## Posts Tagged ‘Math 124’

### An example of why lecturing does not work very well

March 2, 2013

We just started discussing confidence intervals in probability and statistics. As expected, students had a difficult time with it.

As usual, they read the section, answered some questions online, and came to class. In class, we worked on clicker questions. The first was basically:

Q: The 95% confidence interval for the population mean $\mu$ is [x,y]. Based on this interval:

1. There is a 95% chance that $\mu$ is in this interval.
2. 95% of the observations are in this interval.
3. This method of creating intervals works 95% of the time.

This is a tricky idea, but the third choice is the best answer of the three. In my second class, only 2 out 26 students got it correct. This was to be expected, though, since it is a tricky subject.

So I basically gave a 15-20 minute lecture as to why the third one was correct and the first two were wrong. Actually, it is more accurate to say that I repeated a six minute lecture three times about how to think about this.

We had two more clicker questions related to confidence intervals, and then I gave them the following question (perhaps you recognize it):

Q: The 95% confidence interval for the population mean $\mu$ is [x,y]. Based on this interval:

1. There is a 95% chance that $\mu$ is in this interval.
2. 95% of the observations are in this interval.
3. This method of creating intervals works 95% of the time.

The class was completely split into thirds as to which of the three answers was correct (to be fair, the question was only isomorphic to the first question, not equal).

I re-gave my two more variations of my six minute lecture explaining how to think of each of the three choices.

Then I re-gave the question, only with the following choices:

1. There is a 95% chance that $\mu$ is in the interval.
2. The probability that $\mu$ is in the interval is 0.95.
3. 95% of the observations are in this interval.
4. Exactly two of these answers are correct.
5. Each of the first three answers are correct.
6. None of the above answers are correct.

The correct answer is “None of the above,” of course. Three of the 26 students got it correct, even though I had literally just told them why the first three choices were wrong two minutes prior to voting.

This means one of two things. Either

1. Either learning is incredibly complex, and lecturing is not a good tool to help people understand, or
2. I suck at lecturing.

To be fair, Peer Instruction was not working, either. But it is surprising to me that Peer Instruction works as well as it does, and it is surprising to me that lectures fails as miserably as it does. The confidence interval lesson is a good reminder of the latter.

The point is not that my students are dumb—they are not. Nor is it that they are bad students—they are not. The point is that learning is difficult (especially with tricky ideas like “confidence intervals”), and one must be sensitive to this fact.

### Grading for Probability and Statistics

January 23, 2013

Here is what I came up with for grading my probability and statistics course. First, I came up with standards my students should know:

“Interpreting” standards (these correspond to expectations for a student who will earn a C for the course.

1. Means, Medians, and Such
2. Standard Deviation
3. z-scores
4. Correlation vs. Causation and Study Types
5. Linear Regression and Correlation
6. Simple Probability
7. Confidence Intervals
8. p-values
9. Statistical Significance

“Creating” standards (these correspond to a “B” grade):

1. Means, Medians, and Standard Deviations
2. Probability
3. Probability
4. Probability
5. Confidence Intervals
6. z-scores, t-scores, and p-values
7. z-scores, t-scores, and p-values

(I repeat some standards to give them higher weight).

1. Sign Test
2. Chi-Square Test

Here is how the grading works: students take quizzes. Each quiz question is tied to a standard. Here are examples of some quiz questions:

(Interpreting: Means, Medians, and Such) Suppose the mean salary at a company is $50,000 with a standard deviation of$8,000, and the median salary is $42,000. Suppose everyone gets a raise of$3,000. What is the best answer to the following question: what is the new mean salary at the company?

(Interpreting: Standard Deviation) Pick four whole numbers from 1, . . . , 9 such that the standard deviation is as large as possible (you are allowed to repeat numbers).

(Creating: Means, Medians, and Standard Deviations) Find the mean, median, and standard
deviation of the data set below. It must be clear how you arrived at the answer (i.e. reading the answer off of the calculator is not sufficient). Here are the numbers: 48, 51, 37, 23, 49.

Advanced standard questions will look similar to Creating questions.

At the end of the semester, for each standard, I count how many questions the students gets completely correct in each standard. If the number is at least 3 (for Creating and Advanced) or at least 4 (for Interpreting), the student is said to have “completed” that standard (the student may opt to stop doing those quiz questions once the student has “completed” the standard).

If a student has “completed” every standard within the Interpreting standards, we say the student has “completed” the Interpreting standards. Similarly with Creating and Advanced.

Here are the grading guidelines (an “AB” is our grade that is between an A and a B):

-A student gets at least a C for a semester grade if and only if the student “completes” the Interpreting standards and gets at least a CD on the final exam.
-A student gets at least a B for the semester grade if and only if the student “completes” the Interpreting and Creating standards and gets at least a BC on the final exam.
-A student gets an A for the semester grade if and only if the student “completes” all of the standards, gets at least an AB on the final exam, and completes a project.

The project will be to do some experiment or observational study that uses a z-test, t-test, chi-square test, or sign test. It can be on any topic they want, and they can choose to collect data or use existing data. The students will have a poster presentation at my school’s Scholarship and Creativity Day.

I would appreciate any feedback that you have, although we are 1.5 weeks into the semester, so I am unlikely to incorporate it.