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When do college students learn rigorous proofs?

Mathematics Educators Asked by user13395 on September 6, 2021

I teach in a regional university.
In my department, students take their "proof course" (a course that sole focus on writing proofs) in the third or even fourth year.
All the courses before that have minimum proof component.
E.g., even linear algebra is taught without requiring students to produce non-trivial proofs.

Is this normal? What are the common practices?
Do students in other universities learn proofs in first or second year?

11 Answers

In my experience (U.S.), that's on the boundary between 2nd and 3rd year -- either the end of sophomore year or the start of junior year.

Two years ago I did a survey of Associate in Science (2-year) Mathematics degree programs. It's not common to have a dedicated only-proofs course, but I think many use a Discrete Mathematics course as a vehicle where proof-writing is taken seriously for the first time, and one of the core focus points of the course (I could be biased, but that's how it's used at my institution, following the Rosen text; previously Ross/Wright with similar themes). About half (6 of 12) of the programs I looked at have either a Discrete Mathematics or dedicated Introduction to Proofs course.

That also matches my own undergraduate experience, where the Introduction to Proofs course was again taken either sophomore or junior year.

Correct answer by Daniel R. Collins on September 6, 2021

At my school, UIUC (https://illinois.edu/), we have a dedicated proofs course for CS majors (CS 173: Discrete Mathematics) and a dedicated proofs course for math majors (Math 347: Fundamental Mathematics). CS majors will typically take CS 173 in the first or second semester of their first year, and Math majors will take Math 347 before the end of their second year, as it required or highly recommended for almost all math classes after Calc 3.

Answered by Reed Oei on September 6, 2021

This answer is meant to supplement Daniel R. Collins' answer, which is excellent—my goal is to draw a little more distinction between the "levels" of education and training in the US.

Short Answer

In the United States, "proofs based" courses (and formal proofs in general) are typically regarded as topics in "higher mathematics", and are taught to mathematics majors (rather than a more general audience). Undergraduates in the US typically don't specialize into a major until their third year of college. Thus most US undergraduates never take course in mathematical proof, and mathematics majors typically don't see such topics until their third year.

Academia in the US

Eduction in the United States is roughly broken into three levels:

  1. Primary and secondary eduction, which consists of grade school (Kindergarten to 5th or 6th grade; ages 5 to 11–12), middle school (or junior high; grades 5–6 through 9–10; ages 11–12 to 14–15), and high school (grades 9–10 through 12; ages 14 to 18 or so). The following descriptions are based on (1) my experience as a public school student in three states as a child, (2) the experiences of my siblings in two other states, and (3) my training as a classroom instructor in Nevada (in particular, I have probably spent more time reading and learning the Common Core than most folk, though I would imagine that there are folk here more knowledgable than I—I welcome their comments). Thus it is anecdotal, but (I think) broad enough to make some general statements.

    • In grade school, students are taught by in a single classroom by a single teacher throughout most of the day, with (perhaps) excursions to other classrooms for specialized instruction in art, must, PE, technology, and so on. Instruction at this level is very general—with respect to mathematics, it is mostly basic arithmetic, plus related "life skills" like how to count currency, tell time, and use a ruler.

    • In middle school, students start to take classes from more than one instructor—they will often have a "homeroom" class, where organization and directed studying take place, but will move as a cohort from one classroom to another, where they receive specialized instruction in mathematics, social studies (history, civics, etc), science, language, and so on. At this stage of instruction, students might be exposed to some very minimal mathematical reasoning, but it is typically informal. The emphasis is on rote memorization and algorithms.

      I'll note for clarity that not every middle school has the same structure; indeed, the structure of middle schools is quite variable. The common theme is that these institutions are transitional places where students go from being children in grade school to being young adults (teenagers...) in more rigorous high schools.

    • In high school, students start to have some choice in the classes which they take, and when they take them. Typically, math, language, science, and social studies classes are compulsory every term, but students take 6–8 classes every term, and there is quite a bit of flexibility in how those extra classes are filled out. There are also generally multiple "tracks" which can be taken. In short, there is some specialization which takes place in high school.

      It used to be that students were first introduced to proofs (in the style of Euclid, via the study of geometry) in the 10th or 11th grade (15–17 years old). My experience of these classes is that they are very scripted and algorithmic—students learn how to write "two column proofs". My impression is that the integrated math approach has gained significant traction in the US. This approach downplays mathematical proof, though doesn't completely eliminate it.

      In any event, American students very rarely see very much in the way of mathematical reasoning before they graduate from high school. American high schools are very general—there is very little specialization in high school. Beyond a handful of students who take Advanced Placement or "dual-enrollment" classes, most students take essentially same curriculum, and there is very little "tracking" (e.g. there are not really "college tracks" and "vocational tracks" in US schools, which I think is different from much of how the rest of the world operates—my understanding is that, for example, European students often start to specialize into academic topics as early as age 16).

  2. Tertiary education (or post-secondary education, or post-high school education), which consists of college-level studies through the completion of a bachelor's degree. I think that a lot of the confusion about the American education system is related to the transition from high school to college. As noted above, high school is very general, so the first two years of college are generally devoted to more general studies which are part of a classical liberal education. There are typically two tracks:

    • Many students spend their first two years of post-secondary education at "community" or "junior" colleges. These institutions teach general education classes, which are often taught by folk with masters degrees who are not required to conduct research. Community colleges generally offer only 2 year associate's degrees, which are not terribly specialized, and are seen as preparation for work, i.e. the majority of community college attendees obtain a degree and enter the workforce (and leave academia). However, a significant number to transfer to bachelor's granting institutions.

    • The other track is to matriculate at a bachelor's granting institution right out of high school. The first two years of study at such an institution are typically very similar to what a community college offers, but the classes tend to be larger, and taught either by PhD'd faculty, or graduate teaching assistants.

    Whether students start at a bachelor's granting institution or not, the first two years of post-secondary education tend to be fairly general. These "lower division" classes are meant to give students a broad foundation of knowledge, and to help them determine a course of specialization.

    After a student's second year of post-secondary education, they are expected to declare a major. That is, they are expected to pick a field in which they will specialize. In most majors, there is a major change in the type and style of courses offered at the "lower division" (which are taught to non- and potential-major) and at the "upper division" (which are taught primarily to majors). For example, lower division courses in anthropology tend to focus very broadly on the four fields (cultural, linguistics, physical, and archaeology), whereas upper division classes will be more specialized (people's and cultures of southeast Asia; archaeology of the Puebloan southwest, etc).

    In mathematics, the transition is marked by a change from "cookbook" classes (e.g. introductory calculus) to "proofs based" classes (e.g. introduction to analysis). It is common in American institutions to offer some kind of course which is meant to offer a transition to higher mathematics as part of this advancement from lower- to upper-division classes, though the nature of this transition is far from uniform.

    • At my bachelor's institution, the first proofs based course that many students took was undergraduate real analysis (limits, continuity, elementary metric topology, differentiation, and integration from a more formal point of view). This class was mostly attended by students in their third year (calculus and differential equations were prerequisites). This class had a very high fail rate (as it amounted to tossing novice students into the mathematical deep end), so the institution eventually started offering a specific transitional class which focused on proof techniques and logic as applied to set theory (the class essentially followed the skeleton of Halmos' book Naive Set Theory, though I think that some other text was used).

    • At my PhD institution, there was also a transitional class, called "discrete mathmatics", which is offered (typically) at the end of a student's second year of post secondary education. The class is meant to teach the basics of proof via combinatorics, modular arithmetic, and naive set theory, and is a "recommended" prerequisite for all upper division mathematics classes. My impression is that this is not an uncommon approach.

    In any event, mathematics majors (i.e. those students who have declared that they will pursue a bachelor's level specialization in mathematics) and minors (i.e. those students who are specializing in something other than mathematics, but are taking on a secondary specialization in math) are typically first exposed to "formal" proofs in the transition from lower- to upper-division coursework, which roughly corresponds to the end of their second year of college, or the beginning of their third year.

  3. Post-graduate education, which consists of masters and doctoral level training. My feeling is that an American masters or doctoral degree (particularly in mathematics) is quite similar to a masters or doctorate awarded by a European (or other) institution. As I don't think there is much difference, and as post-graduate students are expected to know how to read and write a proof from their first day, I'll stop here.

Answered by Xander Henderson on September 6, 2021

I think, in the US, it is the norm for classes focusing on proofs to be nominally at the 3rd-year level. This is because the third year of college is generally the first year that students have committed to their academic specialty.

More advanced students can still take those classes in the first or second year. This may end up being the norm for students expecting to go on to an advanced degree.

Answered by Mark Foskey on September 6, 2021

Yes.from the first day we learned proofs. But i dunno which approach i correct. We had a course called fundementals of mathematics which introduced us to mathematical proof and college level math. Also in every course including calculus we used to study proofs for every theorem. But it was frustrating and hard for us as freshmen and led to disappointment for some students. I guess it's better to have a lighter program which teaches how to THINK right instead of introducing students to a lot of proofs.then u can teach them what you think suits best for them.

Answered by Parmida Granfar on September 6, 2021

I'm in Scotland and did rigorous proofs in the first year of my degree (Physics) as I sat the full first year maths syllabus. Although, in the final year of high school we were introduced to some simple proofs such as sqrt(2) irrational, and had discussed logic and different proof methods such as contradiction and proof by induction.

Answered by Rowan Alethea on September 6, 2021

From my experience in French Classes Préparatoires, we learn proofs during year one without a specific course about it, just while we learn calculus and linear algebra, starting on day one (we actually start a bit in high school).

Maths in these classes are very rigorous, and everything that is taught gets proven (with few exceptions), even requiring to re-define all types of numbers from the ZFC set theory in first year.

However, these classes are quite elitist and may not reflect what happens in all French universities.

Answered by gdelab on September 6, 2021

I can only speak to my personal experience, but during my time in Undergrad there was a dedicated proof writing class ("Introduction to Higher Mathematics") that was coded as a 330 course. All courses coded above 330 required 330 as a prerequisite, while everything below it did not. Courses below 330 were Calculus 1, 2, 3; Linear Algebra, and (I believe) Differential Equations -- along with a ton of other options for non-STEM majors. During my time there, I think all of the math majors took 330 after taking Calc 2. Sometimes after Calc 3, or Linear Algebra, or Differential Equations (or in the same semester as these).

Answered by Andrew Sanfratello on September 6, 2021

In the UK, students usually learn proofs in the first year of a mathematics degree. My experience is similar to Sumyrda's answer.

They also gain some exposure to proof techniques before university in A-Level Mathematics and Further Mathematics, which include proof by contradiction, trig proofs, elementary algebraic proof and proof by induction.

Answered by A. Goodier on September 6, 2021

From day one.

In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus (which is a part of the high school curriculum here) and half an introduction to proofs.

And the very first homework assignments in analysis and linear algebra already included some simple proofs as well.

Answered by Sumyrda - remember Monica on September 6, 2021

Naval Academy has their proofs class spring of sophomore year. It is required for either applied math or pure math.

https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf

https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMP.pdf

They have it before linear algebra.

Answered by guest on September 6, 2021

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