I’ve learned so much math and strategy by listening carefully to students who haven’t yet “learned” that some things are impossible.
If you’ve read many of my posts, you know I’m a major proponent of the power of CAS for teaching and learning. Following is an unexpected solution to a problem I asked on a quiz. It is also one example of why I believe we should be giving our math and science students access to CAS technology.
I’m teaching (for the first time) a senior Calculus-Statistics class at my school this year. It’s been over a decade since I’ve taught a stats course, so I was pretty excited with this assignment.
This is the second year of the course, all of the students have TI-Nspire CAS calculators, and my high school is in its rollout year of a 1:1 Macbook program. Because of the laptops, all students now have Nspire CAS software.
Using the Nspires, it is possible to avoid all use of normal distribution tables. For the first time in my career as a student and teacher of statistics, I haven’t spent a single moment on the statistical tables in the back of the book, and I think my students are actually stronger and understand better for it. Basically, there are two calculator/computer commands we are using for working with normal distributions: normCdf(a,b,m,s) and invNorm(c,m,s).
- For normCdf, a and b are the respective upper and lower bounds of the normal curve with mean m and standard deviation s. The function returns the percentage of the normal area between a and b.
- For invNorm, c is the area of the left-tailed region bounded by and b under a normal distribution curve with mean m and standard deviation s. The function returns the value of b.
This software is so much faster, more powerful, and more flexible than the tabular computations that comprised all of my prior teaching and learning of statistics. While my students still have a long way to go in their learning, I think their performance is clear evidence that it is time for normal distribution tables to follow into extinction the tables for roots, logarithms, and trig values.
The Quiz Question (Lifted from a stats textbook from which I used to teach at a local community college):
As reported in Runner’s World magazine, the times of the finishers in the New York City 10K run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. … If my time was at the 70th percentile, how fast did I run?
OK, it’s not a deep question at all, but I like what my student did with it.
What I expected:
Given the area under the curve, my students should have invoked the TI-Nspire’s inverse normal command with the area, mean, and standard deviation for a quick solution. This image of K’s response is a quick example of what I expected.
What I got from one student:
I was very surprised when I read T’s response.
The first thing to note here is that T obviously made a mistake in his computations which led to his wrong answer–but his insight was grand. Rather than memorizing and using the two computer commands, T recognized that his CAS might let him get away with a single command. The heart of his syntax says normCdf(x,,61,9)=0.7 which means there is some region bounded on the left by some unknown x and on the right by which has area 0.7.
I think T’s insight is great for several reasons:
- He came up with this alternative solution all on his own!
- Rather than memorizing “given area means use invNorm“, T clearly understood that this problem gave the area under a normal curve for some unknown boundary–arguably a deeper understanding of inverses than any memorized algorithm.
- He recognized the power of his CAS to solve equations that are otherwise algebraically complex or impossible. Particularly nice is his comfort with normCdf as a function and his use of that function in an equation that can be solved.
- I doubt T actually considered this point, but any CDF is automatically a 1:1 function, so an nSolve command can be safely used without worrying about missing potential multiple solutions.
In the early 1990s when I was using TI-89s to teach a statistics course at a local community college, I routinely used a solve command on CDF commands with great success, so I naturally tried solve(normCdf(x,,61,9)=0.7,x) on my Nspire prior to starting this unit. I was disappointed that my TI-89 capability from a decade earlier no longer worked, and I was too “trained” to consider nSolve (numeric solve) when solve had already failed. This is a major reason why I believe all teachers should encourage all of their students to be creative and to search for solutions that make sense to them.
I’ll try to offer in a future post some more technical explorations I’ve done on this technique inspired by T’s approach.