See the activity around page 8 of this,
http://www.vpri.org/pdf/m2007007a_revolution.pdf

To me they all involve looking at patterns in how things change, looking at “increase by” patterns i.e. differential relationships.

Since they deal with relative change, they also have a “kinesthetic” quality to them. You can understand the rate of change in a graph the way you feel your body in motion.

See also this way of understanding of parabolas,

#1formula for slope
#2pythagorean formula
#3counting formula
#4doesn’t matter what you do, subtracting what you started with gives everyone the same answer.
#5function that gives the linear, usually a factor of the actual function.
These are all skills on the way to somewhere else, like the steps needed for DOK 3 type problems.
They are solvable by algorithms that don’t require understanding, which points me back to the Chinese parable in one of your past posts.
I have used similar problems as starters, intros to ideas where I want to see what my Ss know (the formulas) or some, like #4, to gain my students’ attention. Nothing like sleight of hand to explain how math isn’t magic. Even though they sometimes feel like it is.
I think these are good think starters, if used well.

What jumps out at me here is the “open-beginningedness” of the problems that allow students to do some (not a lot) of work and will likely spark some curiosity and/or a debate. Offers a ‘hook’ without necessarily offering any sort of relatable context.

I agree with David: there’s some danger in using these problems because if a student misses the aha or trick or issue, there’s not much to pull them back to, and they may feel like they missed a joke.

These problems are tricky business but can pay off BIG when they hit.

My personal favorite was in Alg2/Trig when kids were learning to graph sine and cosine variants, and I had them graph the evil function y = sin 666x on the TI-83’s standard trig window. If you’ve got a TI-83, try it!

You are asking students to look and think first, before strategies are shown. These problems also pose opportunity for developing conceptual understanding of fundamental algebraic principles. Here is a link on how #4 can used to reinforce basic arithmetic and algebraic skills: http://bit.ly/1tBPcVl

In my opinion, the “Number Tricks” lesson linked to should also include some opportunity for students to work without the overhead of algebraic expressions. Then, the concept of using and evaluating algebraic expressions can derive from the natural question: why did this trick work?

I agree that it reinforces basic arithmetic and algebraic skills, while also providing some necessary connections in the links between the two: it becomes more clear that n stands for a number, because you just worked it through for several numbers.

There’s some detail on elementary-school thinking around this type of task here: http://thinkmath.edc.org/resource/early-algebra-little-kids

I apologize, because this is a kvetch that I’ve offered before, but I’m somewhat concerned that a curriculum or teacher could overuse this technique. If kids came to expect these weird surprises in class, I suspect that we’d see diminishing returns on both the learning and (especially) the engagement potential of this type of problem.

Epiphany problems? http://math-frolic.blogspot.com/2014/12/an-epiphany.html

If so, Varignon’s Theorem is a favorite in the category.

I’m surprised that you all see #2 as simple. Only my quicker witted students know that the answer is “somewhere between 7 and 17”.

I wouldn’t say #2 is simple as much as it offers a “snap” answer of 13. Then you show the kid a 5-12-10 (or better yet, someone else in the room thinks of it) and they go …. ooohhhhhhhhhhh

Disequilibrium! And then the question evolves to “what lengths are possible”, with some kids forgetting about non-integer lengths, how big 16.99999… can be, and a potential debate about whether 7 and 17 form triangles.

But it interests me how many of you suggest teachers should use these surprises sparingly. Each of these problems attempt to provoke disequilibrium in a student…This is disequilibrium, which Piaget argued is essential for learning, which many of you argue we should strictly ration.

I don’t think that this is the right way to characterize some of the above objections. There’s more than one way to provoke disequilibrium than with “twist” problems. You can get protein from hamburgers, but you don’t need to get protein from hamburgers.

I provoked disequilibrium this past weekend (at least for some of the students) with an activity I did with 8 to 10 year olds.

I asked them to count out 100 snap cubes and prove that they had 100 cubes.

One group was counting out the blocks and snapping them together and not paying attention to the total. They just thought they had to get to lots of blocks first. I asked them how many blocks they had. They had NO IDEA. So they decided to count their blocks.

It turns out that all three of these students thought that in order to count out their 4x5x5 block structure, they just needed to count the outside. First they thought they had 20 + 20 + 25 + 25 blocks. I asked them how many sides the shape they had constructed had (first disequilibrium – I’m offering them information that directly contradicts their 4 side model).

Once they had established that they had six sides, they now thought they had 20 + 20 + 20 + 20 + 25 + 25 blocks (counting each face). I then pointed at a block on a corner and asked how many times this block would be counted (second disequilibrium – the fact this individual block gets counted three times contradicts their model of counting the outside).

One of the kids then tried a different way to count and said he was going to slice up the shape. He then counted five 20s and said the number of blocks should be 20 + 20 + 20 + 20 + 20. Another students, upon seeing this method (third disequilibrium – he’s been prompted to use a different method to think about counting) decides that they could also count it as 4 twenty-fives (25 + 25 + 25 + 25).

In this case, I think that the task itself, without my prompting and the structure with which kids did the task (prove that you have 100 blocks to someone else rather than just count out 100 blocks, work together to construct this proof, have me around to ask pointed questions) is critical to this disequilibrium.

Magdalane Lampert recently asked me why we were separating our website resources into instructional resources and curricular resources. “Curriculum resources are used with instructional routines. I never know if something is supposed to be curricular or instructional. It’s all the same pile to me.”(paraphrasing).

It seems to me that asking if any problem will promote disequilibrium depends on the knowledge of the students and the way the task is actually enacted in their context. Counting to 100 isn’t necessarily going to be a task most people would learn from, but because of the way that I framed the task, the way the students acted on the task, the ways in which I responded to how they acted, it ended up being something to learn from.

I am still not seeing the purpose of #2. If students have the Pythagorean Theorem, then it is dirty pool to provide a triangle that appears to be a right. You are not providing a surprise so much as setting the audience up to be the dupe — gotcha, it doesn’t have a right angle marker. There is a certain mean spiritedness to this one that I don’t see in the others.

Is the punchline really that the side could be between 7 & 17?

Maybe it would be useful to compare these to other problems and construct a taxonomy of task type based on examples and non-examples?

Also, would it be useful to see how other people have constructed taxonomies for this purpose? For example, Bloom’s taxonomy has much more nuance to it (if you actually read the associated book, which I have not yet read, but know about because of another book I am reading) and is the result of many assessment and curriculum people talking about a very similar issue.

I agree that being able to talk fairly narrowly about mathematical tasks and label them in certain ways is likely useful for those of us who use tasks, share tasks, and create tasks. I’m just wondering if we can build on someone else’s start to this work?

I Hodge: I suspect that what I use #2 for is different from what Dan uses it for.

I use similar questions to stop students making poor assumptions. To try and stop them thinking they know what a question is about before reading it properly.

A far too common mistake in geometry is to assume any angle that looks right angle must be right angle (and anything that looks isosceles must be isosceles, anything that look parallel must be parallel).

Over the years I’ve done it, it generally leads into interesting asides as a “practical” introduction to greater than that is not equal to, one we agree that a triangle of no volume is a line. But that’s just an aside for me.

There’s no duping involved in my case, as I warn them at the start of the topic very strongly not to make assumptions. If someone puts a hand up and asks if it is right angled, I say “we can’t tell”.

When we do parallelogram areas I also do one with stated outer lengths (perimeter) and ask what we can say about the area. That gives a lower bound of greater than zero, but an upper of greater than or equal to the area of the equivalent rectangle, which is a nice difference from the triangle.

It helps start them on the process of moving away from every problem has one, exact, answer.

Areas/volumes is also where I introduce accuracy versus precision, and the effect of measurement errors (although generally only the clever really get a proper hold on that).

In response to Dan:

This is disequilibrium, which Piaget argued is essential for learning, which many of you argue we should strictly ration.

Rather than ration disequilibrium in our mathematics lessons, we should be thinking long and hard about how we can find more opportunities to bring this technique into our classroom. If we choose to ration, in an attempt to somehow reserve the effectiveness of this type of questioning, we are forced to resort to less interesting questions.

I think a common mindset amongst math teachers is that we have to work hard to “trick” our students. On quizzes, tests and every once in a while during our lessons. Are we concerned that by introducing disequilibrium in our lessons too often that we won’t be able to trick our students as effectively? Should we be trying to trick them in the first place?

If we use this type of question in our lessons as often as possible, I think that students will begin to look at all problems more critically as if they are searching for a hidden trick somewhere.

Would it be all that bad if students became such great problem solvers that they manage to foil our “trick” before we anticipate?

Here is a common example from the first year of algebra (after doing a bit of factoring).
Demonstrate fantastic mental calculations orally such as 17^2 — 16^2 equals 33, and 23^2 — 22^2 equals 45, an on and on. As students see the pattern they begin to join in.

In middle school and high school students can then explain why this works. With 5th graders they are tickled to discover the pattern and the room is alive with a math teacher’s favorite sound “ahh!”

The same with the pattern 19×21 = 399, 41×39 = 1599, 59×61 = 3599, …

There are a number of calendar patterns that work their “aha” magic on students as well.

In advanced algebra and precalc I use to begin one lesson by mentioning that “every polynomial function has another function associated with it call its “cousin function.” I would then ask students to call out a polynomial function and I would give then another function that was the cousin to their function (by taking the derivative). They would continue giving me functions and I would give them their cousin. Eventually someone would give the cousin function for me. This would really get the other on the edge of their seat. 15-20 minutes later the majority of students are taking derivatives. A few days later, I’d mention that a particular polynomial function is a cousin function to some function. What is the original function? And off we’d go again…

Dan, I think it could be useful and clarifying to the discussion if you could offer some important and varied non-examples of the kind of problem that you’re talking about and looking for.

Here’s one I love and have used it many times, especially when teaching combining like terms. It also helps kids generalize and understand that a variable’s value can change depending on the situation. It’s called “Two Sums” and can be found in Mathematics Teaching in the Middle School, volume 16, no. 2, September 2010, pg. 68.

I like the remainder theorem as an example of a WTF moment.