I’m happy to release video of the talk I gave throughout the 2016-2017 school year, including at the NCTM Annual Convention in San Antonio, TX.
This is my best attempt to tie together and illustrate terms like “intellectual need” and expressions like “if math is aspirin, how do we create the headache.” If you’re looking for an elaboration on those ideas, or for illustrations you haven’t seen on this blog, check out the video.
The Directory of Mathematical Headaches
This approach to instruction seriously taxes me. That’s because answering the question, “Why did mathematicians invent this skill or idea?” requires a depth of content knowledge that, on my best days, I only have in algebra and geometry. So I’ve been very grateful these last few years to work with so many groups of teachers whose content knowledge supplements and exceeds my own, particularly at primary and tertiary levels. Together we created the Directory of Mathematical Headaches, a collaborative document that adapts the ideas in this talk from primary grades up through calculus.
It isn’t close to complete, so feel free to add your own contributions in the comments here, by email, or in the contact form.
Dawn BurgessDecember 1, 2017 - 3:34 am -
Dan, in the video you mention a paper teachers should read, but I didn’t see a link at the end. Could you point me in the right direction?
Joanne WardDecember 1, 2017 - 6:30 am -
Thank you for sharing the video!! :))
Scott McDanielDecember 1, 2017 - 11:06 am -
Stacie BenderDecember 2, 2017 - 10:46 am -
Here’s a recent headache of mine. My regular geometry students have a hard time with the “exterior angle of a triangle is equal to the sum of the remote interior angles” theorem. I don’t know if 6 problems is enough of a headache for my students to want the cure, but it’s a start. You’re welcome to copy to your directory and share.
Dan MeyerDecember 4, 2017 - 10:41 am -
Dig it. You’re provoking a combination here between the need for computation (“This would be easier if I had a general rule.”) and the need for certainty (“How can I be sure this general rule will always work.”)
Added to the document.
Chester DrawsDecember 4, 2017 - 9:07 pm -
Is there any need at all for this theorem though Stacie?
It follows from two simple rules that students don’t usually have issues with — sum of a triangle and then adjacent angles on a line. As a result I have crossed this theorem off my list of ones to teach, since it can be quickly worked out from first principles — and I therefore I also don’t have to teach them when not to apply it (when the angle given is vertically opposite, not exterior) and have more time to teach the ones I do correctly.
I also don’t teach alternate exterior angles on parallel lines are equal. Again it follows from two simple rules we are already teaching, and is one less thing for them to confuse themselves about.
A plethora of rules they have to learn puts students off Geometry — so that’s where their headache tends to be. So I make it easier by cutting out surplus rules. If that means that some questions take an extra step, then so be it.
(I do actually teach both rules, because they are simple introductions to proofs. What I don’t do is expect them to add them to the list of ones they have to memorise.)
ScottDecember 5, 2017 - 8:52 am -
It took me a minute to figure out why you needed a much bigger hammer to pound a stylus into the ground. Then I realized it was a stake…
I really like the coin problem and how it showed why arrays come in handy for counting. I’ve been working with our elementary math coach (I’m middle school) on using arrays. I am definitely sharing this with her.
Ivy KongDecember 8, 2017 - 1:18 pm -
I was trying to come up with ways to teach the distributive property to my 7th grade intervention class. In my last few years, I tried many different ways, like using an area model, drawing the arrows, telling corny stories about talking to everyone inside the parentheses house, using algebra tiles, etc, etc. It dawned on me one day that the students probably might not even understand what was meant by 3(2x-5). So, whatever models I trying to teach were all non-sense steps to them.
Then I tried this this year: I told the kids that 3(2x-5) meant 3 sets of (2x-5) and made them expand the expression by writing out three separate sets of (2x-5). Then we combined like terms and figured out that it was 6x-15. I pounded in the idea that the ‘3 times’ meant ‘three sets’. The kids got the idea they needed three sets of 2x and three sets of -5. I hadn’t even mentioned the distributive property yet.
Then your talk came in… I wanted the ‘set’ idea to stick, so I gave them different headaches.
Then I taught them the convention and stuff. All in all, it was great and the kids felt super successful. (I also felt successful by using this strategy–I had a headache and you gave me an aspirin!)