my contribution to the Convention of Mathematical Flavors – check it out!
Every teacher has tics. I don’t mean facial twitches (though a few rowdy, beloved classes have driven me there). I mean teaching tics: instincts, fallbacks, go-to-moves. The habitual ways that we frame questions, or receive answers, or structure lessons. I’m now going to recount one of my tics—one that crops up, for better or worse, in every grade, on every topic, in rain or in sun or in humidity so damnable that my sweat is sweating.
The tic of “give me.”
Give me six different triangles whose area is 10 cm2. Give me a pair of functions so that f(g(x)) is always bigger than g(f(x)). Give me five data points so the mean, median, and mode are all different. Give me, give me, give me…
Am I selfish, asking for so much? Am I selfless, giving my students the gift of giving (which, after all, is better than receiving)? No and no. I’ve converged on this question style because it’s quick, it’s carefree, and it encodes several of my mathematical and pedagogical priorities.
What am I hoping to convey with “give me”? Here’s a partial list:
- In math, we create just by thinking.
It’s worth asking: How do you “give me” a rectangle? Do you forge it from iron? Sculpt it from clay? Buy it off Amazon Prime?
No: you just describe it. Specify a length and a width. Maybe sketch a wobbly picture. No construction permits needed.
In mathematics, description is creation. A description is the blueprint of a thought, transferred from your brain to mine. That’s the principal activity of mathematicians: the conjuring of ideas in one another’s skulls. “Give me” is an invitation to join that process.
- Answers live in multiplicity.
I ply my students with plenty of closed-form questions. “Compute this area,” “simplify those fractions,” “go wrassle them integrals.” Such questions are easy to assign, easy to grade, and easy to tailor for specific skills. Some even make for thick, juicy puzzles.
But mathematics is not a kingdom of single answers. It’s a realm where truth comes in families, where answers live in glorious multiplicity. Take this standby: “Give me a pair of rectangles, the first with a bigger perimeter and the second with a bigger area.” A dozen students could supply a dozen replies. Your answer needn’t look anything like your neighbor’s.
- A good technique is a gift that keeps on giving.
Here’s another favorite: “Give me a fraction between 10/11 and 11/12.”
Finding your first example may take some elbow grease. But once you’ve got one, it’s easy to generate others, because good mathematical techniques aren’t one-trick ponies. They’re fertile, birthing answer after answer.
For example, I saw one student note that each original fraction has “one piece missing.” The first is missing a bigger piece (1/11) than the second (1/12). To find something in between, pick a middle-sized piece (like 1/11.5, or 1/11.4, or 1/11.78). My student didn’t just find an answer; he found a path that led to infinite answers.
- The math is for you, not for me.
The way school is structured, students seem to perform for the teacher’s amusement. I say “jump”; they jump. I say “differentiate these functions”; they plug them into Wolfram Alpha. I say “studentwithextrahomeworksayswhat”; they say, “What?”
So it’s liberating—for me and for them—to do work that’s a little freer. Work that’s not graded by resemblance to some master key. Work with room to swing your elbows and let your mind romp, work that even lets you—dare I say it—express yourself.
I’ve rarely seen my 7th-graders so energized as when writing their own challenges and questions, to be shared with the rest of the class. Instead of working to please a single authority figure, they now play for a crowd of 25 peers. “Give me” questions—selfish as they may sound—help flip the usual script.
- In math, we sift the possible from the impossible.
Sometimes I’ll make a trollish demand. “Give me three numbers,” I’ll say, “so that the sum of any two of them is odd.” Before long, students are yelping in outrage. And that’s where the learning happens.
First—and most important in life—they learn never to trust me.
Second, they learn that math is obsessed with possibility and impossibility. More than any other subject, it’s the dancefloor of the possible, where we see how far logic can bend without breaking. (Caution: do not try this on an actual dancefloor.)
In the sciences, “impossible” is a contingent idea, subject to revision. Try asking Newton about gravitational lensing, or Einstein about whether God plays dice. But in math, “impossible” is a tangible presence. There are no three numbers whose pairwise sums are all odd. Period. The attempt to find one brings you into contact with impossibility—and thus with certainty, with proof, and with the other qualities that make math so delightfully and epistemically math-y.
I could go on. But I’ve given you enough, haven’t I? Now that I’ve got you here, I want you to give me your “give me” questions. I’m always looking for fresh ones—and I suspect that you’ve got ‘em. Transfer them from your mind to mine, please. That’s what math is all about.
“Give me three numbers,” I’ll say, “so that the sum of any two of them is odd.”
{0.5, 0.5, 0.5}
Oh, you meant three different numbers?
{0.5, 2.5, 4.5}
He probably meant 3 integers
He wrote “numbers”, not “integers”, twice. I’m sure Ben would appreciate a student coming up with a solution like the above.
A favorite is roll 2 d12. Roll 3 d6. Use the 3 d6 to equal the product of the 2 d12. Can add, subtract, multiply, divide, use parenthesis, turn them into exponents. (Trick is they can only use the numbers that are there. Inevitably if the numbers are 3, 4, 5, a couple students will try to do 3 squared, but that’s not legal because there is no 2.) Great game for reviewing order of operations, “punishment” when they all get quiz answers wrong because of order of operations. Or some classes just love it and want to play it as a reward game.
Could also be a time filler, “Give me 5 (or 10) different answers using the numbers 3, 4, and 6.” or whatever numbers you choose.
Give me pictures from your cell phone that show parabolas in the real world.
Give me 3 careers and one example of a task for each where people must use exponents. (Example: scientists calculating population, mortgage banker, etc.)
Give me 3 examples of patterns in your day to day life.
Give me a pattern in this song (music notes, melody, words, beat, pitch, a specific sound that repeats every so many beats)
Draw a picture that shows…
This one is common homework for my students.
Give me a 6 word summary that explains… (or 8 word, 10 word, etc. Kids get really feustrated when I say 6 words, they write 8 and I tell them to try again.)
Give me a sketch of every type of graph you can think of.
Give me 3 similarities between ___ and ___.
Give me 3 differences between ___ and ___.
Give me 3 things you notice about these problems.
I suppose I did a lot of explaining and reasoning with my math students. It helped them make connections between ideas, though. I will miss teaching math this year.
Happy new year by the way! Have a great school year!
Reblogged this on ilripassinodimatematica and commented:
Un bellissimo articolo di una grande prof di matematica! Per chi se la cava con l’inglese!
Wow. Love this post!
Give me a function that is differentiable only at finitely many points and non-differentiable everywhere else.
Typo in 3) you want 21/23 not 11/23.
gimmie an equilateral triangle with vertices on lattice points of the coordinate plane.
Maybe I’m strange, but then I came to mathematics after doing doctoral work in English. So I pay very careful attention to language. How you frame a question can have an enormous impact on how the question/task is perceived by those facing it.
So I have tried for decades to avoid “Give me” phrasing in my teaching and encourage others to eschew it as well.
Instead, I prefer to ask, “Give us. . . ”
“Give us this triangle, our daily isosceles.” (Okay, I’ve not tried to put math questions into the form of psalms, but I suppose I could).
The point, which I hope is obvious, is that I prefer to have students see themselves as part of a community to which their answers, conjectures, wild ideas, inklings, questions, etc., are meaningful contributions. And I’m just a member of that community, so my inquiries are requests for the greater good of the whole (in theory).
Maybe it’s just bilge and they still know that I’m the dancing master to whose tunes they must perform. But I hope that the linguistic shift helps to create a psychological one in support of the classroom culture I’m trying to build with them.
Take a sphere. Drill a cylindrical hole through the center of the sphere so that the altitude of the hole is 6 inches. What is the volume of what’s left?
Give me a few ways to count (assign natural numbers) to all points with integer coordinates on a 2D plane. On a 3D plane. To all possible chess positions. To all possible valid chess games. To all possible polynomials with integer coefficients. To all finite strings of digits. To all infinite strings of digits. (Discuss Kantor’s diagonalization. Discuss continuum.)
Give me a few numbers which are hard to describe. (Bring up the “smallest number that cannot be described in nine words” paradox. Discuss Kolmogorov complexity. Discuss Knuth’s arrow notation.)
Give me a few programs that never stop. Give me a few programs for which nobody in this room can say for sure whether they ever stop. Give me a limited programming language in which every program stops. Give me a limited programming language in which you always know whether a given program stops. Give me a few functions that cannot be computed in that language. (Discus Turing completeness. Discuss halting theorem.)
What I have trouble with sometimes is getting my college students to give me an example I haven’t given them. Some of them seem to think I want to hear what I or the book already said. It can be hard to persuade them otherwise.
I agree, but you can convince them. Sometimes I just say “give me an example you haven’t seen in class”. Sometimes I say “give me three examples” when I know I’ve only shown them one. If you do it often enough, they figure out you really mean it. (Also I think sometimes they are just trying to play it safe; they don’t want to risk hearing their example doesn’t work.)
Give me a function which differentiable at a point, but it’s derivative is not continuous at that point.
So, I have found my Geometry kiddos are the ones who most rebel against these ‘open’ questions. An early in the year example is to ask for some points that are at least 10 units but no more than 50 units from the point (3, 7). Another one I like is to ask for an equation of a circle whose center is in the first quadrant and whose graph does not intersect either axis. At first, this feeling of ‘no RIGHT answer’ is dismaying, but many students come around to appreciating this type of exercise.
Seems like coordinate geometry is a perfect place for gimmes! The vertices of a triangle that is similar to triangle ABC, a circle with it’s center in the _____ quadrant that is tangent to the x and y axes, the endpoints of a line segment that has it’s midpoint at (x,y), a rhombus ABCD such that the slope of AB is ____ times larger/smaller than the slope of BC, and whatever else you/your students can dream up.
This is a fantastic post! (I also appreciate Michael Paul Goldenberg’s comment on “us” rather than “me”). Would you by any chance have a cache of these open-type questions that you wouldn’t mind sharing?
Thanks, slevydc.
Excellent post. We (math teachers) would do well to ask more “give me/give us” open questions. Thank you.
For #3 a nice way to do this is to use the “freshman sum.” The “freshman sum” of 10/11 and 11/12 is just the sum of the numerators over the sum of the denominators: (10+11)/(11+12) = 21/23. It happens that this method always gives a fraction between the two original fractions. A nice proof of this can be found here: https://www.youtube.com/watch?v=HMwhuKfM4xc
You might like this song (Gimme by Fatcat & Fishface, aka the birthday song): https://fatcatfishface.bandcamp.com/track/gimme
Wow.. Love this concept. Particularly, it gives the concept of creating your own world as a pupil. I always want that independent spirit for my pupils….
Give me the largest number you can make out of three 1s.
Give me the largest number less than 100 that you can make out of three Is only.
Give me an equation of a line that has a positive slope less than one.
Create a scatter plot for which the equation y = -2x+5 could be a reasonable trend line.
Give me an irrational number between 7 & 8.
Give me an equations of a line for which (-2, 7) is a solution. Now give me another one.
Give me a sketch of something that isn’t a function.
Give me the side lengths of right triangle that has a hypotenuse that is irrational and legs that are rational.
Give me the side lengths of a tight triangle that has a hypotenuse that is rational, and one leg is rational and one is irrational.
I love this strategy! Thanks for sharing! And thank you to Dylan Kane for bringing it to my attention.