**Gabriel Cunningham** (Department of Mathematics)

STEM EDX Fellow, AY 20-21

### Overview

In “Mastery-/Standards-Based Grading” students have a concrete set of standards they are working toward, and there are opportunities to re-assess standards. (See this blog post from the American Mathematical Society for example.)

Mastery-/Standards-Based Grading is appropriate to use in STEM and beyond. Advantages include:

- Clear articulation of the concepts and skills that we want students to master.
- Encourages student meta-cognition: students see where they need extra work and can focus their studying more effectively.
- Provides opportunities and incentives for students to read our feedback on homework and exams and act on it.
- Helps ensure that students have mastered material that is needed for later material in the course.

There are many approaches to Mastery-/Standards-Based Grading, and it is possible to build in a small component of a course that uses these principles to support student success.

### Description of Activity and Assessment

Weekly written work (homework and/or quizzes) is essential in Calculus so that students get practice solving problems and learning how to organize their work.

To help give students multiple opportunities to display mastery, I used a “quiz/requiz structure”:

- The quizzes were asynchronous and unproctored. They were available on Gradescope over several days, so students could find a time that worked well for them and study at their own pace. They were also untimed.
- For the requizzes, each problem was posted separately. I typically had about 50-60 students taking the requiz for a particular problem, and at least half of the students who requizzed (and usually many more) achieved full credit on each requiz problem.
- I usually give a worked-out answer to students whose answer was assessed as “Getting Started”.

The description below is from my Fall 2020 syllabus:

*Quizzes*

*Roughly every week, you will have a take-home quiz to test your mastery of key concepts, calculations, and problem-solving strategies. These quizzes are the primary method I have of giving you feedback and directing your learning, so it is important that you complete the quizzes on your own without outside assistance. Different quizzes may have different guidelines for what materials are allowed, so please read each quiz’s instructions carefully.*

*Quiz problems will be marked as one of the following:*

*“Mastered” – The answer is correct, or it contains only a small error that is not directly relevant to the topic being tested.**“Progressing” – The answer is substantially on the right track, but it is missing some key pieces.**“Getting Started” – The answer has only the beginnings of the right concept or method, or the answer is blank.*

*Your score on a quiz is equal to the number of problems you have Mastered. There is no partial credit for a score of Progressing, but it shows that you are on the right track and just need to push a little more to Master it next week. (Keep reading.)*

*Each quiz also has a “requiz” which will be given in the following week. A requiz covers the same basic topics as the previous quiz, but the problem types may be different. Requizzes are optional and are open to every student. If your requiz grade is higher than your quiz grade, then I will replace the quiz grade with your requiz grade.*

### Sample Question and Assessment

“Suppose ** V(t)** is how many liters of water are in a bathtub

*minutes after 5pm. Suppose we graph*

**t****and zoom in on the point at**

*V(t)**= 3, and we notice that the slope appears to be 6. Write a one-sentence interpretation of what the physical situation is. In other words, what can we say about the bathtub, knowing that the slope of*

**t****at t = 3 is 6?”**

*V(t)*Learning Objectives

- A student can determine the units of the slope (rate of change) of a function graph. In this case, the slope of
has units of liters per minute, so I am looking for an answer that includes “6 liters per minute”.**V(t)** - A student can demonstrate that the slope is different at different points, and in particular that knowing the slope at one point does not necessarily tell us anything about the slope at other points. In this problem,
= 3 corresponds to 5:03pm, so the answer should say something like “At 5:03pm, the volume of water in the bathtub is increasing by 6 liters per minute.”**t**

#### Typical “Mastered” answer

“The water in the bathtub is changing by 6 liters per minute at 3 minutes after 5:00 or 5:03 pm.”

This is awarded full credit with no further comment. (Sometimes, if an answer is close enough to fully correct, I will mark it as full credit but leave a comment.)

#### Typical “Progressing” answer 1

“At the end of 3 minutes (close to 3 minutes) the water left in the bath tub will be approximately 6 liters.”

This meets Learning Objective 2 but not 1. I left the following comment: “What you said corresponds to V(3) = 6. (The height of the graph is 6.) What does it mean if the slope is 6?” I usually try to just give a bit of a nudge to students who are Progressing.

#### Typical “Progressing” answer 2

“From 5 pm to 5:03 pm, the bathtub will be filled with 18 liters of water since the slope is 6.”

This meets Learning Objective 1 but not 2. I left the following comment: “You’re assuming that the flow is constant — what if it isn’t? (What if we turn off the bathtub at 5:01 and then on again at 5:02?)”

#### Typical “Getting Started” answer

Usually blank or mostly off-base, such as “At 3 minutes, the volume in liters of the bathtub is half full.” I left the following comment: “The slope has units of liters per minute.

t = 3 means 5:03 pm. So: At 5:03pm, the tub is filling at a rate of 6 liters per minute.”