Designing a University Math Unit In Real Time, part 2: Learning Goals

Yesterday, I wrote about my motivation, goals, and guiding principles for my redesign of the classes in our differential equations unit. Today, I am pinning down the in-class learning objectives.

It is tempting for me to jump right into lesson design: I’m drawn to creating examples and activities, which help me to “see” how the unit is going to turn out. As with any decision or project (especially curricular ones), spending time at the beginning writing down the “big picture” learning objectives and referring to them often not only saves time later on, but results in a more cohesive and better plan.

Photo by Marten Bjork on Unsplash

My lesson learning objectives are not all of the learning objectives that I have for students. They are simply the ones that I am going to focus my class on. Students are expected to come to class knowing the “basics” from their pre-class reading and WeBWorK assignment and will need to consolidate their learning after class too. The overall learning objectives are the same as those I set-out at the beginning of the year. But the things that I choose to emphasize during a class will change.

How do I choose my learning objectives for class? I use two criteria:

1. What will students need support in learning? These are usually learning goals that are difficult or that students often have misconceptions about.
2. What are the absolute essential components of the story that I’m trying to tell? These are points that link topics together and weave a common narrative through the course and through the Unit.

The story for this unit is centred around building mathematical models for the spread of the coronavirus. As we add assumptions that better reflect the “real-world” we will add complexity to the models that we develop as we go through the unit, moving from single differential equations in 2 variables to systmes of differential equations. We will integrate it into the Story of Calculus by representing changing quantities verbally, algebraically, numerically, and graphically. To me the BIG STORY is pretty clear, so I need a lot of reminders to share this with my students regularly.

Learning Goals for the Unit’s Lessons

Day 1: Verbal Representations

• Explain the assumptions behind 3 key disease transmission models: SI, SIS, and SIR models
• Translate between verbal and algebraic representations of differential equations. a) involving elements of disease transmission models; b) using the word “proportional”
• Determine if proportionality constants are positive or negative based on the meaning of the model

Day 2: Slope Fields

• Explain why there is an infinite number of solutions to a differential equation (without initial conditions); why does this feature increase the power and applicability of disease models?
• Given a differential equation, plot a portion of its slope field; use equilibrium solutions as a guide
• Predict & investigate the influence of changing parameters on the slope fields and solution curves of differential equations: a) involving the SI disease model; b) using technology

Day 3: Euler’s Method

• Visualize Euler’s method as “walking along a slope field”
• Starting with a system of differential equations modelling the spread of a disease, produce numerical estimates for the number of infected over time in 4 different scenarios (fixing different proportionality constants and different numbers of other variables)
• Explain why the assumption of fixing variables is not reasonable for the SIR disease model and preview the phase plane

Day 4: Separation of Variables

• Decide whether a differential equation is separable, and recognize that many differential equations fail to be separable. a) explain why a differential equation involving 3 or more variables cannot be separable; b) recognize the limitations of this algebraic technique
• Construct algebraic solutions to families of differential equations (use disease model examples)

Day 5: Growth & Decay

• Explain what growth in one variable means for the growth or decay of other variables in differential equations and families of differential equations
• Approximate proportionality constants in differential equations modeling disease progression (e.g. SI model) given numerical data of the disease progression
• Construct and analyze mathematical models similar to Newton’s Law (e.g. the amount of medication in a patient’s bloodstream given the assumption that the amount of drug remaining in the bloodstream is proportional to the amount or concentration remaining)
• Compare verbal, numerical, graphical, and algebraic representations of simple growth and decay models

Days 6 & 7: Systems of Differential Equations & Analyzing Phase Planes

Usually on these days we would study modelling problems and the logistic equation. Instead, we are going to study systems of differential equations

• Plot a portion of the phase plane of a differential equation, given different initial conditions; describe differences and similarities between phase planes and slope fields
• Describe the progression of a disease model out-loud and in writing, given a phase plane and different initial conditions
• Approximate proportionality constants in the disease model given numerical data of the coronavirus progression
• Using differential equations modelling the coronavirus progression, make numerical predictions for populations over time using Euler’s method
• Compute threshold values for the coronavirus in different populations, and use it to make recommendations about the spread
• Compare verbal, numerical, graphical, and algebraic representations of systems of differential equations

Next, I’m starting to dive into lesson planning: what models do I want to highlight each day? What types of activities should I use to best achieve the learning outcomes above? I’ll also need to write some new homework for Days 6 & 7 since I’ve changed the overall learning outcomes for these days too.

Question: Do you have any ideas for how I can respond to the racist attitudes that the coronavirus has fuelled in the unit? I think it will come out in our analyses that staying away from people who appear to be Chinese or Asian has absolutely no factual basis, but I’m also hoping to attack the topic earlier.