### "Discussing statistics: using problems with more than one correct answer to facilitate students' understanding"

Måns Thulin, Uppsala University

#### Abstract

In this short webinar I will describe a homework-based classroom activity that encourages the students to think of statistics as a way to solve problems rather than a set of fixed methods. The homework problems in our statistics courses for engineering students had long been textbook-type problems where there was a single correct solution which we wanted the students to find. Rather than thinking about what a reasonable model and statistical analysis for the given data set would be, our students tended to focus on trying to figure out how the clues in the problem text related to the methods discussed in the lectures. In a sense, they were solving the wrong problem. We decided to turn things around and to give the students homework problems where there were multiple solutions, neither of which was clearly superior to the others. Instead of handing in written solutions, the students discussed their solutions with the teaching assistants in small groups. I will give some examples of problems that we have given (covering modeling, probability and statistics), discuss how the students solved them and what they and we learned from the discussions.

#### Materials

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#### Comments

**Marc Isaacson:**

Very nice presentation Mans! I am also one who is very interested in developing student quantitative reasoning skills and that includes seeing that there are multiple possible answers depending on how the problem is approached. Keep up the good work!

**Måns Thulin:**

Cheers Marc! I think that moving away from the idea that there always is a single correct answer is important when learning statistics (and mathematical modeling in general), as it forces (or encourages, if you're not feeling to cynical) students to think more about why they should believe in a solution. Which they will have to know how to do once they graduate, since there is no back-of-the-book in real life.

**Bethany White:**

Thanks Mans for a very interesting talk! Many of my students want to know "the right answer". How did your students react to these more open-ended questions? Did they push for instructor/TA input about the answers (or assumptions)? If so, how do you handle these types of questions?

**Måns Thulin:**

Thanks Bethany! At first the students would, when they were more or less finished with their group discussions, ask the instructor for "the right answer". In those situations we tried to show that there was no (single) right answer: "one the one hand... but on the other hand...", which often amounted to verifying and repeating facts that the students themselves had already discovered in their discussions. So in some sense we told them that they had reached the "right answer" by concluding that there were multiple solutions. After a while the students got used to this type of open-ended questions and became comfortable with the idea of there being "no right answer": this was when they started to focus more on understanding the question and their solution, which was what we were aiming for.

**Nicholas Horton:**

I really liked this poster. Teaching students to wrestle with problems without one (or any!) correct answer is a great way to get them thinking about weighing evidence and drawing conclusions. Joan Garfield and colleagues at the University of Minnesota have utilized a similar approach with their Model-Eliciting Activities. For the elevator problem, is it reasonable to assume independence? I liked the part about possible mixture distributions. It might be interesting to see what happens if students are asked explicitly about possible dependence. @askdrstats

**Måns Thulin:**

Thanks Nicholas. I forgot to mention that the first thing that the students usually discuss about the elevator problem is that the mean weight should depend on what type of people walk into the elevator: if they are all teenage girls then their mean weight should be different than if they all were grown male hockey players, and teenage girls and more likely to walk into the elevator together with other teenage girls rather than hockey players. We then try to help them to translate this reasoning into a statistical model, where there is a single population mean weight but the weights may be correlated (although we do not perform the calculations under this assumption).