Estimations and Models

Our main objective in this class is to use the tools of mathematics to make better decisions. We’ll explore where math is and isn’t useful, how to fit a real world situation into a mathematical model, and perform an accurate and legible computation for others to follow.

This section of the text introduces the idea of defining a decision, the estimation needed to support the decision, and a mathematical model to make the estimation. After we have made the estimation, we can evaluate its impact on our decision.

An important part of the estimation process is letting go of the idea that your answer will be “correct”. We often must make important decisions without complete information. Your answer will be an imperfect but useful estimation that will give us insight into a decision we are making. You can ask yourself as you begin the estimation how accurate do you need to be? This is a very different practice than many math problems which have exact answers.

Decisions

Our scientific progress and the strength of our democracy our based on a shared understanding of how the world is and what we should do to improve it. Many of these claims about how we should change our world are based on quantitative evidence. Often the most challenging part of creating estimations and models is determining what question you are trying to answer and what decisions you should make based on the evidence.

• Are plant-based diets a useful tool against climate change?
• Are we spending too much money on health care?
• What are the best ways to promote human health?

We are also seeing that people are putting deliberately false but plausible news on the web because it will earn them money. How will you use your knowledge of mathematics to sharpen your critical thinking and separate falsehoods from legitimate reporting? Will you be able to look at quantitative data and see any problems of justice that the data show? You are frequently presented with arguments persuading you to think, act, spend, or vote in a certain way. How will you evaluate the validity of these arguments, especially when mathematics is involved?

Critical Thinking and Arguments

There are several important concepts from critical thinking that we will use in this class.

• What makes a strong argument?

• What are logical fallacies?

• Many arguments rely on mathematical claims

• Are the arguments sound?

• Are the math claims made to support the argument valid?

• What assumptions to these claims make? Would you agree?

• What alternate explanations are possible?

Evidence vs Allegations

It is important to distinguish between a quantitative allegation and evidence. Some questions to consider when you hear an allegation.

• Is the allegation believable?
• Is evidence being presented that supports the allegation?
• Is there any evidence that could refute the allegation?

The Principle of Charity

In this class we will always use the principle of charity to evaluate arguments even if we do not agree with the argument. The principle of charity says we should:

• Consider an argument to be rational and worthy of exploration
• If we see an quantitative analysis we disagree with, it is only through understanding it that we can seek to find its flaws

Limits of Quantitative Argument and Reasoning

• Mathematics can only describe the world as it is
• It isn’t a good tool for claims about how it should be
• Positive statements are about how the world is
• Normative statements are how we think the world should be