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As someone who has a relative deficit in quantitive reasoning (63%ile) vs. verbal (97%ile), but a strong interest in applied math, is it possible to be successful and competitive at the PhD level?

I am committed to putting in the effort, and I want to believe that this is possible - but it is very daunting.

EDIT1:

To flesh out the q: I’ve taken Calc 1, Calc 2, Calc 3, Stat/ w Calc & Discrete Math. I got a B+ in the first 2, A in the third and 4th, and B in the 4th. Taking Linear Algebra/Real Analysis and expecting an A or a B. Calc 1&2 and Discrete math were taught at the liberal arts college I went to where I received a BA in the humanities. After my BA in the humanities I pursued ad-hoc post-bac in math so that I’d be able to apply to grad school. Calc III and Stats with Calc were taken at Columbia and LA/RA At Harvard.

I will say, I have improved dramatically since I first began taking math (w/ Calc I & II).

The test I took was WAIS-III full scale IQ test, designed to be taken with no preparation and administered at a world renown neuropsych facility.

Regarding the GRE, I expect I’d be able to score in a percentile on par with other prospective applicants.

My plan is to apply to a terminal masters, ace that, and the go one to apply to PhD programs.

Lastly, I have significant adhd for which I was unmedicated in undergrad and may but may not account for relative split in grades before and after.

EDIT2:

I'm specifically interested in applied math. I want to pursue it because I find it challenging and interesting and immensely rewarding when I grasp a concept -- and I want to go into a field that helps people.

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    I'm not in math, but I've heard on this site that a typical pure math PhD student is expected to easily ace the math GRE. I'm not sure about applied math (or really even what it is, but I digress). – Azor Ahai Dec 12 '19 at 5:15
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    What is your current education level? Have you taken college level math classes (in actual college, not high school)? How did you do in those? I would be hesitant to allow a standardized test to decide my life and career path, and there is nothing wrong with taking a few classes to "test the waters" prior to making a significant commitment. – Pete B. Dec 12 '19 at 13:32
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    Is “quantitative reasoning” assessment anything more than pseudoscience? This is a genuine question (despite, admittedly, scepticism on my part). I know that IQ tests are backed by solid evidence, but this concerns broad correlations to general intelligence not, as far as I know, specific (and somewhat arbitrary) subsets. – Konrad Rudolph Dec 12 '19 at 13:49
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    Tests don't mean anything. I mean, ask yourself how motivated you were to do well on that test. How much effort did you put into understanding the types of questions it was going to ask and why? All it tells you is that you're probably not that interested in quantitative reasoning tests or the types of problems they throw at you. But maybe there are applied math problems that get you excited, and you'll see your aptitudes change dramatically when you're driven by passion to stay up until 11PM scratching equations onto paper. If you have that passion, then do it. – J... Dec 12 '19 at 14:24
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    I don't have the exact quote, but I remember reading Paul Halmos (perhaps somewhere in his autobiography) where he said that in his experience the verbal score was a better predictor of success in pure mathematics than the mathematics score. – John Coleman Dec 12 '19 at 19:27
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To know whether you can do an applied math PhD, the question to ask is: did you succeed in difficult math, science, and computer science courses in undergrad, and do you have professors in relevant fields willing to write you a strong recommendation? General aptitude tests are dramatically less informative. This is not to say that, if you’re correct that you have average quantitative reasoning ability, a math PhD might not be the wrong choice for you. It’s just that you have other, much more refined and relevant, information available to judge by.

EDIT: Based on your edits, I’d say an A in Calc III at Columbia is more informative than the IQ test. That said, you have few advanced courses-only one, once you finish real analysis. So there’s not much evidence yet of your prospects. You’re in a plausible position to apply to master’s programs, and if you get into a good one, you’ll a get a much clearer idea of yourself there.

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    General aptitude tests are dramatically less informative --- Especially since we are given no information about the test. Also, the comment under the answer by @Allure doesn't really help much. What age are you, what is your math background, how well did you do in various math courses, etc.? For example, among other such things, saying something like "one of the top 3 students in a class of 28 students taking an honors level course in beginning calculus using Spivak's book at an Ivy League school" would be an example of what might be specific enough to begin trying to answer your question. – Dave L Renfro Dec 12 '19 at 8:17
  • Thanks for the input. I refined the question. – Cdela Dec 12 '19 at 14:42
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I certainly wouldn't give up hope. The reason is that "number" per se is a very small part of mathematics. Quantitative reasoning helps in many things, of course, but there are lots of areas where it is less important than, say logical reasoning and a geometric sense (among other things).

Mathematics is largely about abstraction and relationships, much more so than about "quantity". Most people get their feet wet in quantitative studies and Real Analysis is grounded in numeracy. But even Complex Analysis has many important results that are non-intuitive to a novice.

So, just assume that there are still large segments of both pure and applied math that you may be well suited for. I suggest you explore rather than give up.


Note: I have a doctorate in Mathematics. Real Analysis specialty but a strong interest and some accomplishment in Topology. Computer Science, which most of my career was involved with draws on the same ability of abstraction and composition. They aren't exactly the same, but intersect in many ways.

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Looking at example questions in the quantitative reasoning section of the GRE, the questions are genuinely easy, and I would expect good high school students to be able to do well. A math PhD student scoring in the 63rd percentile is therefore very strange.

I've never worked in math PhD admissions, but my reactions would be 1) why did you do so poorly? (63rd percentile is not poor in general, but for a math PhD student it appears to be very poor), and 2) are you sure you have the background to study math at PhD level? Surely a math PhD student should have no problems solving the linear equation 3x-2 = 2x+5 for x - in fact, I'd expect a math PhD student to be able to solve quadratic equations easily. If the score genuinely reflects your ability, I would double check your transcript to see how much math you've done.

This doesn't mean that you cannot do a math PhD, but I'd make doubly sure that you do indeed have the basic skills needed to attempt it.

Edit: the question makes clear now that the scores weren't the GRE quantitative & verbal tests, but rather some kind of general IQ test. In that case it's a much, much weaker sign (if it is a sign at all) of inability to succeed at PhD level. If IQ implied mathematical ability, then it'd show up in your transcript, but it's the transcript that matters more than your IQ. If you score very well in your undergraduate courses, then what your IQ is is unimportant ... conversely, if your IQ is very high but you don't score well in your undergraduate courses, it would still be a red flag.

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    I believe the "63rd percentile" figure that the OP mentions is regarding the IQ test mentioned in the fifth paragraph, since they later say "[r]egarding the GRE, I expect I’d be able to score in a percentile on par with other prospective applicants", which implies they haven't taken the GRE yet. – mai Dec 12 '19 at 15:38
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    @quarague I don’t think that’s very accurate. 167/170 is the average math GRE score for admitted MIT engineering PhD students; I think it is very close to true that “the rest score slightly below that” for top programs. I would say the math phds who aren’t very good-to the moderate standard of the GRE-at high school math are few and far between, and the majority of the latter could become very good by reviewing the boring basics for a few hours. – Kevin Carlson Dec 12 '19 at 16:17
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    @quarague If someone’s going for a PhD, I assume they’re going for the top programs. In pure math weaker programs have a rather poor career outlook. I suppose someone might get an applied PhD at a weaker program to qualify for a government or industry job, but my sense is that even outside of academia most PhD-level positions are so competitive that if one can’t get into a top program, it’s likely not worth it on career grounds-assuming that’s the basis of the decision, of course. In the same way, I advise people who can’t get into a top-14 law school not to go to law school. – Kevin Carlson Dec 12 '19 at 16:27
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    @quarague The rest of your comment is fine, but the point is that the calculations on the general GRE are very easy. Math PhDs are almost always pretty good at all of math. – Kevin Carlson Dec 12 '19 at 16:29
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    @quarague I cannot speak for Math, but there is good evidence that at least the majority of Law students have made a poor decision. – Chuu Dec 12 '19 at 16:35
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First of all, I don't think anyone knows if "quantitative reasoning" WAIS subscores are indicative of talent in higher mathematics. It's possible that you would make a lousy accountant but a good mathematician. You probably already know whether you're "good with numbers," and you surely already know that what you do in advanced mathematics doesn't have a lot to do with what you're asked to do on a WAIS. It's unfortunate that even people who should know better tend to conflate quantitative or computational fluency with mathematical talent.

Speaking from my own experience, I reliably score around the 30th percentile for "arithmetic fluency," although there is nothing wrong with my ability to reason mathematically. But I'm not the person you should ask to do your taxes.

Secondly, ADHD often shows up as irregular profiles in skills and aptitude testing. Whatever the WAIS-III is worth, it needs to be interpreted carefully and in context. You can't really conclude much from a single subscore.

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I just want to point out one example, it is a somewhat of a common story but I couldn't find any decent sources. Garry Kasparov is a titan of chess. He is one of the very bests if not the best. He had been the world champion for many years. Apperantly people used to estimate his IQ was between 180 and 190. He allegedly took an IQ test in the 80s and scored 135. Now, 135 is quite low considering his extreme success. Above 130 IQ seem to be estimated to be 2% of world population. Which might still sound high but there are billions of people.

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    IQ scores follow roughly a Gaussian distribution. There are roughly 17% above 115, 2% above 130 and 0.1% above 145. Anything above that is essentially just guesswork. These few lucky people are by now means all the same but the difference cannot be easily squeezed into a x point IQ difference. Estimating the IQ of anyone at 180 or 190 just shows that whoever did the estimate has no clue what these numbers mean. The fact that Kasparov plays better chess than any other random grand master does not imply that he has a higher IQ. – quarague Dec 12 '19 at 16:11
  • @JohnDoucette wut? The abstract says the disparity between 20 and 70 is "as much as 2.3" sd, "at least 85%" of which is due to the Flynn effect. Since 1 sd = 15 points, that would mean any decline is at most 0.34 sd = 5 points. All of which is irrelevant anyway, since the post says Kasparov supposedly took the test in the 80s, not his 80s. I think Kasparov is in his 50s. – Elizabeth Henning Dec 12 '19 at 21:45
  • @JohnDoucette, again dubious sources claim the test was taken in 87 or 88, Kasparov would 24 or 25 years old at that time. He became a world champion in 85 at the age of 22. – Boaty Mcboatface Dec 12 '19 at 22:34
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    @BoatyMcboatface Ah, I misunderstood, and thought this was in his 80's, not the 1980's. I'll remove my comments. – John Doucette Dec 12 '19 at 22:35
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As someone who has a relative deficit in quantitive reasoning vs. verbal, but a strong interest in applied math, is it possible to be successful and competitive at the PhD level?

You could succeed in fields like topological algebra or mathematical logic.

I'm specifically interested in applied math.

Consider also computer science. You should view it as some applied mathematics. Look into π-calculus for a good example. And the job opportunities could be better (e.g. around artificial intelligence, semantics, static program analysis, machine learning, cybersecurity, etc...). See these slides on the future of mathematics.

Consider also computer programming and software development. It is tightly related to math, and I tend to believe you'll need some of it during your math PhD. I know several persons (Roberto Di Cosmo, Xavier Leroy, or Roberto Bagnara, or the late Jacques Pitrat, or Emmanuel Haucourt to name a few) who had an exceptional academic career while being expert in both math and programming.

Look also into economics, bioinformatics, computational chemistry, and theoretical physics (as science with a lot of math, so much that you might view them as applied math).

PS. I got a PhD in 1990 but not in math, just in AI. I am French, so your GRE things make no sense at all for me. In France, some equivalent could be the baccalauréat. I graduated from ENS Cachan which probably has no real US equivalent.

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I would say with quite strong confidence that the score on a test is insufficient to determine your ability in such a vast field as maths.

Consider the following:

  • All tests are imperfect. Some very intelligent people score badly and some very "average" people score highly

  • A single sub-test bares little resemblance to most you will be doing in a PhD. E.g. reading papers, communicating with colleagues, writing eloquently, researching questions, doing formal maths, programming, etc. I would particularly stress some underappreciated skills such as communicating with colleagues

I did a PhD in Applied Maths and spend most of my time in a laboratory doing fluid dynamics experiments, setting up cameras, writing code to do image analysis, build experimental apparatuses etc.

Finally, I have done quite a bit of maths at University and I found that maths doesn't come easily. In my first semester as a under-grad I was studying for my first exam and read all the lecture notes, attempted all the examples and understood zero. At that point I was a little bit panicking. Next day, I came back and tried it all again. Again, I felt I understood nothing. Not knowing what else to do, seriously worried, I came back on the third day. Finally I understood a little bit. On the fourth day, suddenly everything fell into place in a couple of hours. I've repeated this experience for every lecture course. A period of incomphresnshion that eventually gives way to understand by repeatatly reading the same notes an examples.

This is of course a personal story and your experience will vary but I think determination can go a long way, particularly in a field like math.

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