How Can Parents and Teachers Identify a Student's Potential for a Maths Degree?

  • #1
ParentUK
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Hi all.

I'm looking for some insight from parents/teachers/uni lecturers about predictors that might indicate a student is talented enough to do a Maths degree and excel at it. My little girl is now in year 10 and about to do her GCSEs next year and is trying to figure out what she wants to do at uni. She's in a grammar school and pretty much at the top of her year group in maths and all the sciences (triple science + computing science). She is contemplating doing maths at uni pretty seriously. The problem is that imo maths at GCSE (and even at A level) is more about practice than talent? Whilst she is doing well and her teachers assure me she has a good mathematical mind, I remain unsure whether it is the best choice for her given her predicted grades in sciences also open up many potential degree courses.

So, I would be grateful if anyone could give me some clues as to how you can spot a student's potential talent in maths. I just want her to enjoy whatever she chooses to study and would hate for her to start studying something and feel like she wasn't good enough to keep up with the best i.e. not graduate with a first. Perhaps it is hubris to look so far ahead, but like any parent I would prefer she did something she truly excelled in.
 
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ParentUK said:
She's in a grammar school and pretty much at the top of her year group in maths and all the sciences (triple science + computing science). She is contemplating doing maths at uni pretty seriously. The problem is that imo maths at GCSE (and even at A level) is more about practice than talent? Whilst she is doing well and her teachers assure me she has a good mathematical mind, I remain unsure whether it is the best choice for her given her predicted grades in sciences also open up many potential degree courses.
This is a good point, IMO. When I retired ten years ago, I started with revision of A-level maths and, to be honest, the material seemed pretty good to me. Perhaps it's GCSE maths that's dumbed down to the point where it's not even maths at all! How she gets on at A-level will be a much better test.

There is lots of good stuff on the Internet these days. There are some instructive videos of real or mock Oxbridge maths interviews! Also, engineering and science interviews. And, what they are doing is precisely trying to figure out whether the student really understands stuff or has just parroted their way through A-levels.

And, first year lectures that have been recorded and are available on line.

In the meantime, you could find some suitable mathematics that isn't just "plug and chug". That could be a project for the holidays!

Good luck!
 
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I'm not sure there's any litmus test that will reliably indicate whether a student will be successful with a degree in mathematics. Certainly ingredients for it would include:
- a love of mathematics
- a desire to pursue such a degree
- the self-discipline to spend long hours working through problems and proofs (to the exclusion of other opportunities), and
- a plan for what they're going to do with the degree and skill set they're building.
 
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Please forgive me, but this comment concerns me a little: "I would prefer she did something she truly excelled in."

I recommend you try to be satisfied with the implication of your earlier comment: " ... she is doing well and her teachers assure me she has a good mathematical mind."

I.e. if she enjoys it and qualified teachers say she has the mind for it, you might want to back off a little. I myself, as a parent, made the mistake of trying to choose my child's path. It is usually unhelpful.

As an answer to your actual question, I suspect there is no way we can tell you how to spot real mathematical talent in your child, unless you are yourself a mathematician. The fact that her mathematics teachers think so is about as good an indicator as you are likely to find at the moment.

She'll find out pretty quickly how she stacks up when she encounters real mathematicians at uni, and top fellow students. Even then, if she loves it and can make it, it is actually ok not to be the absolute best. I am a retired research mathematician/university professor, and the only time I can recall in my career when I was the absolute best, was in one lower level college course which I soon realized was not challenging me enough, and I immediately transferred to an elite honors course where I had to struggle hard for an A-/B+. In my research life too, being regularly around stronger mathematicians was inevitable, and always stretched and benefitted me.

Good luck to you both, I know this is important and stressful for you. But remember it is important for her to choose her own way. Even if one is talented, scientific training is so long and hard that it is crucial to be going in a direction you really enjoy, in order to have the stamina to continue successfully.

Sincere best wishes.
 
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  • #6
It's probably worth thinking about what her post-degree plans are. One path is to go on to get a doctorate and become a professor or research mathematician. Another is to get a degree in math because you need a degree in something, and then go on to the workforce. Many, many more people take the second path.

These are the ends of a spectrum, and there are paths in between. But it helps to know where you want to go in order to chart a path there.

Not all math majors go on to become mathematicians. Bud Herseth, who was the principal trumpet player for the Chicago Symphony Orchestra majored in math.
 
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  • #7
mathwonk said:
I myself, as a parent, made the mistake of trying to choose my child's path. It is usually unhelpful.
Once they asked a mathematician if he will insist that his children follow his footsteps in choosing a career. He said: "No, no, of course not, they can do whatever they want to. They can do algebraic geometry, or number theory, or differential equations, or..."
 
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  • #8
PeroK said:
This is a good point, IMO. When I retired ten years ago, I started with revision of A-level maths and, to be honest, the material seemed pretty good to me. Perhaps it's GCSE maths that's dumbed down to the point where it's not even maths at all! How she gets on at A-level will be a much better test.

There is lots of good stuff on the Internet these days. There are some instructive videos of real or mock Oxbridge maths interviews! Also, engineering and science interviews. And, what they are doing is precisely trying to figure out whether the student really understands stuff or has just parroted their way through A-levels.

And, first year lectures that have been recorded and are available on line.

In the meantime, you could find some suitable mathematics that isn't just "plug and chug". That could be a project for the holidays!

Good luck!

Muu9 said:
https://ukmt.org.uk/intermediate-challenges/intermediate-mathematical-challenge

https://www.drfrost.org/worksheets.php?wdid=46

https://www.drfrost.org/worksheets.php?wdid=50

Dr frost maths is a great website for learning and reviewing A levels maths and further maths. I highly recommend taking Further maths if possible. Beyond that are the MAT and STEP exams, which are more challenging than a level FM

Thank for the helpful ideas and worksheets. She has DrFrostMaths and I will send her the links and see how she gets on. She has done UKMT in the past and got silver medals including for the last (intermediate). STEP and MAT are likely beyond her at this point (as are the uni lectures?). Will revisit this in the futyre but I was hoping to get an idea before she picks her A levels (see below)

Choppy said:
I'm not sure there's any litmus test that will reliably indicate whether a student will be successful with a degree in mathematics. Certainly ingredients for it would include:
- a love of mathematics
- a desire to pursue such a degree
- the self-discipline to spend long hours working through problems and proofs (to the exclusion of other opportunities), and
- a plan for what they're going to do with the degree and skill set they're building.

mathwonk said:
Please forgive me, but this comment concerns me a little: "I would prefer she did something she truly excelled in."

I recommend you try to be satisfied with the implication of your earlier comment: " ... she is doing well and her teachers assure me she has a good mathematical mind."

I.e. if she enjoys it and qualified teachers say she has the mind for it, you might want to back off a little. I myself, as a parent, made the mistake of trying to choose my child's path. It is usually unhelpful.

As an answer to your actual question, I suspect there is no way we can tell you how to spot real mathematical talent in your child, unless you are yourself a mathematician. The fact that her mathematics teachers think so is about as good an indicator as you are likely to find at the moment.

She'll find out pretty quickly how she stacks up when she encounters real mathematicians at uni, and top fellow students. Even then, if she loves it and can make it, it is actually ok not to be the absolute best. I am a retired research mathematician/university professor, and the only time I can recall in my career when I was the absolute best, was in one lower level college course which I soon realized was not challenging me enough, and I immediately transferred to an elite honors course where I had to struggle hard for an A-/B+. In my research life too, being regularly around stronger mathematicians was inevitable, and always stretched and benefitted me.

Good luck to you both, I know this is important and stressful for you. But remember it is important for her to choose her own way. Even if one is talented, scientific training is so long and hard that it is crucial to be going in a direction you really enjoy, in order to have the stamina to continue successfully.

Sincere best wishes.

As an honorary lecturer at uni myself, I sometimes question whether the students I teach actually love what they do. It is easy to spot the few where the subject is a real passion but in a class of 400 this might only be 10% of the cohort. For the rest, I would sadly say they truly lack passion. It doesn't prevent success of course but passion makes success easier and more fulfilling.

If you ask me whether she has a real passion for anything she is learning, then I would say no. Is she self motivated and driven? For sure. I am very much a hands off parent with respect to her studies. She identifies her needs and I help to provide them but any additional resources on top of what the school provide is very much driven by her asking me for it. I am confident she has a good brain. But having had a very chequered career path myself, I think it is easiest to forge a career in something you love doing and are good at. Therein lies my concern - whether her success in maths is a result of general intelligence vs a real mind for maths. FWIW, she has earmarked Maths/FM/Physics/CS for A levels and as mentioned those might be a better marker of ability though I still think too much learning is rote and practice as opposed to real understanding. My other concern is the A levels she wants to study would close off some uni degree choices though repeating a year and getting other A levels isn't the end of the world. Still it would be nice to get some reassurance of her math ability.

Vanadium 50 said:
It's probably worth thinking about what her post-degree plans are. One path is to go on to get a doctorate and become a professor or research mathematician. Another is to get a degree in math because you need a degree in something, and then go on to the workforce. Many, many more people take the second path.

These are the ends of a spectrum, and there are paths in between. But it helps to know where you want to go in order to chart a path there.

Not all math majors go on to become mathematicians. Bud Herseth, who was the principal trumpet player for the Chicago Symphony Orchestra majored in math.
That's a good question. Probably something with real world math application and less likely research but that would depend on what areas of math she enjoys and chooses to study more closely. Something that involves programming? She seems to really enjoy that (maybe even close enough that I call it a passion) but her degree would be so far removed from mine that she would have to make her own way as my experience would not be helpful.
martinbn said:
Once they asked a mathematician if he will insist that his children follow his footsteps in choosing a career. He said: "No, no, of course not, they can do whatever they want to. They can do algebraic geometry, or number theory, or differential equations, or..."
This is quite funny if only because a friend of mine said the way medicine is going, we'll soon have doctors that only specialize in treating disorders of the big toe :D. We truly live in the era of the superspecialist.

Thank you all who have taken the time to reply. Appreciate your thoughts and wishes!
 
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  • #10
Ok, I want to say I have been, and am, severely handicapped by not knowing anything about education in the UK, or your daughter. After some research, I have concluded that your daughter is about 14 years old, and studying pretty elementary maths. This is very young for making predictions. Also I suspect her lack of passion may be explained by the possibility she is not seeing any real maths so far. Thus I would suggest exposing her to some maths books written by actual mathematicians. Without meeting her it is very hard to specify particular books, but I will mention some candidates. These are found online and in libraries and one might catch her interest.

1) What is Mathematics, by Courant and Robbins. This is a very substantial book written for the intelligent general public. Not at all easy, but potentially very interesting and wide ranging, assuming very little background.

2) The Elements, by Euclid. My all time favorite geometry book, although I only got into it in my 60's. A bias against substantive books caused this to be banned from schools in US 100 years ago. Geometry instruction has gone downhill steadily since. I recommend this as the best place to learn Pythagoras' theorem. Quadratic equations are solved using only geometry. Includes construction of a regular pentagon, something omitted from my own high school class. I recommend the Green Lion Press edition.

3) Elements of Algebra, by Euler. Another great classic book from a master, written for his butler, hence assuming absolutely nothing. Wonderful problems. After teaching the topic of cubic formulas in graduate school algebra, and even writing my own book covering it, I finally learned how easy it really is in this book, and then began to teach it to brilliant 10 year olds.

4) Mathematical methods of science, by George Polya. This is one of those books from the brief revival of math education in the 1960's US, by a modern master. Very interesting problems are studied, such as how Eratosthenes measured the radius of the earth.

5) Introduction to modern algebra, by John Kelley. Another attempt from the 1960's to present more advanced material to younger students. A little bit on vector geometry ...

Let me emphasize that my intent is to try to help her find what she may find interesting and enjoyable, and not to quantify her aptitude, at this stage. In particular it its no reflection on her ability if she does not find any of these books attractive or accessible.

These books can all be found, used, on the website abebooks. E.g. here is Polya's book, several lines down the page:
https://www.abebooks.com/servlet/SearchResults?kn=polya, mathematical methods of science&sts=t&cm_sp=SearchF-_-topnav-_-Results&ds=20

and here is Courant - Robbins:
https://www.abebooks.com/servlet/SearchResults?sts=t&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&an=Courant, Robbins

Euclid: https://www.abebooks.com/servlet/Se...esults&ref_=search_f_hp&tn=elements&an=euclid

Kelley: https://www.abebooks.com/servlet/SearchResults?kn=john kelley, modern algebra&sts=t&cm_sp=SearchF-_-topnav-_-Results

Euler: https://www.abebooks.com/servlet/SearchResults?kn=euler, elements of algebra&sts=t&cm_sp=SearchF-_-topnav-_-Results


notice these great books are all under $20 each, most under $10.

added later: If she wants to look at a real, no-holds-barred, university level book, there is Courant's Differential and Integral Calculus, vol. 1, the book that gave me my "I'm pretty sure we're not in Kansas any more Toto" experience in college math. Michael Spivak's version, Calculus, is more fun and maybe more approachable.

and here is Courant for under $20, a bargain.
https://www.abebooks.com/servlet/SearchResults?kn=courant, differential and integral&sts=t&cm_sp=SearchF-_-topnav-_-Results

and here is spivak for under $40:
https://www.abebooks.com/servlet/SearchResults?kn=spivak, calculus&sts=t&cm_sp=SearchF-_-topnav-_-Results

Especially given that even a poor college education can cost tens of thousands of dollars, I highly recommend any of these books. It is only a partial oversimplification that the difference between the professor and the students is the professor reads the book.
 
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I feel like trying to work through Courant at this stage (prior to any calculus exposure) might be discouraging - "If this is what real math is, then maybe I'm not cut out for it after all".

Apostol would be the best option for a rigorous first book on calculus, IMO
 
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  • #12
this reaction is exactly why it is so hard to give advice of this kind, I repeat:

"Let me emphasize that my intent is to try to help her find what she may find interesting and enjoyable, and not to quantify her aptitude, at this stage. In particular it its no reflection on her ability if she does not find any of these books attractive or accessible."

the point is not to try to "work through Courant", but just to look at it.

In fact I myself have never in the last 60 years of my career, ever worked entirely through any book. But I have learned a lot from dipping into them. Don't be afraid, you have a lot to gain, if you just don't judge yourself by how far you get.

Of course Apostol is excellent, but it is very dry for a young person. But still, I agree to suggest it.

I myself am still trying, as of tonight, in my 80's, to get through the section on blowing up in Mumford's redbook. But I just advanced another page! the difficulty is apparently mostly psychological.
 
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Appreciate your effort to look up a completely different schooling system! Year 10 would be 4 years into secondary (high) school having had 7 years of primary school. And yes the maths is still very basic of course. Basic enough that I can help her with pretty much 90% of it which partly explains my concerns regarding her choice of maths at uni.

Thank you too for the list of books. It will likely be something for her to look at over summer. I glanced through the Euler and the syllabus has covered basic algebra up to quadratic equations though there's insights to be gained from the book with regards to mathematical rigor. That and "What is Mathematics" might interest her enough for her to work through bits independently. I like the idea of her dipping in to areas she might find interesting.

She actually had a mini maths camp last summer with a tutor I found just for 2 months or so (lessons twice a week) where she was introduced to some calculus. The remit I gave the tutor was to cover the basic rules as well as to try and show how calculus relates to some of the equations in physics. She enjoyed it and the tutor gave me some feedback again that she seemed to grasp things easily. Still 2 months isn't a lot of time to delve deep into calculus and it was my intention to do something similar this summer. I would have liked to continue the tutoring longer term but being in a grammar school and having to do 11 GCSEs (she is driven enough to want to get the best grades possible in all subjects and has several other extracurricular activities to boot) means she has very little time to explore maths.

In any case, the suggestions here have helped to clarify where next to explore so again thank you for your time and efforts.
 
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  • #14
Euler is perhaps the least likely of my suggestions to be easily accessible. My copy has over 576 pages, and quadratic equations occur on page 244, with cubic equations and the "rule of Cardano" for solving them on page 262.

There is also a supplement by LaGrange including a discussion of continued fractions, so it is a bit more advanced than just quadratic equations. Indeed the first 250 pages or so are quite elementary topics, but discussed by a master. Still it is probably not the best choice for a young person.

Courant and Robbins discusses an introduction to calculus in the last chapter, chapter 8, which might interest your daughter after her tutorial in calculus.

In fact a glance at Courant's calculus book might be of interest. I myself was hooked as an 18 year old student, by the footnote on page 27, where he shows how to derive a formula for the sum of the kth powers of the first n positive integers, if one knows such formulas for all powers < k.
 
  • #15
My recommendations for sources in math will be almost entirely devoted to understanding concepts rather than learning rules. On my own personal webpage there are a number of free sets of notes for various courses, some of which should be accessible to your daughter. In particular, the class notes in paragraphs 4. (math4000), 9. (math5200),10. (epsilon camp), 11. (math 2200). on this page:
https://www.math.uga.edu/directory/people/roy-smith

the notes in 11.a, are on the concept of a tangent line for the graph of a polynomial, and may interest your daughter. They are from a little different point of view than in most calculus classes.

The "elementary algebra" notes in 4. (math 4000), should be accessible as well, beginning from familiar material like induction, infinitude of primes, and binomial theorem, to less familiar topics like extensions of number systems, Gaussian integers, and results such as Fermat's theorem that the primes p which can be written as a sum of two squares are exactly those of form p = 4k+1: E.g. p = 89 = 4(22)+1 = 5^2 + 8^2, but p = 83 = 4(20)+3, is not a sum of two squares.

The notes there under 10.a, are actual notes handed out to a class of brilliant 10-12 year olds to whom I taught Euclid's first 4 books, in a 2 week intensive summer math camp, "epsilon camp". They go as far as comparing Archimedes' and Newton's approaches to calculating volumes, and explaining how Archimedes could have computed the volume of a 4 dimensional ball, using the concept of center of mass.

Your daughter might find those notes interesting as an expansion of her study of geometry. They begin at the beginning with simple properties of triangles, but assume the reader has at hand a copy of Euclid.


I want to make a remark as to why I recommend books by great masters, even If one cannot expect to work all through them. One can learn something unique just from a single sentence by a master that is not available elsewhere. As an example, one may have been told the volume of a ball of radius r is given by the formula (4/3)πr^3, and that the surface area of that ball, i.e. of the sphere, has formula 4πr^2. But how many of us were taught that either one of those formulas suffices to derive the other? This is learned from one sentence in Archimedes: roughly; " a ball can be thought of as a cone whose vertex is at the center of the ball, and whose base is the surface of the ball." If you can visualize this, you will realize that the formula for the volume of the sphere is thus obtained from that for a cone, namely 1/3 (base area) x height. Hence the volume of the ball equals (1/3)(4πr^2)Xr = (4/3)πr^3.
 
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mathwonk said:
I want to make a remark as to why I recommend books by great masters, even If one cannot expect to work all through them. One can learn something unique just from a single sentence by a master that is not available elsewhere.
"It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils." - Niels Abel
 
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  • #17
mathwonk said:
My recommendations for sources in math will be almost entirely devoted to understanding concepts rather than learning rules. On my own personal webpage there are a number of free sets of notes for various courses, some of which should be accessible to your daughter. In particular, the class notes in paragraphs 4. (math4000), 9. (math5200),10. (epsilon camp), 11. (math 2200). on this page:
https://www.math.uga.edu/directory/people/roy-smith

the notes in 11.a, are on the concept of a tangent line for the graph of a polynomial, and may interest your daughter. They are from a little different point of view than in most calculus classes.

The "elementary algebra" notes in 4. (math 4000), should be accessible as well, beginning from familiar material like induction, infinitude of primes, and binomial theorem, to less familiar topics like extensions of number systems, Gaussian integers, and results such as Fermat's theorem that the primes p which can be written as a sum of two squares are exactly those of form p = 4k+1: E.g. p = 89 = 4(22)+1 = 5^2 + 8^2, but p = 83 = 4(20)+3, is not a sum of two squares.

The notes there under 10.a, are actual notes handed out to a class of brilliant 10-12 year olds to whom I taught Euclid's first 4 books, in a 2 week intensive summer math camp, "epsilon camp". They go as far as comparing Archimedes' and Newton's approaches to calculating volumes, and explaining how Archimedes could have computed the volume of a 4 dimensional ball, using the concept of center of mass.

Your daughter might find those notes interesting as an expansion of her study of geometry. They begin at the beginning with simple properties of triangles, but assume the reader has at hand a copy of Euclid.


I want to make a remark as to why I recommend books by great masters, even If one cannot expect to work all through them. One can learn something unique just from a single sentence by a master that is not available elsewhere. As an example, one may have been told the volume of a ball of radius r is given by the formula (4/3)πr^3, and that the surface area of that ball, i.e. of the sphere, has formula 4πr^2. But how many of us were taught that either one of those formulas suffices to derive the other? This is learned from one sentence in Archimedes: roughly; " a ball can be thought of as a cone whose vertex is at the center of the ball, and whose base is the surface of the ball." If you can visualize this, you will realize that the formula for the volume of the sphere is thus obtained from that for a cone, namely 1/3 (base area) x height. Hence the volume of the ball equals (1/3)(4πr^2)Xr = (4/3)πr^3.
Thank you for the link to your page. It's an embarrassment of riches. I looked through math 4000 :) and I will point her to this for starters. If she doesn't take to the logic of maths then perhaps she isn't cut out to studying this at uni! 10.a also looks interesting so that's going her way as well. If she takes to these and can work her way through it without much help then perhaps she has the makings of a mathematician :D.
 
  • #18
Thank you for taking the trouble to look at my webpage. I hope something is of use there. The geometry notes 10.a were inspired by Euclid, and our reading of Euclid was guided by the wonderful book Geometry:Euclid and Beyond, by Robin Hartshorne.

[I just realized that Archimedes' remark about viewing a ball as a cone, is just a generalization to one higher dimension, of something I was taught in my high school geometry book, namely that a disc is just a (limit of) triangle(s), with vertex at the center of the disc, base (on) the circumference, and height equal to the radius. Since the area of a triangle its (1/2) base x height, thus the area of a circular disc equals (1/2)circumference x radius; or if the circumference is defined as 2πr, the area is (1/2)2πr x r = πr^2.]

@martinbn: yes, but it seems not everyone takes Abel at his word. This question was discussed a while ago on mathoverflow, where I argued for it.
https://mathoverflow.net/questions/28268/do-you-read-the-masters/51868#51868
 
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dear parentUK: so far we are still firing in the dark, as we do not know if any shots have hit with the target audience. hence I will stop now and wait for feedback from the person who really counts, your daughter. if you or she lets us know what, if anything, has caught her interest, we can perhaps go further.

one additional challenge for girls, at least as reported, is they are often shy. people say the boys raise their hands faster and dominate discussions. My granddaughter too reports having to put up with "man-splaining" in her uni courses. So your daughter will also have to deal with this. A beginning might be for her to weigh in here on her own. just a suggestion.

Be assured, this advisement is our pleasure, since at an advanced age, our enjoyment stems largely from passing the torch.

best wishes,
mathwonk

PS: have you heard of this place? I have not, but maybe it would offer something.
https://www.kingsmathsschool.com/summer-schools

they seem to prioritize helping girls get a start in maths/physics, and it seems to be inspired by some impressive Russian programs and scientific names.

here is the US based program where I taught in its inaugural year, 2011, and for which I wrote the geometry notes 10.a, on my webpage. this program is for very young students who already know math is their passion, which would not include most of us, but it does a fine service for that small and very underserved population.
https://epsiloncamp.org/#about-section

I mention it only to emphasize that programs do exist at many levels to help young people feed their interest in maths, or in case it appeals.

one more remark: although we teachers love to lecture, bright students prefer to do math themselves over listening to us talk about it, so I recommend posing some problems for your daughter, and not just recommending readings.

e.g. from my notes on epsilon algebra:

"7. The first recorded instance I know of solving quadratic equations, is in the book Arithmetica by Diophantus, but much of his book apparently concerned linear equations. When he died his tombstone was reportedly inscribed as follows (in Greek):
“This tomb holds Diophantus. Ah, what a marvel. And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled him the light of marriage after a[nother] seventh, and in the fifth year after his marriage He granted him a son. Alas! Late begotten and miserable child, when he had reached the measure of half his father’s (full) life (span), the chill grave took him. After consoling his grief by this science of numbers for four years, he (Diophantus) reached the end of his life.”
What was the length of Diophantus’ life?

Next I want give Euler's explanation of how to solve cubic equations.
first he shows that any cubic equation can be transformed by a trick to change the cubic into one with no X^2 term. So it is only necessary to be able to solve cubics like this one:
X^3=pX+q. E.g. suppose that we have X^3=9X+28.
Then Euler explains that to solve this all we need to do is find two numbers u,v such that 3uv=9 and u^3+v^3=28. Then X=u+v will solve the cubic.
14. See if you can use Euler's method to solve X^3 = 9X + 28.
(Essentially this solution method was found apparently by Scipio del Ferro, and later Tartaglia, who explained it to Girolamo Cardano, who eventually published it. It is often called "Cardano's method".)
15. Try this one as well: X^3 = -18X + 19."

The explanation for why this works is in Euler's book, part I, section IV, chapter XII, page 262ff in my copy.
 
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  • #21
mathwonk said:
Ok, I want to say I have been, and am, severely handicapped by not knowing anything about education in the UK, or your daughter. After some research, I have concluded that your daughter is about 14 years old, and studying pretty elementary maths. This is very young for making predictions. Also I suspect her lack of passion may be explained by the possibility she is not seeing any real maths so far. Thus I would suggest exposing her to some maths books written by actual mathematicians. Without meeting her it is very hard to specify particular books, but I will mention some candidates. These are found online and in libraries and one might catch her interest.

1) What is Mathematics, by Courant and Robbins. This is a very substantial book written for the intelligent general public. Not at all easy, but potentially very interesting and wide ranging, assuming very little background.

2) The Elements, by Euclid. My all time favorite geometry book, although I only got into it in my 60's. A bias against substantive books caused this to be banned from schools in US 100 years ago. Geometry instruction has gone downhill steadily since. I recommend this as the best place to learn Pythagoras' theorem. Quadratic equations are solved using only geometry. Includes construction of a regular pentagon, something omitted from my own high school class. I recommend the Green Lion Press edition.

3) Elements of Algebra, by Euler. Another great classic book from a master, written for his butler, hence assuming absolutely nothing. Wonderful problems. After teaching the topic of cubic formulas in graduate school algebra, and even writing my own book covering it, I finally learned how easy it really is in this book, and then began to teach it to brilliant 10 year olds.

4) Mathematical methods of science, by George Polya. This is one of those books from the brief revival of math education in the 1960's US, by a modern master. Very interesting problems are studied, such as how Eratosthenes measured the radius of the earth.

5) Introduction to modern algebra, by John Kelley. Another attempt from the 1960's to present more advanced material to younger students. A little bit on vector geometry ...

Let me emphasize that my intent is to try to help her find what she may find interesting and enjoyable, and not to quantify her aptitude, at this stage. In particular it its no reflection on her ability if she does not find any of these books attractive or accessible.

These books can all be found, used, on the website abebooks. E.g. here is Polya's book, several lines down the page:
https://www.abebooks.com/servlet/SearchResults?kn=polya, mathematical methods of science&sts=t&cm_sp=SearchF-_-topnav-_-Results&ds=20

and here is Courant - Robbins:
https://www.abebooks.com/servlet/SearchResults?sts=t&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&an=Courant, Robbins

Euclid: https://www.abebooks.com/servlet/Se...esults&ref_=search_f_hp&tn=elements&an=euclid

Kelley: https://www.abebooks.com/servlet/SearchResults?kn=john kelley, modern algebra&sts=t&cm_sp=SearchF-_-topnav-_-Results

Euler: https://www.abebooks.com/servlet/SearchResults?kn=euler, elements of algebra&sts=t&cm_sp=SearchF-_-topnav-_-Results


notice these great books are all under $20 each, most under $10.

added later: If she wants to look at a real, no-holds-barred, university level book, there is Courant's Differential and Integral Calculus, vol. 1, the book that gave me my "I'm pretty sure we're not in Kansas any more Toto" experience in college math. Michael Spivak's version, Calculus, is more fun and maybe more approachable.

and here is Courant for under $20, a bargain.
https://www.abebooks.com/servlet/SearchResults?kn=courant, differential and integral&sts=t&cm_sp=SearchF-_-topnav-_-Results

and here is spivak for under $40:
https://www.abebooks.com/servlet/SearchResults?kn=spivak, calculus&sts=t&cm_sp=SearchF-_-topnav-_-Results

Especially given that even a poor college education can cost tens of thousands of dollars, I highly recommend any of these books. It is only a partial oversimplification that the difference between the professor and the students is the professor reads the book.
may I add another book, that is probably more accessible, and easier to work through, while introducing fun mathematics?

There is the number theory book by Pommersheim? [A Lively Introduction To The Theory Of Numbers].
Very basic in its coverage of number theory, and easy problems. Has interesting pictures.

Some of the proof exercises may be to hard if someone does not know the basic of proofs, but they could be looked at, tried, and reasoned, then skipped when honest effort.

The issue is that it is pricey. I think $80 last time I checked.

On the cheaper end, an even more elementary, is the high school geometry book by Edwin E. Moise and Downs, titled Geometry. A book by a top notch mathematician.
 
  • #22
I would also recommend reading biographies of mathematicians (and scientists). They can be interesting and motivating and can show what mathematicians do and give a gimps of what mathematics actually is and how it is done.
 
  • #23
To the OP:

You made the following comment which concerned me: "The problem is that imo maths at GCSE (and even at A level) is more about practice than talent?"

As far as I'm concerned, that is the wrong way to think about this, because the study of mathematics is cumulative, and to gain proficiency require consistent practice through the chain of reasoning and the various problems. From what I gathered in your initial post, it does appear that your daughter has shown an interest in mathematics and proficiency in her coursework thus far.

I understand your concerns about wanting to ensure that she truly enjoys and excels in her future university studies, to a certain extent these are not really under your control. I would think that if your daughter has the interest, and her coursework thus far established a foundation, then she should consider pursuing the studies.
 

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