# Problem Sets

Problem sets are great ways to provide students with the practice necessary to gain mastery of new skills that you have introduced in class. Problem sets are also great at reflecting the nature of the scientific process, which so often involves problem solving, and in so doing help reinforce the explanatory power of your discipline. What kind of questions are students prepared to answer because they have taken your class? How are they better prepared to understand and explain the rules that govern the universe? How are they prepared to predict phenomena if you give them a particular scenario?

1. Some of the most motivating problem sets are those in which students learn something new by doing them. Are there questions that you can ask in which students can synthesize course concepts to understand something novel to them? For example,
1. are there any recent research publications that use the knowledge and skills that you teach to discover something new? If so, can you use such a paper to walk students through a multi-step problem to achieve the same discovery (even if you have to simplify)?
2. is there a skill students learn late in a class that can be used to reinterpret a concept taught earlier in the semester, or understand that concept more deeply?
2. Can you design problem set questions that prompt students to make back-of-the-envelop calculations that allow them to get a sense of “scale” in your discipline; to help students understand which types of approximation are typical and important in your discipline?
3. Problem set questions often serve two functions:
1. they reinforce a skill or concept that was taught in class by giving students the ability to practice using that skill in a variety of different contexts. In particular, you can ask how a concept can be applied when given different things or circumstances in the question prompt, or you can ask how that skill or concept can be used to provide different kinds of answers. The best way to help students build a discrete skill is often by asking very short, direct questions that only ask students to execute one skill at a time. For example,
• Draw a Lewis dot structure of a molecule given a chemical formula;
• Determine the pKa of a monoprotic acid given a simple titration curve;
• Calculate the momentum of an object given its velocity and mass;
• Draw an arrow pushing mechanism of a simple reaction
• Identify the chemical properties of a given region of a molecule;
• Take the derivative of simple equation in which only one technique is needed;
• Determine the directionality of an induced magnetic field given a current through a wire;
• Describe the steps of elongation during ribosomal protein translation
• Identify the types of molecules that can diffuse across a biological membrane
2. they synthesize multiple skills and concepts together to ask complex questions.
Synthetic, complex questions are often the most engaging, intellectually-stimulating, and internally-motivating questions for your students. They are the ones that most often address the strengths of the discipline: how these skills and concepts allow us to design a new drug to treat a disease, or test for a drug’s efficacy? how do the skills and concepts taught in class allow us to understand a problem like climate change, predict its consequences, or allow us to design technologies to minimize its effects? how do the skills and concepts taught in this class allow us to understand the birth of our universe, how stars form, or why we care about detecting gravitational waves? how can we accurately predict future problems and devise proactive solutions?

The best problem sets typically offer a mix of simple, discrete skill-building and complex problem solving skills. While the complex questions are typically more interesting and engaging, students are ill-prepared to address them without developing small, discrete skills, first. It is therefore helpful to design problem sets that ask students to exercise one skill (“how do I use the triangle inequality proof”) before asking them to use that skill in combination with other skills to solve a more interesting, complex question.

Some classes parse these two questions in two halves of the problem sets: part I includes “basic,” single-step questions and part II includes “applied,” multi-step questions. Some other courses mix these two question types throughout the problem set.

Another way of considering the distribution of question complexity is using Bloom’s taxonomy. Bloom’s taxonomy describes questions across a range of “lower-order cognitive skills” (e.g., “recall,” “define,” and “explain in your own words”) and “higher-order cognitive skills” (e.g., “apply,” “analyze,” “evaluate,” or “create.”) See Figure 1 below: Students need to be able to use the lower-order cognitive skills to be able to do the higher-order cognitive work, so problem sets should emphasize a range of testing the comprehension of factual knowledge and the application of knowledge in more complex, analytic problems.

You may notice that many of the different Bloom’s levels correspond to different verbs you can use when writing questions. A large list of verbs that roughly correspond to different Bloom’s levels can be found here.

4. Lastly, we often want students to engage with problem sets in a way that is authentic to how we engage with questions in our disciplines. One component of this is that we tend work collaboratively in our academic fields, and so it is worth emphasizing that we want our students to learn from each other and work collaboratively on our problem sets. To avoid plagiarism and academic dishonesty issues, it is important to emphasize that we want students to work together, but write down their own final answers. For example, we want students working together collaboratively at a blackboard while they figure something out, but we want them to then return to their seats individually to write down their final answers.