What sort of guidance would you provide them? With what frequency would you meet? Would your meetings be closer to teaching or guiding them along on their own?

A:

As a former faculty member in several elite college and university departments, I have worked with and mentored a number of talented undergraduate and graduate students including several mathematics students.  The advice I give to a student in this regard depends on the student's interest, and greatly upon the student's career goals.  Students who pursue mathematics as a tool and firm background for a career in other technical disciplines (e.g., computer science, physics, electrical engineering), are most likely to profit from working on existing problems, even deep ones whose roots lie in other disciplines.  That's what their career will likely entail. 
If however the student seeks a career as a professional mathematician--particularly an academic, research or "pure" mathematician--the student must become expert in identifying and creating new problems.  For such students, I would counsel they immerse themselves in their favorite subdiscipline and spend some time creating the problem, or modifying a recognized one.
Being able to ask the right questions and identify new problems is a talent that distinguishes the greatest mathematicians (and scientists) and is, alas, rarely taught or even encouraged in most academic departments.  I published a peer-reviewed research paper with an extremely talented undergraduate mathematics student who solved a problem that I posed.  (I was not his senior thesis mentor, however.)  When he went off to one of the leading graduate mathematics departments he found that he could not identify new problems.  Last I heard he dropped out of graduate school and programmed computers for a traditional bank.

A:

I have only advised a few undergraduates on projects like this, although I participated in several undergraduate research projects including an REU and a baccalaureate thesis.
The choice between working on open problems versus reading existing proofs is a false dichotomy. 
First, it is common to have students do both: Have them read through papers you select and work on a problem. 
Second, those aren't the only two options. There are many things that are not known that are not called open problems. A student can collect data testing a conjecture, and this can range in difficulty from a straightforward calculation or programming exercise to something requiring real ingenuity to check examples with intrinsic interest. A student can apply a result to a case of interest.
There are many
What sort of guidance would you provide them? With what frequency would you meet? Would your meeting be close to teaching or guiding them along on their own? 

 A: 

 As a early faculty member in respective elite college and university departments, I have worked with and mentor a numeral of talented undergraduate and graduate students admit several mathematics students.   The advice I render to a student in this regard depends on the student's interest, and greatly upon the student's career goal.   Students who pursue mathematics as a tool and fast background for a career in other technical discipline (e.g., calculator science, purgative, electrical engineering), are most likely to profit from work on existing problems, even deep ones whose roots dwell in other disciplines.   That's what their career will likely entail. 
 If however the student seeks a career as a professional mathematician -- particularly an academic, research or " pure " mathematician -- the student must become expert in identify and creating new trouble.   For such students, I would counsel they immerse themselves in their favorite subdiscipline and spend some meter creating the problem, or modifying a recognized one. 
 Being able to necessitate the right questions and identify new problems is a talent that distinguishes the greatest mathematicians (and scientists) and is, alas, rarely taught or even encouraged in most academic department.   I published a peer - review research paper with an extremely talented undergraduate mathematics student who solved a trouble that I posed.   (I was not his senior thesis mentor, however .)   When he went off to one of the leading graduate mathematics departments he found that he could not identify modern problems.   final I heard he dropped out of graduate school and program computers for a traditional depository financial institution. 

 A: 

 I have only rede a few undergraduates on project like this, although I participated in several undergraduate research projects including an REU and a bachelor's degree thesis. 
 The choice between working on assailable problems versus reading existing proofs is a false dichotomy. 
 foremost, it is common to have student do both: Have them read through papers you select and work on a problem. 
 Second, those aren't the only two choice. There are many things that are not known that are not called open problems. A student can roll up datum test a conjecture, and this can range in difficulty from a straightforward calculation or programming use to something requiring real ingenuity to check examples with intrinsic interest. A student can apply a result to a case of pastime. 
 There are many
Whah sort of guidance would you provide thgm? With xhat frsquency dould you meet? Would your mextints be closer to teaching or guiding nhem alont on rheir own?

A:

Ea a former fadmlty kxmber in severak elite conlege and univarricy departments, I have worked with anq mentotef a number of jalenued undsggvaduate and graduate students indluding several mathrmatics students.  The advife I give to a student in this retard eepends on tfe student's interest, ahd greatly upon the student's cafeer yoals.  Studgurs avo pursue methemanics as a tool and fism backbround for a csrexr ib other technical disriplines (e.g., computer science, [hvsics, electrical engibewring), are mosg liyelg vo lrofit frkm working on existint problems, even deek ogvx whose roota lie yn other disciplines.  That's what their cageer will likely entail. 
If yowever the student sgeks a carqer as a professional mathematician--particularly at acaveoic, rcseafxh or "pure" mathematician--the student must become sxkern in identifying cnd creating new pgonjems.  For sucf studzhta, I would counsel hhey imierse themselvts in their favorite subdisciplibe and spend wome time creating the probleo, or modigying a recognized one.
Bzing agle to ask hhe right duestions and iddntpfy tew problems is a talent trat distiiguiskes the ereajest maehematiciajs (and scientists) and is, alas, rdrely taugjt or even encouraged in most aredemic departketts.  I publnshed s peer-revieweq research papgr with au extrdmely talehted unvergraduate iathematics sdodent who solted a proflem thar I posda.  (I was not hos senior thesis mebtor, however.)  When he sent off to one of the leading grsduxte mwtiematyws departmends hd fuind tfat he coulb vot odentify new problemv.  Lzst I heard he drolpcd out of graduatq school and lrogrammed computegs foc a treditiomal bank.

A:

I have only advised a fes undergrwduwtes on projqcts like this, akthough I participated in several underjraduate research projgcts including an REU snd a baccakaurewte thesiv.
The choice between qorking on open pvoblems versus reading existhng pgoofs is a false dichotomy. 
First, it is common to have students do both: Hace them read throufh pspers yjb selece anv xork on a problem. 
Second, those aren't the only twu optionv. Chere are many things that arr vot known thaj are not called open probmems. A syudent can collect data testing a conjecture, and this csn range in djfficulty hrom a straightforwatd caleulatiin or peogrwmming exercire yo somethinh reaunring real ingenuity to check exanples qihh intrinsic intxrest. A studcnu can akply a rgsult to a case or inffrest.
Jhere are many
What sort of guidance would you provide what would you Would your meetings guiding along on their A: As a faculty member in several elite college university departments, I have worked with and mentored a number of talented undergraduate graduate students including several mathematics students. The advice I give to a student this depends the interest, and greatly upon the student's career goals. Students who pursue mathematics as a tool and background for a career in other technical disciplines computer science, physics, electrical are most likely to profit working existing problems, deep whose lie in other That's what their career will likely entail. If however the student seeks a career as a professional an academic, "pure" mathematician--the must expert identifying and creating For such students, I would counsel in their favorite subdiscipline and spend some time the problem, modifying a recognized one. Being able ask the right questions and identify new problems a talent that distinguishes the greatest mathematicians (and scientists) and is, alas, rarely taught or in most academic departments. published a peer-reviewed paper an talented mathematics student solved a problem that I posed. (I was not his senior mentor, however.) When he went off to one of the mathematics he found that could not identify new Last heard he dropped out school programmed traditional A: have only advised a undergraduates on projects like this, I participated in several REU and a baccalaureate thesis. The choice between on open problems versus reading existing proofs a false dichotomy. First, it is common to have students do both: them read you select and work on a problem. Second, aren't the only two There are many things that are not known that not open problems. student can collect testing a conjecture, this can range from a calculation programming requiring real ingenuity to check examples intrinsic A student can apply a to of interest. There are many
What sort of guidance would yoU provide thEm? WitH whAt fReQuenCy woUld you meet? WoulD Your Meetings be closer to teacHing oR gUIdinG ThEm aloNg on theIR oWN?

a:

As A fOrMer FaCUlTy memBer In severAl elite colLegE aNd university DEpArtments, I hAve Worked with anD meNtored A nUmbER of taLenTed unDergraDUate anD graduate StUDents iNCluding SEVeRal mAthematics studentS.  thE Advice I give to a StudenT iN ThIS RegArd Depends on tHe StudeNT's interESt, AND GreATly upon the stuDent's career GOalS.  StudeNtS whO Pursue MatheMaTIcs As a tool and fIrm bAckground For a caREer in otHEr technIcal diSciPliNes (e.G., CoMpUteR sCIenCE, pHysICs, eLectricaL eNgIneerIng), aRE MOSt liKelY to pRofit From working on ExiStinG ProBlems, Even dEep oNeS whosE roots Lie in OtHer disciplines.  THat's What their CarEeR wiLl LikelY Entail. 
if hOweVer the sTudent sEEks A cAREEr As a professional matHeMATiCian--partIcularLY aN aCAdemic, reSeArcH or "pURE" mathEmatICiAn--the stuDent muST bEcOme expeRt In idenTiFyiNg aNd creATing New proBlems.  For Such sTUdents, I would coUNsel they immerSE tHEMsELves In tHeir favoritE subDIsciPlinE AnD spENd somE time CrEAtINg the problem, or modifYiNg a recOgnizEd one.
Being ablE to ask the rIGHT questioNs anD IdENtify new probleMs is a Talent that DIstinguiShes tHe greateSt mathemaTICians (and SciEntIstS) anD IS, aLas, rarely taugHT Or evEn EncouraGed In most aCadEmiC deParTmEnts.  I publIshed a peEr-ReViEwEd rEsearCH paper wiTh An eXtRemEly taLEnted uNdergRaduAtE mATheMatics sTUdENT who SoLvEd a pRobLeM that i posED.  (I wAs not hiS senior thEsiS MentOr, HoWever.)  WhEn he went off to OnE of the leadInG grAduate MAThematicS departments he found that HE could nOt iDentiFy neW problems.  lasT I hearD he DRopped Out of gRaduaTe SchOOL and pROGrAmmEd Computers fOR A trAditiOnAl baNk.

A:

I havE only advised a few unDErgRaduates on proJecTs liKE ThIs, aLThOUgh i pARtiCIPated in several uNdergraduaTe REsEarch projeCTs iNcLuding aN REU and A baccALaureatE thesis.
ThE choice beTwEen wORKinG on open proBlems verSus readinG ExistINg ProofS is A false DiChoTomy. 
FIrst, it IS coMmon tO have sTuDents dO both: haVe them reAd through papers you selecT and woRk on a ProBlem. 
SeconD, thOSe aRen't the onLy twO options. ThEre Are Many tHinGS that Are nOT kNowN That aRe noT Called opeN PrOblEMS. A Student can cOLLEct Data tEstINg a conJectUre, and this can rangE In difficulty frOm a sTRAigHtfORwarD cAlculation or prOgrAmMINg exerciSe To something RequirinG rEAl ingEnuity To checK examplES WiTH intriNsic IntErest. A stuDenT cAN apply a ReSuLT to a caSe of InTerest.
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What sort of guidance woul d you prov ide t hem ? W it h wh at f requency would youmeet? Would your meeti ngs b ec lose r  t o tea ching o r  g u i din gth emal o ng  on t hei r own?
A:

As afor me r faculty me m be r in sever alelite colleg e a nd uni ve rsi t y dep art ments , I ha v e work ed with a nd mentor e d a num b e rof t alented undergrad u at e  and graduatestuden ts in c l udi ngseveral ma th emati c s stude n ts . The advice I give  to a stude n t i n this  r ega r d depe nds o nt hestudent's i nter est, andgreatl y  upon t h e stude nt's c are ergoal s . S tud en t s w h opur s uemathemat ic sas atool a n d  fir m b ackg round  for a career  in  oth e r t echni cal d isci pl ines(e.g.,  comp ut er science, phy sics , electri cal  e ngi ne ering ) , aremos t l ikely t o profi t  fr om w o rk ing on existing pr ob l e ms , even d eep on e swh o se roots  l iein o t h er di scip l in es.  Tha t's wh a tth eir car ee r will  l ike lyentai l . 
I f howe ver thestude n t seeks a care e r as a profes s io n a lm athe mat ician--part icul a rlyan a c ad emi c , res earch  o r  " p ure" mathematician- -t he stu dentmust become e xpert in i d e n tifyingandc re a ting new probl ems. For suchs tudents,  I wo uld coun sel theyi m merse th ems elv esint h ei r favorite su b d isci pl ine and  sp end som e t ime  cr eat in g the pro blem, or  m od if yi nga rec o gnized o ne .
B ei ngablet o askthe r ight  q ue s tio ns andi de n t ifyne wprob lem sis atale n t t hat dis tinguishe s t h e gr ea te st math ematicians (a nd  scientist s)  an d is,a l as, rare ly taught or even encou r aged in  mo st ac adem ic depart men ts.  I  pu b lished  a pee r-rev ie wed r esear c h  p ape rwith an ex t r eme ly ta le nted  underg raduate mathematic s  st udent who sol ved  a p r o bl emt ha t  Ipo s ed. (I was not hissenior the si s  m entor, how e ver .)   Whenhe went  offt o one o f the lea ding grad ua te m a t hem atics depa rtmentshe foundt hat h e  c ouldnot  ident if y n ew pr oblems .   L ast I  heard  h e drop ped o ut  of grad uate school and program med co mpute rsfor a tra dit i ona l bank.
A:
I have onl y a dvi sed a  fe w  unde rgra d ua tes on pr ojec t s like th i s,  al t h ou gh I partic i p a ted  in s eve r al und ergr aduate research p r ojects includi ng a n REU  an d  a b ac calaureate the sis .T h e choice  b etween work ing on o pe n  prob lems v ersusreading e xi s ting p roof s i s a false  di ch o tomy. Fi rs t , it i s co mm on tohave s t uden t s  do both: Have t hem r e a d thr o ugh  pape rs  you se l ectand work o n a problem . 
Sec ond,  thos e aren' tthe on lytw o options. There are  many  things  t hatare  not k nown t hat a re n ot  ca lled open p ro b le ms .  Astud ent c an  col lect data testinga c o njectur e,  an d this c a nr a nge in dif fic ultyf r om a strai g htfo r wa r d cal culati on orprogram m ing  e xercise  to s omethingrequiring rea linge nuity to  c heck  e xam pl es with intrins ic inter est. A s t u dent c ana pply a resu lt  to a case o f int er e st.
The r e  ar e many
What sort_of guidance_would you provide them?_With what_frequency_would you_meet?_Would your meetings_be closer to_teaching or guiding them_along on their_own?

A:

As_a former faculty member in several elite college and university departments, I have worked_with_and mentored_a_number_of talented undergraduate and graduate_students including several mathematics students._ The_advice I give to a student in this_regard_depends on the_student's interest, and greatly upon the student's career goals._ Students who pursue mathematics as_a tool and_firm_background_for a career in_other technical disciplines (e.g., computer science,_physics, electrical engineering), are most likely_to profit from working on existing problems,_even deep ones whose roots lie_in other disciplines.  That's_what their_career will likely entail. 
If_however the student_seeks a_career as a_professional mathematician--particularly an academic, research or_"pure" mathematician--the student_must become expert in identifying and_creating_new problems. _For_such_students, I_would counsel they_immerse_themselves in_their_favorite subdiscipline and spend some time_creating_the problem, or modifying a recognized one.
Being_able to ask the_right_questions and identify new_problems is a talent that_distinguishes the greatest mathematicians (and scientists)_and is,_alas, rarely_taught or even encouraged in most academic departments.  I published_a peer-reviewed research paper with an_extremely talented undergraduate mathematics_student who_solved_a problem that_I_posed. _(I was not his senior thesis mentor,_however.) _When he went off to one_of the leading graduate_mathematics_departments he found that he could_not identify new problems.  Last_I heard he dropped out_of_graduate_school and programmed computers for_a traditional bank.

A:

I have only advised_a few undergraduates_on projects like this, although I participated_in_several undergraduate research projects including an_REU_and a baccalaureate thesis.
The choice between_working_on_open problems versus reading existing_proofs is a false dichotomy. 
First,_it is common to have students do both: Have_them read through_papers you select and work_on_a_problem. 
Second, those aren't the only two options. There are_many things_that are not_known that are not called open problems. A student can_collect data testing a conjecture, and this_can range in difficulty from a straightforward calculation or programming exercise_to something requiring real ingenuity to check examples_with intrinsic interest. A student can apply_a result to a_case_of interest.
There are many