13
$\begingroup$

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?

$\endgroup$
  • 6
    $\begingroup$ I would hope that all these students did proofs in high school geometry. $\endgroup$ – Ben Crowell Jun 24 at 22:44
  • 8
    $\begingroup$ @BenCrowell. No, or if they did, most didn't do it very well considering at least around half of high school math teachers are pretty poor at coming up with proofs themselves. $\endgroup$ – Alexander Woo Jun 24 at 23:18
  • 4
    $\begingroup$ Note that outside of the US, it may not be clear what a "regional university" is and why that might make a difference. Also, in some countries, a Bachelor's degree is often nominally three years in duration and specialising in a subject might begin as early as in the first year. In other words, the major is chosen at the outset. $\endgroup$ – J W Jun 25 at 11:21
  • 5
    $\begingroup$ @BenCrowell: The "proofs" I learned in (U.S.) high school geometry were all of the two-column variety, and offered little (no?) insight into how practicing mathematicians write (and more importantly - think about) proofs. That said, this was back in...2000? So maybe the curriculum has changed. $\endgroup$ – Jason DeVito Jun 25 at 15:51
  • 3
    $\begingroup$ Of course in a "regional university" in the US, during the first two years most of the math courses include also students who do not intend to get a degree in mathematics. For example, science and engineering students. Of course those are not proof-intensive courses. $\endgroup$ – Gerald Edgar Jun 25 at 21:34

11 Answers 11

12
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 4
    $\begingroup$ on the boundary between 2nd and 3rd year --- That's been my experience also. Of course it could easily be a year earlier for those in stronger programs who also plan on going to graduate school, but this is a minority among all (mathematics) students in U.S. colleges collectively, and even those not doing this until the Fall of their 3rd year should still not be seriously disadvantaged in getting into a graduate program (except maybe the top 20 or so programs) if the primary reason is circumstantial rather than ability and potential. However, in some other countries, it is very different. $\endgroup$ – Dave L Renfro Jun 24 at 23:11
  • 1
    $\begingroup$ Is your survey publicly available somewhere? $\endgroup$ – ssquidd Jun 29 at 2:00
  • $\begingroup$ @ssquidd: It wasn't published, but I'd be happy to share it. Email me at dcollins at kbcc.cuny.edu if that's good for you. $\endgroup$ – Daniel R. Collins Jun 29 at 4:13
21
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 13
    $\begingroup$ I was about to share the same experience – in most German universities, a math lecture (for math students, this doesn't apply to lectures such as “math for engineering students”) without rigorous proofs would be unthinkable even in year one. That being said, the dropout rate for math is extremely high – this source reports it as 80%. $\endgroup$ – Sora. Jun 25 at 8:10
  • 8
    $\begingroup$ @Sora. Yes. Our professors framed this as "better to discover whether you can wrap your head around proofs during the first semester when there is still time to switch to another major without much difficulty than to discover that it's not for you after 2-3 years of study". $\endgroup$ – Sumyrda - remember Monica Jun 25 at 8:28
  • 4
    $\begingroup$ @Tommi: Mathematicians didn't start with epsilons and deltas either. They came up with that formality only when they realized they (occasionally) needed it. If you would show it to students after they realize they need it, you would probably have a very different experience regarding how "demanding" they find it. Starting with epsilons and deltas is like trying to teach Peano's axioms before teaching how to add and multiply. $\endgroup$ – Matt Jun 25 at 13:40
  • 3
    $\begingroup$ In Germany, it would typically begin even before that, in the last years of the Oberstufe. I vaguely remember a proof being part of my Abitur, but I might be misremembering. $\endgroup$ – Jörg W Mittag Jun 25 at 14:12
  • 4
    $\begingroup$ @Sora.: Baden-Württemberg, 1998, Mathematik-Leistungskurs. Note that this was a particularly hard exam, which was actually successfully challenged in court and resulted in an automatic +1 point bonus for every student taking the Mathe-LK Abitur, if I remember correctly. Even if we exclude high school, proofs were definitely a big part of the voluntary "maths refresher" prep course that older students offered to freshmen over the vacation days before the first semester at my university (University of Karlsruhe, now Karlsruhe Institute of Technology) $\endgroup$ – Jörg W Mittag Jun 26 at 9:20
5
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ And indeed prior to that; simple deductive algebraic proofs (e.g. 'prove that the sum of the squares of two odd numbers is even') feature in the GCSE course, normally taught over two years and examined at age 16. $\endgroup$ – dbmag9 Jun 26 at 13:19
5
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 3
    $\begingroup$ Most students at a regional university are like the ones who barely passed their bac, not the ones who are considering a grande ecole. $\endgroup$ – Alexander Woo Jun 26 at 19:35
  • 1
    $\begingroup$ True, I've added an edit. $\endgroup$ – gdelab Jun 27 at 13:25
5
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Great explanation! Is there any way to find stats on the departments that offer first proof-centric course in the 2nd year vs 3rd year (or even vs 1st). My suspicion is that offering the first proof course in the 3rd year will lead to poor preparation for grad school in mathematics which is the problem I'm seeing in my school. $\endgroup$ – ssquidd Jun 29 at 1:55
  • $\begingroup$ @ssquidd I don't have those data to hand, nor do I know if anyone has studied the issue. In this answer, Daniel R. Collins indicates that he has studied the problem among (essentially) community colleges, so he may have a better idea of the statistics. $\endgroup$ – Xander Henderson Jun 29 at 1:59
  • $\begingroup$ Regarding preparation for graduate school, I don't think that it is a major issue. The best students, who are looking to get into a PhD program right away, will likely come in with some classes already, and may take their intro to proof in their second year. There are also a significant number of students at less prestigious institutions who will go from their bachelors to a masters before applying to PhD programs... $\endgroup$ – Xander Henderson Jun 29 at 2:00
  • $\begingroup$ Finally, even if a student waits until their third year to take intro to proofs and intends to enroll in a PhD program, a year and a half is typically enough to get the prerequisite courses in analysis, topology, and algebra. And don't forget summer classes. :) $\endgroup$ – Xander Henderson Jun 29 at 2:01
  • $\begingroup$ In my school, the way we schedule classes often means just one year of "real math" courses after the proof-centric course. I doubt one year of mathematics gets many students ready for graduate program. Sure, a very motivated student will start early or take summer course, but I'm more interested in how our program actually prepare our students (which is not very well). $\endgroup$ – ssquidd Jun 29 at 2:06
3
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • $\begingroup$ This is also my personal experience. I had double major: CS + Math. And I took both proof courses. I was quite surprised that many math students are not as familiar with proofs as CS students in their 1st and 2nd year. However, those who eventually study math at graduate level usually learn everything much earlier. $\endgroup$ – ssquidd Jun 30 at 16:55
3
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ Naval Academy is a very selective university (though their selection isn't purely for academic ability). As such, it's not very representative. $\endgroup$ – Alexander Woo Jun 24 at 23:16
2
$\begingroup$

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).

| improve this answer | |
$\endgroup$
2
$\begingroup$

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.

| improve this answer | |
$\endgroup$
0
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • $\begingroup$ Can you expand on this some? How is your answer different form that given by Daniel R. Collins? $\endgroup$ – Xander Henderson Jun 28 at 23:37
  • $\begingroup$ The main additional piece of information is the part about the late specialization in American colleges. $\endgroup$ – Mark Foskey Jun 28 at 23:57
-1
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 3
    $\begingroup$ What school or location is this? Without this kind of information, this answer is not really usable by anyone. Some more detail would also be very helpful. Look at the other answers for examples of how to provide more detail. "from the first day we learned proofs"... ? $\endgroup$ – Chris Cunningham Jun 27 at 18:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.