What is “quantitative reasoning”?

If you read this blog regularly, you know that I teach a course with the strange title of “Quantitative Reasoning.” What does that mean?

I’ll describe the course in general, followed by specifics of all four units.

Generally referred to by the abbreviation “QR,” this intensive course extends over two consecutive summers at the Crimson Summer Academy at Harvard University. Together with one or two co-teachers (one of whom has always been Joyce Bunten), I’ve taught QR to rising high-school sophomores and juniors for the past 16 years! As you would expect, it took a few of those years before it settled down to its current form and curriculum; we were tuning it up to be sure that we included topics that were important, accessible, and relevant. Here’s what we tell the students before the summer begins:

During the next two summers you will take various courses at the Crimson Summer Academy. One of them is called Quantitative Reasoning — or QR for short. You may well wonder what you’ll be learning in a course with such a strange title!

Basically, you’ll be applying mathematical thinking to real-world applications by deepening and broadening your knowledge of math. It’s not just a math course, although you’ll definitely learn a lot of math along the way. It’s a course in using math, in developing your ability to reason with numbers (and other quantitative objects, such as graphs, functions, and tables). You’ll be studying algebraic and trigonometric models, demographics, models of voting, and cryptography. In these topics you’ll learn more about the uses of algebra and trigonometry; how local and presidential elections work; how to apply math to other subjects like physics, economics, and government; and how to use math to construct secret codes like the ones used by spies and the ones that protect credit card numbers over the Internet. You’ll use a lot of technology along the way.

So that’s the elevator pitch (as long as you’re in a tall building, or at least a building with a slow elevator, like Sever Hall). If you’re interested in more specifics, here are the topics, listed sequentially in the order in which we teach them; the first two units are for rising sophomores, the last two for rising juniors.

Unit I: Algebraic Models

In the Algebraic Models unit we study real-world applications of three families of algebraic functions —linear, exponential, and quadratic — along with writing computer programs for various mathematical purposes. Students learn how to perform regressions with the wonderful Desmos app, including comparing and applying coefficients of determination.

  1. Brief review of linear functions.
  2. Stamp prices (continuous or discrete?).
  3. Coding (getting started with Blockly).
  4. More on linear functions.
  5. Algebraic language and other delights.
  6. Fitting equations to data.
  7. Coding (digging deeper).
  8. Finding the best fit.
  9. Coding (turning blocks into code).
  10. Which model works best? (linear, quadratic, and exponential models).
  11. Radioactivity (extended problem set on exponential decay and real-life interpretations).

Unit II: Models of Voting and Demographics

In Unit II we study a variety of models of voting and demographics, along with associated statistics. This unit incorporates a substantial project, in which each group of four sophomores visits and studies a Boston neighborhood, produces a mostly quantitative analysis of that neighborhood’s demographics and voting patterns, and presents it to the class. Because a lot of lead time is required for completing this project in one short summer, it is necessary to begin it well before Unit I is completely finished. As a result, the two units unavoidably end up overlapping.

  1. Intro to the role of the states in the Electoral College, as well as the role of districts, wards, and precincts in City of Boston voting.
  2. A demographic questionnaire.
  3. Very brief analysis of conceptual hierarchy of Western (Abrahamic) religions.
  4. Intro to election methods in other states and countries.
  5. Demographics I (racial and religious distributions).
  6. Electoral College.
  7. Two-round runoff method.
  8. Demographics II.
  9. Presentations of projects.
  10. Boston’s version of two-round runoff.
  11. Cambridge model of voting (basically Ranked Choice Voting).

Unit III: Crypto

In the Crypto unit we study six different cryptosystems, along with some associated number theory.

“Crypto” is short for both “cryptology” and “cryptography” — a convenient abbreviation, as most people do not distinguish these two words. Cryptography is a branch of applied mathematics (mostly using algebra, number theory, and linear algebra) that focuses on constructing ciphers and codes and using them to encipher and decipher messages. Cryptanalysis is a different branch of applied mathematics (mostly using statistics and linguistics) that focuses on breaking ciphers and codes. In other words, if Alice sends an encrypted message to Bob, they are both using cryptography; if Eve is eavesdropping and trying to break the code, she is using cryptanalysis.

So what is cryptology? That’s merely an umbrella term including both cryptography and cryptanalysis. Hence the simplicity of just saying “crypto” to cover the entire field.

Our POV is that crypto deals with functions. Just like traditional algebraic functions, each one can take an input and produce an output, where a given function always produces the same output for a given input. Most functions in algebra use numbers for the inputs and outputs; in crypto we usually use letters. As in algebra, a function can be represented as an algebraic rule, or a table, or a graph, or a description in words. As in algebra, the inverse of a function is important when solving an equation — in this case when deciphering a message. We call the input message (written in English or some other language) the plaintext; we call the output message (written in code) the ciphertext. A particular function may process one letter at a time, or a string of letters.

A cryptosystem is just a family of functions — like quadratics in algebra. In this course we study six cryptosystems:

  • Substitution, which requires a complete table of input letters (the plaintext) matched with output letters or other symbols (the ciphertext).
  • Caesar, which requires a shift; just shift the letters the agreed-upon amount, wrapping around if you pass the end of the alphabet. In other words, add a constant and express your results in mod 26.
  • Affine, which is just a linear function. We always number A as 0, B as 1, etc. Agree on a multiplier and an adder, write the result in mod 26, and convert back to letters.
  • Vigenère, which requires a key word or phrase that we agree on. Write this key under your plaintext, one letter at a time, repeating as often as necessary. Then add each plaintext letter to the corresponding letter of the key (mod 26).
  • Matrices, which require agreeing on a square matrix (we use 2-by-2). Use matrix multiplication and mod 26 to multiply it by a 2-by-n transcription of the plaintext after writing the plaintext in pairs vertically. Then read the product back (vertically) to form the ciphertext.
  • RSA, which we teach in a slightly simplified version, using WolframAlpha as a necessary calculator.

Along the way, we explore a bit of number theory — primarily factoring, modular arithmetic, and reciprocals. You will see the term “invertible and non-invertible” ciphers from time to time; an invertible cipher is one that can be deciphered by the intended recipient. In the case of affine ciphers this means that the multiplier must have a reciprocal in mod 26. In other words, the multiplier and 26 are relatively prime (we say “share no prime factors”).

Speaking of mod 26, we also do a fair number of examples from other languages, so that students don’t get frozen into this particular mod. We don’t want anyone to think that there’s something magical about 26 or its factors.

Unit IV: Trigonometric Models

In this unit we study real-world applications of trigonometric functions, interpreted as the x- and y-coordinates of a point moving around a unit circle.

  1. Geometric background/brief review of Pythagorean Theorem and special right triangles.
  2. Circular motion.
  3. (Co)sines of n π/6 and n π/4.
  4. Periodic functions.
  5. Graphs and transformations.
  6. Putting it all together.
  7. Models of periodic events.
  8. Intro to using trig and exponential functions to model musical notes.


Categories: Math, Teaching & Learning