When I taught a 100-level math class last semester where the compound interest formulas were a big part of the course material, we started with understanding plain old r^x, then moved into a*r^x (now with a starting point), then a*(1+r)^x (now with a percent ratio/multiplier where >1 grows the values and <1 shrinks them), then a*(1+r)^xt (now with a multiplier for converting time units), and finally, a*(1+r/t)^xt (now with a divider to convert the APR to some other time unit).
When students viewed the compound interest formula as a special case of the more general exponential functions (repeated multiplication with a starting point), it was a lot easier to use the formula correctly and to make intelligent changes to make things easier.
Interestingly, most of my students really loved this, but I would say maybe 1/6 absolutely hated the in-depth discussion about how the algebra works and found little value in it, even if they understood it. When I asked why they didn't like it, every one of these students replied that they would rather just plug the numbers in from the formula because it's easier that way.
That’s a beautifully rendered paragraph you wrote saying what the compound interest says. Especially how you described the relation between n times a year and 1/n th the rate. That relation is often obscure in the comprehension of students, textbooks and teachers. Nice.
Makes me wonder why we don’t ask more often, ‘what does this expression say about the quantities?’
Excellent food for igniting curiosity, "ts beauty comes not from glamor, but from the work it does. Every letter is famous, but its fame is earned; each has a job, and you can read the job from the first letter, a convention that rewards you for paying attention."
When I taught a 100-level math class last semester where the compound interest formulas were a big part of the course material, we started with understanding plain old r^x, then moved into a*r^x (now with a starting point), then a*(1+r)^x (now with a percent ratio/multiplier where >1 grows the values and <1 shrinks them), then a*(1+r)^xt (now with a multiplier for converting time units), and finally, a*(1+r/t)^xt (now with a divider to convert the APR to some other time unit).
When students viewed the compound interest formula as a special case of the more general exponential functions (repeated multiplication with a starting point), it was a lot easier to use the formula correctly and to make intelligent changes to make things easier.
Interestingly, most of my students really loved this, but I would say maybe 1/6 absolutely hated the in-depth discussion about how the algebra works and found little value in it, even if they understood it. When I asked why they didn't like it, every one of these students replied that they would rather just plug the numbers in from the formula because it's easier that way.
Love this story, Joshua. And I'm gonna say, 5/6 is pretty good!
This was such a great read to start my morning ; it got all the neurons firing. Thanks, Dr McCallum
That’s a beautifully rendered paragraph you wrote saying what the compound interest says. Especially how you described the relation between n times a year and 1/n th the rate. That relation is often obscure in the comprehension of students, textbooks and teachers. Nice.
Makes me wonder why we don’t ask more often, ‘what does this expression say about the quantities?’
Thanks Phil. I used to give question like that when I taught college algebra.
Excellent food for igniting curiosity, "ts beauty comes not from glamor, but from the work it does. Every letter is famous, but its fame is earned; each has a job, and you can read the job from the first letter, a convention that rewards you for paying attention."