|Follow the center of gravity.|
Students who are taught in an appropriately scaffolded fashion will understand better, retain material understanding longer, and apply the material more intelligently in non-standard situations. They will see the action, realize the math behind it, and "use this in RealLife" in ways that neither you nor they can possibly predict right now.
Throwing them immediately into the deep end, however, by giving them the incredibly messy RealLife questions with all of the ramifications and qualifications, coupled with seven different solution methods, actually blocks acquisition and development. The Gateway Arch is not a parabola but saying that in the quadratics chapter won't help the students.
The takeaway: Make the numbers fit the first few times. Your R²-value should be 1 until the kids get the hang of things. Your numbers should work out evenly for a while. Give them the messy stuff after they've mastered the simple.
That's why math books tend to look simplistic and abstract, why they have false-seeming questions ... they know that the kids can't handle the complex stuff yet. Word problems are deliberately simplistic - to make them realistic would lead only to needless frustration. That's why the students don't see five years down the road, why they can't see the long-term implications and utility of what they are seeing right now ... because they're still learning to drive this car and haven't gotten the basics of its operation down yet. Watch a young driver and think about that teenager learning algebra: they are overly focused on the minutiae of the work and not yet ready to drive fast and react to changing road conditions. If there's another teenager in the car, they'll get themselves killed ... that's why learners permit and first-year drivers are so restricted.
FalseRealLife is even worse. If you attempt to overlay a fourth-order function on a photograph of a cloud formation or a transcendental function on ivy-covered bushes, you destroy, in the minds of every student in the room, the utility and purpose of a regression -- there is no possible natural reason for that edge of that cloud to have any relationship with a fourth-order function or for the series of leaves to follow that curve. If you attempt to overlay a parabolic function on a Norman Window (semicircle on rectangle), you demonstrate the uselessness of mathematics by being somehow unable to come up with an example of data that is appropriately modeled by the regression you chose.
The takeaway: If you want students to see the utility of math, you have to use math in a useful way, using actual numbers and actual results. Bringing in your extensive knowledge of physics and other sciences here is worth the time.
For this reason, I include Bad Numbers in the FalseRealLife category as well. When "Johnny" winds up to be 71 meters tall, you have done yourself and your students a true disservice. This gem from a worksheet encapsulates this perfectly: "Math Problem 16. My sister greedily lapped up 483 liters of maggot juice from a saucer. I slurped up 5 times as much creamy maggot juice as she. How much delightfully delicious maggot juice did we drink altogether?" Besides the stupid attempt to engage the boys with "gross" humor, we have to ask ourselves how we expect the students to ever get a handle on the metric system or their own sense of "reasonableness" if we give them problems with numbers like this.
The takeaway: If you want students to relate, it has to be relateable, it has to be understandable, and it has to be real.
|When it's clearly not a fourth-order and you try |
to put a quartic on it, what message are you sending?
That your math is bullshit.
But, then there's FantasyLife which uses crazy situations and wildly weird math to model it. If you use this at the very end of a section, it can lighten the mood while opening minds. It's so bizarre that the students can't apply their RealLife knowledge to it and they then can focus on the math. The math is so inappropriate to the situation is draws laughter and silliness.
I bring you "Love Mathematically."