https://www.researchgate.net/publication/282947903_How_Math_Can_Be_Taught_Better
In this paper I write, “Practical problems will hold the attention of students going into the trades. They will also hold the attention of college-bound students because bright college boys live in fear of being embarrassed by a ‘dumb’ blue-collar worker when it comes to mathematics.”
For instance, in a geometry class, I might ask this question:
“I am building a warehouse out of which I will sell 30-gallon (o.d. 18 ⅝”) and 55-gallon (o.d. 23 ⅛”) steel drums of petroleum products. Most of my customers have pickups and so my loading dock must be about two feet high. I say ‘about’ because there will be a steel plate on a hinge to fold over the pickup’s tailgate and it can reach up or down a few inches.
“The problem is that some of my customers will arrive on foot intending to buy a barrel and roll it home. I cannot just drop a steel drum off the dock because that would damage it. Resting planks against the dock is not strong enough, so my first thought was to construct a wooden ramp in the shape of a right triangle with internal braces that can be set against the dock.
“But a permanent wooden structure would be in the way. So, when a customer arrives on foot, I am going to assemble and then disassemble a ramp by rolling an empty 55-gallon drum against the dock, then rolling an empty 30-gallon drum against the first drum, and then resting planks on top of both drums with hooks on the ends of the planks to catch on the edge of the loading dock. The planks must touch the level driveway, both drums and the lip of the loading dock with no gaps or deformation.
“How high should I build the loading ramp?
“How long should the planks be?
“What angle is the incline?
Can anybody here suggest some similar practical problems that will challenge high school math students? What is your opinion of asking students questions like this, whose solution can actually be tested by having them each cut a plank to the length they think is correct and then resting their planks against two barrels? Or do you feel that it is better to just train them for the multiple-choice questions found on standardized exams and which typically take about three to four minutes apiece to answer?
Article How Math Can Be Taught Better
I think you are correct that teaching math from the perspective of real problems is important and I do not believe that teaching toward standardization is the way to go. What I would suggest is to do the experiences as you are and then connect them to the standardization. What you are accomplishing by doing that is to show that there is a theoretical world of algebra and symbolic manipulation and the like that is fundamentally related to the problems we face in the world. Math as a field helps us to answer those questions.
Calculus is of course the prime example because of its intense connection to physics based on Newton's development of both areas.
As for examples, there are a huge number of practical examples that can help to exhibit many of the concepts in high school mathematics. A lot of them are fun and have bigger lessons to them:
My personal favorite is the idea of card games like blackjack or poker used to teach probability concepts. By showing that the probability of success in card games is relatively low, you can create an attitude toward gambling that is more consistent with the mathematical reality. It grabs people's attention.
I think another good one is the idea of compounding interest. A lot of people will one day take out loans from a bank and will blindly accept the terms of the payment from the bank. A really good experience to work out some algebraic techniques is to profile a loan as if you were the banker offering it. You can explore what happens when you pay extra principal on the loan and so many other techniques. There are many equations that help you do that. This is a common experience that almost everyone will encounter in their lives, so that makes it a really good one to help learn some algebra techniques.
http://hereweplaywithmathematics.blogspot.com/ brings a few problems that could interest you. Please mention the source.
Marcia-
Thanks, I just came back to this question and found your answer. It looks useful and, yes of course, I will mention the source of any that I use.
I am now writing a geometry book and am particularly interested in geometry problems. See the link.
Article Volume One: Geometry without Multiplication
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