# Quantitative Reasoning

## Rationale

Mathematics is one of the great achievements of the human mind, and applications of mathematics pervade today’s society. A mathematical way of thinking, more broadly referred to here as “quantitative reasoning,” is widely used, for example, to justify data-based decisions, encode and protect information, manage the treatment of disease, provide a unified understanding of the forces of nature, and formulate government and international policies. As such, it represents several distinct modes of thinking, which can broadly be classified as analysis, logic, probability and statistics, and modeling. From each of these derive techniques that are applicable to specific classes of problems. Often, a combination of different quantitative techniques is necessary to approach specific situations.

Students completing courses that satisfy the quantitative reasoning requirement should have been exposed to multiple aspects of quantitative reasoning. For example, they should learn how to use deductive reasoning in problem-solving, apply the inductive process to draw conclusions through quantitative analysis, evaluate data and think probabilistically, assess the strength of numerical evidence, and mathematically model processes or systems to be able to predict (or change) their outcomes. In short, the main goal of courses that satisfy the quantitative reasoning requirement is for students to engage in multiple mathematical ways of thinking that will enhance their ability to make informed decisions as citizens and as potential leaders.

## Learning Goals

The quantitative reasoning requirement recognizes that the ability to understand and analyze measured quantities is not the only aspect of mathematical reasoning that is important. Another fundamental mode of reasoning involves the ability to think in the abstract and to switch reasoning and analysis from the abstract to concrete, from the cognitive model to the experienced reality.

The quantitative reasoning requirement thus includes two distinct, but complementary components, **quantitative analysis/inductive reasoning** and **deductive/formal reasoning**. Courses meeting the majority of the learning goals for either component would satisfy the quantitative reasoning requirement, as described below. The rationale and learning objectives for these two components are as follows.

### Quantitative analysis/Induction

Applications of mathematical analysis pervade today’s culture. We live in an era with vast amounts of quantitative information that can be easily accessed. Big data analyses have become indispensable in the operations of business, education, health, and other settings. Policymakers and ordinary citizens increasingly confront issues in science and technology that can be approached using mathematical techniques. For example, quantitative methods are used to analyze personal finances, formulate government policy, justify data-based decisions, encode and protect information, provide a unified understanding of the forces of nature, and manage the treatment of disease. Understanding the scope and power of mathematical analysis and how to draw conclusions from it enables graduates to better make informed decisions as citizens and as potential leaders of the country and of the world.

#### Goals and Perspectives

The main goal of the **quantitative analysis** component of the requirement is to provide students with experience in the use of mathematical and statistical methods in the analysis of real-world problems.

- Students will be able to set and solve numerical or geometric problems in a variety of contexts.
- Students will be able to analyze data with appropriate tools, think probabilistically, interpret results, and assess the reliability and uncertainty of conclusions.
- Students will develop the skills to mathematically model processes or systems so as to be able to predict or change their outcomes. Students will gain an appreciation that models only approximate real-world situations and are therefore imperfect, and will develop the skills to quantify these imperfections.

### Formal reasoning/Deduction

The ability to abstract symbolic representations of arguments or problems and to utilize formal logic in the analysis of their structure is a distinct form of reasoning that empowers the human intellect, enhances critical thinking, and facilitates rational decision-making. Representing ideas in a symbolic manner and analyzing arguments with the help of logic are, first and foremost, mathematical exercises, but also occur in many other contexts. Many disciplines, such as the physical sciences, computer science, cognitive science, linguistics, and even music theory rely on the principles and rules of logic to classify, predict, and analyze.

#### Goals and Perspectives

The main goal of the **formal reasoning** component of the requirement is to provide students with experience in the mathematical way of thinking, especially insofar as this way of thinking fosters the development of disciplined habits of the mind and enhances the power of the intellect.

- Students will learn deductive reasoning in problem-solving through problems in which the system of formal reasoning is itself the object of study.
- Students will learn how mathematics and statistics can be used to abstract key features of our world and reason about these features in a general context.
- Students will be engaged with problems whose goal is to follow a rigorous path of deducing conclusions from simple basic assumptions.

## Criteria

A course recognized as meeting the requirements for quantitative reasoning is one that provides a rigorous basis in logical or analytical thought. In terms of learning goals, a course that meets any three of the six goals from either Inductive or Deductive reasoning would be considered a quantitative reasoning course. Rigorous courses in formal logic, statistics, computer programming, and calculus are expected to qualify for this designation, as are, for example, mathematically intensive courses in specific disciplines, where quantitative methods are applied to analyze and model observational data. Courses based on discipline-specific applications of formal logic may also qualify, given the level of formal logic employed. In all cases, it is expected that a substantial focus on the understanding and the application of mathematical ideas, as defined above, should form the core syllabus of a quantitative reasoning course.