Activities that replace lectures

This page collects together various college mathematics teaching activities which can be printed on paper for students to work on individually or in groups, and which do not need a lecture for students to get started. These are generally modeled on the POGIL approach to inquiry learning, but do not necessarily adhere to that standard. These are not worksheets that you have students work on after a lecture, these are intended to be reasonably self contained and written so that students can make sense of them with only occasional trouble-shooting from the instructor.

Key advantages of this way of teaching:

• Activities such as these get every student in the classroom actively engaged and thinking about new material
• Students benefit from reading and figuring out mathematics by themselves, with discussion with other students where they might sometimes be asking questions and other times explaining their thoughts to other students
• Activities can be edited and improved, so that every time they are used, they are better
• Activities can be used by novel instructors with good effect, since they do not rely on saying just the right things in the right way and in the right order in a lecture or in a lecture mixed with class discussion.
• Activities like this can be used on the first day of class to great effect. Read about first day projects.
• Activities like this can be used by students who miss class, or by students who need to review some mathematics for a different class.

The activities below are organized by course, and sometimes have textbook section numbers from particular textbooks used at Bowling Green State University when the activity was written.

Introduction to Statistics

• 1.1 Statistical versus mathematical questions.docx | Instructions
• 4.1 Make a scatterplot of height-weight data |
• 5.4 Coin flipping activity | Instructions | Web site | Followup page on confidence intervals
• Sampling distribution of a population proportion
• 6.2 Normal percentiles and detailed normal table, where students fill in various numbers under the normal curve
• 6.2 Normal percentiles and z-scores, where students calculate z-scores and look up normal percentages
• 6.2 Normal percentages for coin flipping, where students calculate z-scores for sample sizes 50, 500, 5000
• 7.2 Reporting point estimates and sample sizes, which makes clear how important samples sizes are
• 7.4 Confidence intervals for a population proportion, which briefly explains confidence intervals, then gives exercises which will probably need to be split over a few days; they get harder

Precalculus

• Properties of logarithms

Calculus

• There are many activities for Calculus I posted on a separate page for calculus

Mathematical proof course, sophomore/junior level

• Group activity proof writing activities is a link to a single PDF with an entire course of material. Full LaTeX source code is available in a Github repository. Specific activities include:
  • Even and odd numbers, basic definitions and proofs
  • Sum and dot product of 3-dimensional vectors, rewrite expressions using one property on each line
  • The Division Algorithm, proof of existence and uniqueness and some applications
  • Exploring inequalities
  • Contrapositive, process of elimination, contradiction
  • Inter-valued functions, used for rounding, which is important in the construction proofs below
  • Construction of an object with a property
  • Introduction to set theory
  • Set subsets and equality
  • Quantifiers and nested quantifiers
  • Union and intersection of sets, including unions and intersections of intervals
  • Mathematical induction
  • Infinite unions and intersections, for example, involving increasing and decreasing intervals on R
  • Deriving properties of inequalities, from definition of the positive real numbers
  • Construction of the real numbers, via Dedekind cuts
  • Union, intersection, Venn diagrams, complement, set difference
  • Operations on sets, de Morgan's laws
  • Review exercises

Metacognition

• Studying, Learning, and Getting Exam-Ready is a two-page activity that explores the difference between studying, learning, and really being ready for the exam, using a sports team learning a new play as a metaphor. The activity was inspired by the talks and books by Saundra Y. McGuire, but have diverged a bit from her approach.

Tips for creating activities that replace lectures

• Put a place for students to write their name. That way, you can collect the sheets, read them to see what changes to make, and then return them to students.
• Number the individual steps in the activity linearly from 1 to n. This makes it easier to get the whole class's attention on one particular problem as they work through it, if you want to amplify or clarify some point. I suggest not using subparts like 1a, 1b, 1c, but use your own best judgment.
• Start the activity with a short orientation to what is happening, then introduce some new thing, and then as quickly as possible, start having students answer questions. Long introductions will lose people's attention.
• When leading students through a new technique for solving a problem, I prefer leading them through a problem step by step, showing the correct procedure, and leaving blanks for them to fill in. This is instead of showing a complete solved example and then asking them to solve a different problem. I think it's hard for students to fully engage with a solved example. After leading them through a problem, you can give them another problem with less structure and larger blanks to fill in.
• The challenge in writing an activity is to shift the thinking work from you to the students, by carefully and quickly leading them to the thoughts they need to have to understand what is going on and what they need to do. It really does take effort to write a good activity. Remember, the goal is that you don't lecture at all, so all of what you want your students to understand needs to be baked into the activity, but without including long sections of explanation.