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.