After a riveting session about brain science at the summer program I attended (where I met Sam!), I wanted to read a little bit more about epistemology. I chose a few books that the presenter suggested: I just finished reading “Made to Stick” by Chip and Dan Heath (about why some ideas stick in our mind better than others and how to turn your ideas into some of those better ones) and am about halfway through “Brain Rules” by John Medina (twelve basic rules about how the brain works). Both were fascinating and will absolutely influence my teaching.
One thing that I have really latched onto is the idea of working with students’ previous knowledge about everything and anything in order to guide and improve learning (both books kind of harp on this). Take this example from Made to Stick where they define a Pomelo (an example which the lecturer also talked about at the summer program):
A pomelo is the largest citrus fruit. The rind is very thick, but soft and easy to peel away. The resulting fruit has a light yellow to coral pink flesh and can vary from juicy to slightly dry and from seductively spicy-sweet to tangy and tart.
If you already know what a pomelo is, that should make sense, and if you don’t, you can still get a pretty good picture of what’s going on. But compare that definition to this one:
A pomelo is basically an oversized grapefruit with a very thick and soft rind.
Both define a pomelo, but the second one uses the crazy ideas in your head to build new knowledge, making a much more descriptive and much stickier idea – not only is it easier to learn what a pomelo is, you will remember it much better. AAAND, the big bonus, it’s more efficient! [Here is a picture of a pomelo, by the way, if you need one. They’re kind of gross, but I am still partial to pomelos – in Arabic, pomelo is “bomaly,” and at first the guards at school couldn’t understand my weird sounding name (“Booooowman”), so they chose to hear the closest familiar thing, and started calling me “bomaly.” The origin of one of my many nicknames.]
Using Schemas in Math Education
I was thinking back to my year to see if I used anything like this in to teach math. I thought of one example, which I wanted to share, and then decided to put out a call for others. Can you think of a specific instance where you used anything from students’ prior knowledge to effectively and efficiently make a mathematical concept stick?
Piecewise-Defined Functions and Music Mashups
When reviewing at the beginning of the year in my Calculus class, I found that a lot of students were surprisingly stymied by the idea of piecewise-defined functions, which kind of blew my mind (this was in my first two weeks teaching math, and I was not expecting this to be a tricky concept for seniors in high school). It dawned on me that piecewise functions (which I call “Frankenfunctions”) are a lot of like Music Mashups, like this awesome mashup of the Top 25 songs from 2009 by DJ Earworm:
I played the song for the class and before connecting it to math, we broke down what we were hearing – like, actually had a brief conversation about what a mashup is (basically, one song constructed from segments of many others, though we went into more detail). Then we talked about the piecewise functions with this context:
- A piecewise-defined function is one function made up of pieces of many others.
- Each segment on a piecewise function is just a little part of a much bigger function.
- The segments are broken down into intervals based on the x-axis (or time axis).
- In piecewise functions, only one “song” can be playing at a time for it to be a function.
- Piecewise functions can capture more interesting situations where the relationships between the variables in play changes.
from @bowmanimal
Continuous Everywhere but Differentiable Nowhere 
I have used roller coasters to connect with the idea of rising and falling functions. I showed pictures of roller coasters and drew the direction (left to right) of the car. Every single learner could tell me whether the function was rising or falling.
Then I drew the function over the picture of the coaster. Same thing.
Then I drew the function and took away the picture (using ppt as a way to show a pic on the whiteboard) so the learners could see just the function. They again were able to do so.
These were not calculus classes, but algebra 1 and 2 classes. Funny thing was, on the exams, some drew little roller coaster sleds to help them. It was awesome!
I used the image of rollercoaster too but I showed one that went underground to connect the idea of when a function is positive or negative.
I use stepping stones to discuss terms in a sequence. If you have an arithmetic sequence and want to know how many terms there are in the sequence, you first have to see how far you go until the end (a_n – a_0), then see how many steps you took to get there (a_n – a_0)/d, then you have to add 1 for the original stone you were standing on, but didn’t take a step to.
I guess I usually compare piecewise function to flipping a switch on some toy. It’s doing one thing and then you flip the switch and then all of a sudden it’s doing something completely different that may or may not even be related to the original thing it was doing.
I use meeting friends as a simile for continuity. You call your friend and say you want to meet at the football game at 7pm. So, our definition of a successful meeting is to meet at the game with your friend. What could happen? Well, the first thing to check is if there even is a football game at 7 or maybe it was moved to another time or just called off altogether (f(c) exists). The second condition is that you and your friend actually meet up somewhere; you could go to different places or your friend could get grounded before they even left the house, etc (limit exists). The last condition, if there is a game and you and your friend actually do meet, do you actually meet at the game? Maybe the game was moved to the other high school and you and your friend meet up for a home game, but it’s not happening where you want (f(c) = limit).
I guess I do a lot of them, but I’m forgetting more right now.
love the meeting friends one for continuity – i’ll definitely use that next year. i remember those rules being difficult for them to remember. thanks!
I use hiking up a mountain (a POSITIVE thing) and going down a mountain (my expedition is over, I’m kind of down that my big trip is over, I’m feeling a little NEGATIVE) as a way to help students understand positive and negative slopes. I make it into a big story with drawings, etc. They really like the story and they can quickly see when they look at a line if the slope is positive or negative. “Am i going up or down the mountain?”
These aren’t things I teach, but some thoughts…
A lot of people use dominoes to describe how mathematical induction works. I don’t teach induction, sadly, so I can’t do that.
Inherently, by it’s name alone, we have the pigeonhole principle.
Venn Diagrams and maybe Google+ circles. (Dante put people in circles, too.)
For things that are commutative versus not commutative, one example for non-commutative is: putting on socks then putting on shoes, verus putting on shoes then putting on socks.
Wait, I don’t think the last two examples are what you’re talking about, but I’m too lazy to delete them.
Sam
A little late to the game, but FWIW:
I know the school I will be teaching at this year does a “mini field-trip” to introduce the idea of slope. We go out and walk up and down Potrero Hill in various places with various steepnesses.
This is very much a consructivist approach which makes a lot of sense to me – the trickiest thing being judging the gap…too big a gap and you lose the kids, too small and it all sounds lame. Vygotsky refers to the zone of proximal development (ZPD) and the tricky bit is judging proximity.
More theories I like along this line are Connectionism as well as Tangential Learning (yeah, that’s right, just google them if you haven’t heard of ’em as I ain’t going into them in depth here, ok?)
Here are some examples as per your request above:
Problem vs exercises
Teaching equations big-picture style
and I did blog a bit about Tangential Learning here
So it all boils down to providing a context for learning which I’ve mentioned here before…or maybe that was a comment in another blog.
Anyway, I don’t teach Calculus but if I did, I will use your Frankenfunctions. Hmm, could apply it to composite shapes – Frankenshapes or more complex problems/questions – Frankenquestions. double-edged sword though due to connections with a monster (what many perceive maths is) and sounds a bit like f**ken (again what many perceive maths as)…
ps, I have Brain Rules – but I haven’t read it yet. thanks for the reminder.