Mathematical Maturity Survey

The Mathematical Maturity Survey (MMS) is a placement exam for first-year students interested in starting in honors or advanced mathematics courses, especially 21-242 Matrix Theory. It is intended for students with prior exposure to college mathematics beyond calculus, and students must register in advance to take the exam.

The MMS is only used for placement in the first semester. After the first semester, placement in honors courses is based on performance in CMU courses.

Be sure to also read the corresponding article on CMU’s website!

Outcomes

There are four possible outcomes, listed in increasing levels of difficulty:

No placement (take 21-241 and Concepts)
Placement into 21-242 only (take 21-242 and Concepts)
Placement into 21-242 and receive a waiver for 21-241, Concepts, or both
Receive a waiver for both 21-242 and Concepts

Concepts refers to CMU’s intro discrete math course (15-151 / 21-127 / 21-128).

The vast majority of students either receive no placement or are placed into 21-242 only. The last two outcomes are very rare and require exceptional performance on the discrete mathematics and proof-based linear algebra sections of the exam.

Exam Topics

The MMS lasts approximately 2.5 hours and is held at 8:00 AM on Monday of orientation week. The exam is administered in person on paper and consists of four sections.

(Disclaimer: These sample questions are designed to be representative of the difficulty of the MMS. They are not official, nor should you use them to study anything beforehand. The MMS is a placement test after all!)

Problem Solving (Multiple Choice): This section tests mathematical intuition through short multiple-choice questions in a similar style to those seen on the American Mathematics Competitions (AMC). Questions involve algebra, combinatorics, number theory, geometry, probability, logic, and other olympiad-style topics.

Example Question

An organization has 100 developers. Every developer knows at least one of Python, Java, or Rust. There are 50 who know Python, 40 who know Java, and 45 who know Rust. If 30 developers know at least two languages, how many know all three?
(A) 5 (B) 10 (C) 15 (D) 20 (E) 25
Problem Solving (Short Answer): Similar to the previous section, but you will write short numeric or algebraic answers.

Example Question

You roll a fair six-sided die twice. You gain an amount of money in dollars equal to the number shown on the first roll minus the number shown on the second roll. For example, rolling a 6 followed by a 2 results in a gain of $4. Conversely, rolling a 1 followed by a 4 results in a loss of $3. Find the probability that the first roll is a 3 given that you lose money.
Introductory Discrete Math: This section assesses familiarity with proof techniques and discrete mathematical reasoning. You will be asked to write rigorous, well-organized proofs.

Example Question

Consider the sequence defined by $a_1 = 1, a_n = 2a_{n-1}+2, \forall n \in \mathbb{N}, n \geq 2$ . Derive an explicit formula for the sequence. Then, prove that your formula is correct using the Principle of Mathematical Induction.
Proof-Based Linear Algebra: This section assesses mathematical maturity through linear algebra and proof-writing. It emphasizes proofs, not computation.

Example Question

Let $\mathbf{q}_1,\dots,\mathbf{q}_k\in\mathbb{R}^n$ be an orthonormal list of vectors (using the Euclidean inner product on $\mathbb{R}^n$ ) with $k \ge 2$ , and let $\mathbf{v} \in \operatorname{span}(\mathbf{q}_1,\dots,\mathbf{q}_k)^\perp$ . Prove that if $\mathbf{v} \neq \mathbf{0}$ , then no two distinct vectors in the list $\mathbf{v}+\mathbf{q}_1,\dots,\mathbf{v}+\mathbf{q}_k$ are orthogonal.

You will have the full duration of the exam to work on these sections, and you can move freely between them at any time.

Timeline

Before the MMS

When registering for courses in July, add a backup course such as 21-241 if you intend to take 21-242. You will not be able to register for 21-242 until after the MMS. Make sure your schedule does not conflict with 21-242, or be prepared to replace another course with it later.

If you are interested in taking the MMS and your advisor didn’t send you any information about it, sign up using the form linked on the website. The deadline is usually the first week of August.

If you have testing accommodations, also contact the Office of Disability Resources (ODR) at least a week before the MMS.

On the Day of the MMS

Check your email for your assigned testing location. Arrive by 7:45 AM with your CMU ID and at least two pencils. The proctor will give you scratch paper. Free breakfast will be provided after the exam!

After the MMS

If you receive placement or credit through the MMS, you will get an email a few days after the exam with instructions on how to register for 21-242 and/or accept course credits. If you are invited to 21-242 but decide to not take it, then that is okay as well!

FAQ

Should I take the MMS?

If you are interested in mathematics and have some familiarity with proofs from high school, definitely give it a try! The MMS is not binding, and if you do well, the exam gives you more options to choose from for your math curriculum at CMU. There are many students who take the exam, are invited, and ultimately choose not to take 21-242.

How do I prepare for the MMS?

We don’t recommend preparing for it. The goal of the MMS is to measure your current mathematical background and problem-solving experience, not how much you can study beforehand. Intensive cramming will leave many gaps in understanding that make it difficult to succeed in CMU’s advanced math courses. A quick refresher of topics can be helpful, but any more studying beyond that is not necessary.

Do I need to answer every question?

Nope. In fact, many students earn placement into 21-242 without attempting the final section at all. Typically, doing well on the first three sections is sufficient. The proof-based linear algebra section is used primarily to identify students who can receive credit for 21-241/21-242, and you are not expected to be able to answer these questions.

Should I take 21-242 Matrix Theory?

There are several advantages and disadvantages to enrolling in 21-242.

Pros

You are given a higher expectation on rigorousness of arguments and proofs, which is a great practice for future Math and CS courses.
You get to bond with other first-years who really, really enjoy mathematics. And really, really good at it too.
You learn linear algebra through abstract vector spaces, which may help you in future abstract math courses and research.
You are on track for the Honors degree program, which allows you to potentially obtain a Masters in mathematics in 4 years.

Cons

This course is hard. You will commit significantly more time and effort each week on this course than if you had taken 21-241.
Abstract vector spaces may not be intuitive and may not be applicable or useful for the goals of everyone.
A corollary of this is that with the lack of numbers and computation in this course, you may not get as much practice with working through numerical examples and computing numerical answers.
The course assumes some discrete math knowledge that you may have to study on your own time.

This is just a brief overview of the opportunities. If you have more questions, there are course-specific reviews, and you should ask a current student who took 21-242!

With all that said, don’t be intimidated by the MMS. If you are interested in advanced mathematics and have a strong mathematical background, you are well prepared to succeed. Best of luck, and welcome to CMU!