I have been wondering over the past months about the extent to which this time in history might plant seeds of interest for our children, and how those seeds might sprout. Will the ongoing discussions about the nature of the coronavirus and possible mutations lead children to become more curious about biology? Will public health successes and failures lead to a future with increased demand for education on epidemiology or economics? Perhaps the riots of the past weeks will lead a larger share of our youth into studies of sociology, law, and moral philosophy.
At the same time, I am disappointed to periodically observe people rationalizing arguments based on faulty logic and bad math. (Did you know that observing umbrellas on the street is a great predictor of rain? Yes, it is causal, and the math proves it! Maybe we should just prohibit umbrellas to change the forecast…) So, given my background as neither an educator, scientist, or mathematician, nor engineer or statistician, I feel a bit uncomfortable initiating an idea for a lesson plan. But, I will post online and ask for edits amongst my friends who are some (or all) of these things!
My objective is to help our daughter, Camilla, “connect the dots” between the math she is learning (or will soon learn) and the ways these tools can help her make sense of the world. I would also like her to be able to think critically about arguments – both soundly and dubiously rational. Finally, I think there is a cool opportunity to link ideas to historical figures and ideas that can make the “so what” of these lessons energizing and cool.
I don’t think it is helpful to go into notation beyond addition and subtraction, multiplication and division, and fractions and decimals. Here are some thoughts. Let me know what you think!
Day One | Introduction to the goals
- Goals for this program
- Help provide tools to understand the world around you, especially with COVID-19 and protests unfolding around the USA, as well as much of the world
- Help identify instances when arguments seem based on good, rationale analysis
- Link the ideas and techniques to historical figures and stories to help understand why, collectively, the techniques we are discovering, and the issues we are discussing, are incredibly interesting. We are living in quite interesting times!
- Goals for today
- Discussion of goals for the week
- Review of goals for each individual day
- Some preliminary discussion of how we will compare three counties: USA, Denmark, and Greece
- Some preliminary review of websites that have information about these countries
- We should have a table that we can use to build on for the rest of the week
- Items to review
- Deliverables
- Create a table that demonstrates the population of each country, GDP per capita, spending on health care as a percentage, and household income as a percentage of total
- Things to think about
- What do you think these numbers mean?
- What surprises you?
- What might explain some of those surprises?
- Additional thoughts
- …
Day Two | Introduction to probability
- Review from last time
- We spent time reviewing our goals for the week
- We also spent time together looking at sources of information to compare countries
- Finally, we created a table to begin comparing the USA, Denmark, and Greece
- Goals for today
- Introduce some discussion about probability
- Also, introduce some discussion about per capita analysis
- Items to review
- Discussion of probability
- Coin flipping
- What are the possible results? How do you express likelihood in one trial? Multiple trials?
- Six-sided die tossing
- What are the possible results? How do you express likelihood in one trial? Multiple trials?
- What about expressing the likelihood of an event NOT occurring? What is the relationship with the likelihood of an event occurring?
- Coin flipping
- Coronavirus tracker
- Discussion of per capita analysis
- For comparison purposes…
- Discussion of probability
- Deliverables
- Create a worksheet that demonstrates the probability of flipping “heads” on a coin on a flip, “heads” three times in a row on three flips, and not flipping “heads” at all on three flips
- Create a worksheet that demonstrates the probability of rolling a 1 or 2 on a 6-sided die on one roll, three times in a row on three rolls, or not at all on three rolls
- Add to the prior table information about confirmed coronavirus cases in total
- Create an additional entry for each country about confirmed coronavirus cases in per capita terms per 50,000 people
- Things to think about
- What does per capita mean? How do you compare per capita GDP to the income concentration statistics from Day 1?
- How do per capita infection rates compare across the USA, Denmark, and Greece? Do those numbers seem roughly similar or different to you? Do they seem like large numbers or small numbers?
- How many people have you seen over the past month? How does that compare to the number of people you saw last year during your final days of school? What is the chance you came across someone who was infected with coronavirus this month?
- Additional thoughts
- Some talk about independence
- 100 penny bag example with 1 or 2 colored including implications of repeated draws with or without replacement of the observation (i.e. penny)
- A nice website about probability from Andreas
Day Three | Introduction to growth rates
- Review from last time
- We spent time talking about per capita values
- We spent time talking about probability
- We spent time working on our table of USA, Denmark, and Greece
- Goals for today
- Introduce the idea of simple growth and compound growth
- Determine why compound growth matters
- Items to review
- Simple growth
- You have $1 and, for the next ten days you get an additional $1 per day…
- Compound (exponential) growth
- You have $1 and, for the next ten days, the value doubles each day…
- Simple vs compound growth in finance
- 10% simple interest for 10 years
- 10% compound interest for 10 years
- Simple growth
- Deliverables
- Find a graph that shows confirmed coronavirus infections for the USA, Denmark, and Greece over the past few months
- Find a graph that shows growth rates of confirmed coronavirus infections over the past few months
- Things to think about
- Can you see the way low rates of infection can grow to be very substantial over short-periods of time?
- How do the graphs of infections and growth rates compare? What is similar and what is different? What do these mean?
- Imagine you can grow your savings for retirement. Would you rather have simple growth or compound growth? Why?
- Additional thoughts
- Discussion of R0
- Rule of 72
Day Four | Introduction to conditional probability and causation (and lack thereof)
- Review from last time
- We discussed rates of growth in simple and (exponential) compound terms
- We discussed how small observations can become quite large over periods of time within contexts of exponential growth (interest on money, rates of infection, etc.)
- Goals for today
- We will discuss how probabilities can be conditioned upon other things
- We will discuss how likelihood (or correlation) does not necessary mean causation
- Items to review
- Problem 1:
- Imagine a community named “State College” with 50,000 people
- Imagine 1,000 are infected with coronavirus
- Imagine 1,000 people go to the doctor
- Imagine, of the 1,000 people who to go the doctor, 750 of them are infected with the corona virus
- Problem 2:
- Imagine that, of 365 days during the year, it rains during 65 days
- Imagine that, of the 65 times that it rains, umbrellas are observed at Saint’s coffee shop on 63 days
- Imagine that, on days where it is not raining, umbrellas are observed at Saint’s coffee shop on 15 days
- Problem 1:
- Deliverables
- Demonstrate the probability of a person, picked at random from “State College,” will be infected with coronavirus
- Demonstrate the probability of a person, picked at random from “State College” who also went to the doctor, will be infected with coronavirus
- Demonstrate the probability of it raining on a given, random day based upon the rates described above
- Demonstrate the likelihood of observing an umbrella at Saint’s coffee shop on any given day, and compare that with days when it rains
- Demonstrate the probability of a person, picked at random from “State College,” will be infected with coronavirus
- Things to think about
- When you think about people who have jobs that permit working from home and compare their risk of interacting with a person infected with coronavirus with a physician or dentist, how would you assess the difference in likelihood?
- When you think about the likelihood of observing umbrellas at Saint’s coffee shop, in general and on days when it is raining, what does that tell you about the influence of rain on observations of umbrellas? What about the influence of observation of umbrellas on the weather?
- Additional thoughts
- A nice website dealing with notation and these ideas from Herschel
- Monty Hall? (Probably not!!)
Day Five | Risk and mitigation
- Review from last time
- We discussed how probabilities can be conditioned upon other things
- We discussed how likelihood (or correlation) does not necessary mean causation
- Goals for today
- Identify examples of risks in our lives
- Review and discuss risk management techniques
- Develop an understanding of how we might apply risk management techniques to risks we find in our lives
- Items to review
- What are risks?
- What is the insurance industry?
- Techniques for risk management: avoidance, prevention, mitigation, transfer, retention
- Deliverables
- Create a list of risks we encounter in our lives
- Choose two of these risks and assess whether each of the risk management techniques are applicable to managing those risks
- Things to think about
- What are health risks that we encounter in our lives?
- What is the effect of distancing, hand washing, use of masks, and symptom tracking individually?
- How do the applications of the techniques above, together, impact the risk of infection for each of us individually? What about across a population?
- Additional thoughts
- Some discussion of the K&T work on cognitive bias, including regression to mean in HR work in Israeli airforce
- Work on risk management techniques in the insurance business
Day Six | Populations and Samples
- Review from last time
- We identified examples of risks in our lives
- We reviewed and discussed risk management techniques
- We discussed how we might apply risk management techniques to risks we find in our lives
- Goals for today
- Review the definition of a population and parameter
- Review the definition of a sample and a statistic
- Identify measures of center and spread
- Develop an understanding of the importance of “n” – law of large numbers
- Items to review
- Look at this page on Wiki regarding the Central Limit Theorem, and then get quickly away from it…
- What are examples of distributions? Look here on Page 3 for an example of American population.
- Read this article about an estimate of infection in New York
- Read this article about differences between Denmark and Sweden
- Deliverables
- Create a graph that shows the likelihood of rolling a 1, 2, 3, 4, 5, 6 on a six sided die
- Create a table that shows all possible combinations of rolling a 6-sided die 2x
- Create a graph that shows the likelihood of rolling the sum of each possible pair of rolls (e.g. 1+1=2 which is 1/36, 6+6=12 which is 1/36, etc.)
- Write down observations of pairs of rolls, per above, based on 5x, 15x, and 30x, and insert into an Excel or Numbers chart
- Things to think about
- As you look at the comparison of the graph of probabilities of pairs of rolls to the 5x, 15x, and 30x observations, what do you notice?
- In the NYT article, are numbers referencing parameters, statistics, or both? For what purposes are these numbers being used? Do you have any concerns with the approach?
- In the WSJ article, where are per capita numbers referenced? For what purposes are these numbers being used? What jumps out at you as you read this?
- When we look at numbers reflecting infection rates of COVID within the USA, Denmark, and Greece, are we looking at parameters, statistics, or something else? What do we gather when we see such numbers and how can we interpret what they mean?
- When we think about protests happening in the USA today, how can we apply these ideas to considering whether certain parts of the US population are treated differently than others? How should we think about personal stories people share about injustice? How are personal stories different from statistics?
- Additional thoughts
- Review story of William Gosset
- Danny K thoughts about “the law of small numbers”
- Standards of truth | empirical, logical, emotional, by authority
Some examples of tables and worksheets Camilla and I worked through in this can be found below.
