming to include? The “WTF” in the blue diagram seems to stand in for all moments of mathematical confusion/surprise/wonder/curiosity. Many of those aren’t captured in my characterization.

    So when it comes to reaching the WTF discomfort that Piaget recommends, are questions like these the primary road? The only road? I suspect they’re one road of many, though this road is particularly awesome, efficient, and classroom-ready.

  9. Reply

    I think the example you gave here is also very interesting, because it leads to other problems in mathematics later on. I’ve worked, in my tutoring, to help students see the connections in math by taking a seemingly trivial problem like this and using it in various contexts.

    This problem is great for demonstrating that collinear points have the same slope.

    But what if you do it again, but you assign points to individuals…

    Group A, Person 1: (0,0) (2,2) :: slope =1
    Group A, Person 2: (1,2) (2,3) :: slope =1

    Now they think the points should be collinear! But they’re not…new lesson on parallel lines. Instead of direct instruction, they can discover an algebraic definition of parallel lines!
    It could also include a lesson on logic (how collinear and same slope are not 2 sides of the same coin. One implies the other, but not vice versa).

    Then later you can mix up the points (from 2 parallel lines) and ask what the probability of picking 2 collinear points is. The answer will depend on how many points from each line there are, but it’s an interesting question to ponder and investigate (and it saves probability from being a 2-week unit at the end of the year before it’s tested on the final).

  10. Scott Farrand

    January 7, 2015 - 4:09 pm -
    Reply

    The thing that is first interesting in these problems is the surprise they include, but that should not be a sufficient argument for using them in your teaching. They need to support your lesson, not distract from it. I use them as one kind of warm-up problem for class – the goal of the lesson comes first, and I select a problem that launches the lesson productively. I like a problem with a surprise when the surprise motivates the students to ask the question that the lesson will answer.

    For example, the second problem (with the triangle) was intended to have a surprise when the students assumed that the triangle is a right triangle and responded that the longest side has length 13, and the teacher Janelle then said that it actually has length 14. The next steps were for the students to realize that it can’t be a right triangle, then made enough sense of the situation to decide that the angle they thought was 90 degrees must be more than that, and finally they asked how you can determine the measure of that angle, motivating her lesson on the Law of Cosines. With this quick exercise as motivation, the Law of Cosines appeared because of the purpose it serves, making it more likely that it will be more than just another formula to her students.

    I also like these problems to start class because they implicitly rely on student thinking, rather than just mimicking procedures. This does send a non-traditional message to students, and can change the way that students behave. Teachers who are looking for a way to increase the amount of sense-making that students do in learning mathematics should consider whether these and other sorts of warm-up problems that entice students to extend their thinking might be useful.

  11. Reply

    I’m a big fan of these problems, and I’m glad you gave a name to them. But I see a tension here.

    1. WTF problems create important moments of learning that stick, that show students what it means to make a mathematical discovery, to look for and make use of structure, and to depeen their understanding. But WTF problems require skill (some might call it fluency) in the basics of the concept you’re looking at.

    2. Many math teachers, and most in the MTBoS, seem to believe that a new concept is best introduced by picking an engaging problem, or series of problems, that leads students to new understanding.

    Are these two the same thing? Is there a different type of problem for being introduced to a concept than deepening understanding of a concept? How do these problems fit into a lesson, or sequence of lessons?

    I’ve taught lessons where I start with a problem trying to get students to figure out something new. Students try it, we discuss, come to some conclusions, then apply them and practice a bit. Then I throw a tougher question at them that requires them to apply the concept in a different way. These lessons always feel like a lot — like I’m losing kids by putting too much on them too soon.

    Are there “learning something new” lessons and “deepening understanding lessons”? How are these related to the problems we choose. These are big questions, and there’s lots of gray area, but I’m pretty interested in the answers.

  12. Reply

    @cheesemonkeysf, good reference. Great read.

    @Franklin, good hip check there. Added to the post. I don’t think it changes the substance of my point, however.

    @Mike Lawler, good reminder. Your kids’ reactions are pretty priceless.

    Michael Pershan:

    Using your memorable phrase, I’m worried that “students are engaged by surprises” is a limited theory of engagement. We’ll miss out on lots of good stuff if we let WTF moments swallow all of math.

    Agreed. WTF is a sufficient but not necessary condition for engagement. The theories of engagement that concern me most are the ones that yield false positives. “The real world is require for engaging math,” for example, is a theory that inaccurately codes lots of boring real world problems as engaging. A student experiencing WTF is, almost by definition, engaged.

    @Ben Orlin, love your articulation of the design. I’m actually uninterested in a definition that encompasses “all moments of mathematical confusion/surprise/wonder/curiosity.” I’m looking for design specifications that are just narrow enough to fulfill without being too narrow to be useful.

  13. Reply

    On Frederico’s favourite problem (I too have having a few problems typing WT_ but I love the sentiment) there is a beautiful tie in to visualizing divisibility tests and our place value visualization. When we think of the divisibility test for 9, we visualize what the different place values relationship to 9 is. For hundreds, one less would be 99 and divisible by 9 then. The tens would be lessened by 1 to be nines and divisible by 9. Finally the removed ones and the left-over unit ones need to total and also be divisible by 9 for the whole number to be divisible by 9. Eg. 531 has 5 ones removed from the hundreds, 3 ones removed from the tens and the 1 from the units’ place to have five 99’s, three 9’s, and nine ones. Since all of them are divisible by 9 then 531 is also divisible by 1’s. It would be interesting to have worked with this style of visualizing earlier in the term/month/week to see how students connect these ideas to the differences of reversed digit problems.

  14. Reply

    Love the post. I certainly agree that “A student experiencing WTF is, almost by definition, engaged.” One happened by accident at the beginning of this year when we looked at the answer to this graphing story (graphingstories.com/Axh). Kids practically flipped tables over. Engagement? Check. And a very useful formative assessment.
    I’m going to do the slope question as a warm-up next week. We’ll see what kind of uproar it gets.

  15. Reply

    Response to timteachesmath since his site appears to be malfunctioning.
    Isn’t it time that the stupidity of having to write the standard on the board at the start of the class is stopped? Whose idea was it”? What if the class includes ideas and stuff from 10 “standards”? Why not just write the code number and let the kids look it up?

  16. Reply

    I’ve used Frederico’s problem as a warmup, but his version omits the most WTFy part. After the students move to four digit numbers, tell them to pick a number with at least two different digits. Then have them write the number and its reversal and subtract the smaller from the bigger. Then ITERATE the process. Take their resulting four digit number (including leading zeroes, if necessary), write it and its reversal, and subtract. Take that output and do the reversal-subtraction on it. Have them continue until they see something interesting happen. The students will have discovered Kaprekar’s constant.

    Adding on to Michael Pershan’s first problem on the sigma function, here’s a huge WTF:

    On a computer, compute the sum of 1/i for i = 1 to n for n=120, 55440, and 1441440. Call the sum H_n. Then each time, compute H_n + e^(H_n) * ln(H_n).

    Then for each of the three values of n, have the computer compute sigma(n). In each case, sigma(n) is just slightly less than H_n + e^(H_n) * ln(H_n). If you do the computations, they are pretty WTFing close. A further WTF: anyone who can show that this is true for ALL n would get fame and fortune (this is equivalent to proving the Riemann Hypothesis).

    Also: In Calc II, when we reach “functions whose input is a bound on a definite integral”, I like to have students compute the integral of 1/ln(t) from 2 to N for some large values of N. Then have wolframalpha compute the number of primes less than N for those same values of N. They are pretty astonishingly close, WTF-worthy, I think.

  17. Beth MacDonald

    January 8, 2015 - 7:52 am -
    Reply

    These WTF problems do provide a student an opportunity to accommodate their scheme of operations (from Piaget and von Glasersfeld, 1995). I always described this as promoting cognitive dissonance for students. Students experience a novel situation and seek ways of rectifying their understanding of this situation. The beauty of these problems are they are focused on the “why”, not the “what” in mathematics and students seek prior knowledge when refining what they thought they knew.

  18. Reply

    The start of CME Alg 2 Ch 3 is on problems like this:

    If you know a+b = 10 and ab = 21, what is a^2 + b^2?

    Some of the time you can just compute a and b directly: 3 & 7.

    What if ab = 22? What if ab = 29? It still works, but it becomes a total mess. You get saved by noticing

    a^2 + b^2 = (a+b)^2 – 2ab

    For example, if a+b = 10 and ab = 29, then a^2 + b^2 = 10^2 – 2(29) = 42.

    Heeeeey wait a minute, there aren’t numbers that add up to 10 and multiply to 29. Or … are there …

    Anyway this is intended to match as closely as possible the WTF that caused imaginary numbers to be invented and used: the solution of cubic equations from Cardano. In one case, the formula for solving a cubic gave x = 2 + sqrt(-121) + 2 – sqrt(-121). As long as you believed these weird, “imaginary” numbers behaved like normal numbers, everything worked out in the end.

    I love these sorts of problems, they provide great motivation for doing math. Real world, BAH :)

  19. Reply

    Two thoughts and a request (plea?):
    (1) students have adaptive expectations. If the teacher constantly surprises them, they will come to expect surprises. However, I’ve yet to hear of anyone ever grow out of the pleasure of seeing a new connection or recognizing a pattern.

    What I think this means for a teacher who uses this disequilibrium technique a lot is that students will anticipate the coming surprise and enjoy it even more. Analogous (but opposite to?) the anticipation that builds when watching a horror movie.

    (2) I saw a teacher (ht. David Ott) create this effect with a very simple pattern activity. Prep a bunch of color patterns using linking cubes, then cover them with rolled paper. Reveal the patterns cube-by-cube, taking time to ask what the students think will come next. Sequence the patterns so that the rule gets more complex and where the start seems the same as a simpler pattern they just saw.

    This works with little kids and with parents. All seem willing to vote on what color they think comes next and that sets them up for the surprising reveals.

    BTW, notes here of a time I’ve used this).

    Request:
    Dan,
    I was puzzled why I didn’t see your note on Twitter. Turns out you blocked me. I have a hypothesis about why and I’m sorry about that. won’t and haven’t done it again. Of course, there may be some other reason you blocked me, so could you put me on mute instead? My understanding is that would appear the same as blocking from your side, but would allow me to follow you and benefit from your twitterly wisdom.

  20. Reply

    With my lower level math students, the ones that do not want to be in or do anything even remotely related to math, these WTF problems are the ones that are guaranteed to draw them in for at least that class period. For instance, my juniors and seniors who had still never been exposed to the triangle angle sum, I gave them each a different triangle and had them add up the inside angles and write the sum on the board. I think it took about 10 sums on the board before a few of them caught on to the pattern and there were quite a few audible gasps followed by “they all add up to 180!” The aha moments are what make my heart happy. However, all the students were engaged from that point on and were able to practice this theorem, including the algebra portion. I try to have a least one of these WTF problems a week, which is pretty easily in geometry, as it allows for an easy segue into a more in depth explanation of the concept.

  21. Reply

    Reminds me of Confuse-A-Cat. (Credit: Monty Python)

    http://youtu.be/B2Je1CEPkUM

    “Mrs B: It’s our cat. He doesn’t do anything. He just sits out there on the lawn.
    Vet: Is he … dead?
    Mr A: Oh, no!
    Vet: (to camera dramatically) Thank God for that. For one ghastly moment I thought I was… too late. If only more people would call in the nick of time.
    Mrs B: He just sits there, all day and every day. Almost motionless. We have to take his food out to him. He doesn’t do anything. He just sits there.
    Vet: Are you at your wits’ end?
    Mrs B: Definitely, yes. […]
    Vet: Now, what’s to be done? Tell me sir, have you confused your cat recently?”

    Anyway, I like your theory. Is your WTF hypothesis testable?

  22. Reply

    I noticed that Michael P and I were lumped together with regard to how often one should use WTF problems, but I demur. I love WTF because for me, math is all about these incredible circular realizations. I try to evoke this wonderment every time we explore an idea. This week we are looking for holes – those marvelous places that “x” cannot be, and yet there is the “zero”, the “root”. And they are the same thing! How can that be? Why does the graph suddenly run from itself, Change course and leap the other way? These lessons create lovely questions, exclamations, conversations, and, yes, a little learning! Do not ration such a thing as keeps us learning! (My earlier response to Michael was actually a reason to use these marvelous creatures, so that children would come in “expectant”, and ready to learn something new….like us, when we read your and other blogs!

  23. Reply

    Alan Kay described a WTF problem where kindergardners pick a shape they like — like a diamond, or a square, or a triangle, or a trapezoid — and then they try and make the next larger shape of that same shape, and the next larger shape. They then recorded how many shapes it took and how many they needed to add at each stage to get to the next size shape. The two progressions are a first-order discrete differential equation and a second-order discrete differential equation, derived by six-year-olds!!!

    You can see a video of him explaining this here: http://www.ted.com/talks/alan_kay_shares_a_powerful_idea_about_ideas?language=en

    If you go to 9:30 in the video you will see the explanation.

    Cheers,
    Stephen

  24. Reply

    My favorite aha moment for my students involved a compass, a ruler, string, and a trip to the computer lab. We started by constructing a circle, then marking off the arcs around the circle keeping the compass the same distance open. It didn’t quite make it around 6 times -there was a gap. We took a piece of yarn and laid it on the circle. We measured the radius, then counted how many radii there were when measuring the string. Again just over 6.

    Then we went to the lab and constructed the circle there and measured the arc length of a semi circle divided by the radius. Guess what number we arrived at. Suddenly a few students eyes lit up. They were told what pi was and memorized it to a dozen or more decimal places but had never contemplated where it had come from or those who had struggled with the fact that it wasn’t a nice neat number contemplated getting more and more precise of a measurement. I dug out the old activity and posted it here http://systry.com/how-far-is-it-around-that-circle/. There is nothing like the thrill of discovery.

  25. Reply

    Great activity. I suggest using the yarn (I use thread) to make a diameter. Then, ask the students how many diameters they think it takes to make a circumference. The answer (of course) is three and a ‘little bit’. The ‘little bit’ is the point one four part of pi.

  26. Reply

    Isn’t this just the ol “see the point of the mathematics you’re about to develop”? That necessity could come from a problem students cannot yet solve, or a pattern students are baffled by. Both have driven mathematics forward, historically. Both have their place in mathematics teaching and learning.

  27. Reply

    @Julia, your comment may simplify the matter conceptually but, practically, it’s very hard to baffle students productively. Much easier to baffle them destructively.

  28. Reply

    I was just reading something from Dan Willingham that reminded me of the WTF problems.
    http://www.aft.org/periodical/american-educator/summer-2004/ask-cognitive-scientist
    He points out that four elements of stories (Causality, Conflict, Complications, and Character) increase people’s ability to remember and learn (and elsewhere, he links retention to engagement).

    I wonder if the WTF problems are the math class’s version of a story: there is conflict between assumptions and surprises, and of course, there is causality.

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