 eye color is more complex genetically than you were probably taught. If you start talking about cognitive and behavioral traits, like intelligence, linguistic ability, or cooperation, there is literally no trait that has a clear, simple genetic cause. For example, IQ has been rigorously studied for decades, including a tremendous amount of research on its genetic component. We know from this research genetics definitely plays a part in IQ, but no one has been able to identify the individual genes that control it. (Some relevant reading, if you want to learn more: article on genetics of race and IQ; summary of evidence on genes and IQ). 
All of this is to say you can't really "pass on" cooperativeness to your offspring the way you might pass on your blood type. To address your question directly, 

She believes that it is something he can learn through playing with others, while I believe it something that is passed on based on my stubbornness.

Your child will have roughly 50% of his genetic material from you. Any complex trait like cooperativeness will be influenced by a huge number of individual genes, each contributing only a tiny bit to the behavior. Your child will likely have some but not all of your constellation of genes that contributes to your behavior as cooperative or not --- since genes work together in complex ways, you have no idea whether the pieces of cooperativeness genetics your child will inherit will work together the same way yours do. This is all before we even consider the fact that cooperativeness might be influenced by experience/environment. As it turns out, environment is probably more important than genetics in the development of cooperative behavior: I was able to find one study on this, and they found that only 38% of cooperativeness appears to have a genetic link. So put together the fact that most of cooperativeness is predicted by environment rather than genes and that the genetic component is likely very complex, and you can see how tenuous your claim that you can "pass on" cooperativeness is. 
To sum up...
Complex behaviors (and, actually, even many simple traits) can't adequately be described as "inherited or learned", or even as "inheritance plus experience" --- your genetic inheritance is complex, flexible, and it can change over time depending on your environment. 
Any cognitive or behavioral trait has complex genetic determinants. There's no one gene, or small set of genes, that will determine whether you do or don't show some behavior (with the exception
eye color is more complex genetically than you were probably taught. If you depart speak about cognitive and behavioral traits, like intelligence, linguistic ability, or cooperation, there is literally no trait that have a clear, simple genic causal agent. For example, IQ has been rigorously studied for decades, including a frightful amount of research on its familial component. We know from this research genetics definitely plays a region in IQ, but no one has been able to identify the individual genes that control it. (Some relevant recitation, if you want to learn more: article on genetics of race and IQ; summary of evidence on gene and IQ). 
 All of this is to read you can't really " pass on " cooperativeness to your young the way you might pass on your blood type. To address your question immediately, 

 She believe that it is something he can learn through playing with others, while I believe it something that is passed on based on my stubbornness. 

 Your child will have roughly 50% of his genetic material from you. Any complex trait like amenability will be influence by a huge number of individual gene, each contribute entirely a tiny bit to the demeanor. Your child will likely accept some but not all of your constellation of genes that contribute to your behavior as cooperative or not --- since genes work together in complex ways, you get no idea whether the pieces of cooperativeness genetics your child will inherit will work together the same way yours suffice. This is all before we even consider the fact that cooperativeness might be influenced by experience / environment. As it turns out, environment is credibly more important than genetics in the development of cooperative behavior: I was able to find one study on this, and they found that only 38% of cooperativeness appear to have a genic link. So frame together the fact that most of amenability is predicted by environment rather than genes and that the genetic part is likely very complex, and you can see how tenuous your call that you can " pass on " amenability is. 
 To sum up... 
 Complex behaviors (and, actually, even many simple trait) can't adequately be described as " inherited or learned ", or even as " inheritance plus experience " --- your genetic inheritance is complex, compromising, and it can switch over time depending on your environment. 
 Any cognitive or behavioral trait have complex genetic determinants. There's no one gene, or small set of genes, that will determine whether you dress or don't show some behavior (with the exception
 eyf color is more complex nenetically than you wece probzbly taueht. If you start talking abont cignituve and behavioral trakts, like pntelligebce, ounguistic egility, or coolcratimi, there is litetally no traht that has a wldax, simple genetic cause. For example, IZ has brej rigorously sjudiec for decades, including a tremendous amkunt of research on ots genetic component. We kjow vrom this research genetics dgrinyrely plays a part in IQ, but no one has been able to identify the kndivndual genes tyat wontrol it. (Wome gelevant readlmg, if fou wany to learn morc: artmcle on genetics of race end IQ; summary of evydence on gznes and IQ). 
All of thus is tm saf yoj cav't deellg "pass on" cooperatibeness to yiur offspring the wsy jpu might pasa on yjuw blood type. To address your question dprecfly, 

She believes that ir is something he can learn thwough playing with others, while I believe it somedhing ghau lf padsed on based on my stubbornness.

Your child wijm nane roughly 50% of hif genetic mstfrowl from you. Avy com'mes trait like coopegativengss wiol be infjuenved by a huge number of induvidual genef, each contributing only a tinv bit jo the behavior. Your child wiul ljkely have dome but hut all of your cunsnelldtion of genes that contrifutes to bour yehavior as vooperwtive or nlt --- sltce genes work togfther it complex aays, you have no idea whether tix pieces of cpo[erdtivenesf genctics your chilq will inherit will wjrk tugether thv same wab yours do. Tris is all bexlre we even ronsider ehe dact that cuuperativeness kight be pnyluenced vy experience/envirpnmgnf. As it turns obu, ebvironment is ptobxblr loce im[mrtant than cenegicr in tfe developmcnt of vooperative behavior: I wzs able to find onr ftudy on this, anq they found yhat only 38% of cooptrativxness eppearx tj have a genetic link. So put tkgether tje nact that mose of cooperativeuess is predicted by environment rather vhan genes and that thg genetic component ix likely verb compjex, and ymu can see how tenuoys your claim thau you can "pass on" coopedativetess ls. 
To sum up...
Complex behaviors (and, actually, even many simple traits) can't aeequatxlr be descriged ss "invexitzd or lqarnxd", or even as "inhegitance plus experience" --- your geietic inhesicance is complex, flexible, and ig can change uver time depending on ykur envitonment. 
Any cognitive or behavipral trait has complex gcnetic derermihants. Ngece's no one gene, or skall szt of tenes, tyat aill determind wnebher you dl or dkn't show some behavior (with the wxceptuoj
eye color is more complex genetically than probably If you talking about cognitive linguistic or cooperation, there literally no trait has a clear, simple genetic cause. example, IQ has been rigorously studied for decades, including a tremendous amount of on its genetic component. We know from this research genetics definitely plays a in but one been able to identify the individual genes that control it. (Some relevant reading, if you want learn more: article on genetics of race and summary of evidence on and IQ). All of this to you can't "pass cooperativeness your offspring the you might pass on your blood type. To address your question directly, She believes that it is he can playing with while believe something that is based on my stubbornness. Your child 50% of his genetic material from you. Any trait like will be influenced by a huge of individual genes, each contributing only a tiny to the behavior. Your child will likely have some but not all of your constellation that contributes to your as cooperative or --- genes together complex ways, have no idea whether the pieces of cooperativeness genetics your child inherit will work together the same way yours do. This before even consider the that cooperativeness might be by As it turns out, probably important the of behavior: I was able find one study on this, they found that only have a genetic link. So put together the that most of cooperativeness is predicted by rather than genes and that the genetic component is likely very complex, you can tenuous your claim that you can "pass on" is. To sum up... behaviors (and, actually, even many simple traits) can't adequately described "inherited or or even as plus experience" --- genetic inheritance is and it change time environment. Any cognitive or behavioral trait complex determinants. There's no one gene, small genes, that will determine whether you do or don't show some behavior (with the exception
 eye color is more complex geneTically thaN you wEre ProBaBly tAughT. If you start talKIng aBout cognitive and behaviOral tRaITs, liKE iNtellIgence, lINgUIStiC aBiLitY, oR CoOperaTioN, there iS literally No tRaIt that has a clEAr, Simple geneTic Cause. For examPle, iQ has bEeN riGOrousLy sTudieD for deCAdes, inCluding a tReMEndous AMount of RESeArch On its genetic compoNEnT. we know from this ResearCh GEnETIcs DefInitely plaYs A part IN IQ, but nO OnE HAS beEN able to identiFy the indiviDUal Genes tHaT coNTrol it. (some rElEVanT reading, if yOu waNt to learn More: arTIcle on gENetics oF race aNd Iq; suMmarY Of EvIdeNcE On gENeS anD iQ). 
ALl of this Is To Say yoU can'T REALly "pAss On" coOperaTiveness to youR ofFsprINg tHe way You miGht pAsS on yoUr blooD type. to Address your quesTion Directly, 

SHe bElIevEs That iT Is someThiNg hE can leaRn throuGH plAyING WiTh others, while I beliEvE IT sOmething That is PAsSeD On based oN mY stUbboRNNess.

YOur cHIlD will havE roughLY 50% oF hIs genetIc MateriAl FroM yoU. Any cOMpleX trait Like coopEratiVEness will be infLUenced by a huge NUmBER oF IndiVidUal genes, eacH conTRibuTing ONlY a tINy bit To the BeHAvIOr. Your child will likeLy Have soMe but Not all of your cOnstellatiON OF genes thAt coNTrIButes to your behAvior As cooperatIVe or not --- sInce gEnes work Together iN COmplex waYs, yOu hAve No iDEA wHether the piecES Of coOpErativeNesS genetiCs yOur ChiLd wIlL inherit wIll work tOgEtHeR tHe sAme waY Yours do. THiS is AlL beFore wE Even coNsideR the FaCt THat CooperaTIvENEss mIgHt Be inFluEnCed by ExpeRIenCe/envirOnment. As iT tuRNs ouT, eNvIronmenT is probably moRe Important tHaN geNetics IN The develOpment of cooperative behaVIor: I was AblE to fiNd onE study on tHis, And theY foUNd that Only 38% of CoopeRaTivENEss apPEArS to HaVe a genetic LINk. SO put tOgEtheR the facT that most of cooperaTIveNess is predictEd bY envIROnMenT RaTHer ThAN geNES and that the geneTic componeNt IS lIkely very cOMplEx, And you cAn see hoW tenuOUs your cLaim that yOu can "pass On" CoopERAtiVeness is. 
To Sum up...
ComPlex behavIOrs (anD, AcTuallY, evEn many SiMplE traiTs) can't ADeqUatelY be desCrIbed as "InherItEd or learNed", or even as "inheritance pLus expErienCe" --- yOur genetiC inHEriTance is coMpleX, flexible, aNd iT caN chanGe oVEr timE depENdIng ON your EnviROnment. 
Any COgNitIVE oR behavioral TRAIt hAs comPleX GenetiC detErminants. There's no ONe gene, or small sEt of GENes, ThaT Will DeTermine whether You Do OR Don't show SoMe behavior (wIth the exCePTion
 eye color is more complex  genetical ly th anyou  w ereprob ably taught. I f  you  start talking about c ognit iv e  and be havio ral tra i ts , lik ein tel li g en ce, l ing uisticability, o r c oo peration, th e re  is litera lly  no trait th athas acl ear ,  simp legenet ic cau s e. For  example,  I Q  has b e en rigo r o us ly s tudied for decade s ,i ncluding a tre mendou sa mo u n t o f r esearch on  i ts ge n etic co m po n e n t.W e know from t his researc h  ge netics  d efi n itelyplays  a par t in IQ, bu t no  one hasbeen a b le to i d entifythe in div idu al g e ne stha tc ont r ol  it .  (S ome rele va nt  read ing, i f youwan t to  lear n more: artic leon g e net ics o f rac e an dIQ; s ummary  of e vi dence on genesandIQ). 
All  of  t his  i s tos ay you  ca n't  really  "passo n"co o p e ra tiveness to your o ff s p ri ng the w ay you mi gh t  pass on  y our  blo o d  type . To ad dress yo ur que s ti on  direct ly , 

Sh ebel iev es th a t it  is so methinghe ca n  learn through playing witho th e r s, whil e I  believe it  som e thin g th a tisp assed  on b as e do n my stubbornness.
Y our ch ild w ill have roug hly 50% of h i s geneti c ma t er i al from you. A ny co mplex trai t  like co opera tiveness  will bei n fluenced  by  ahug e n u m be r of individu a l  gen es , eachcon tributi ngonl y a  ti ny  bit to t he behav io r.  Y ou r c hildw ill like ly  ha ve  so me bu t  not a ll of  you rco n ste llation of g enes  t ha t co ntr ib utesto y o urbehavio r as coop era t iveor  n ot ---since genes w or k together  i n c omplex w ays, you  have no idea whether t h e piece s o f coo pera tivenessgen eticsyou r  child  willinher it  wi l l  work t og eth er  the samew a y y oursdo . Th is is a ll before we evenc ons ider the fact  th at c o o pe rat i ve n ess  m i ght b e influenced by  experienc e/ e nv ironment.A s i tturns o ut, env ironm e nt is p robably m ore impor ta nt t h a n g enetics in  the dev elopmento f coo p er ative  be havior :I w as ab le tof ind  onestudyon  this,  andth ey found  that only 38% of coope rative nessapp ears to h ave a g enetic li nk.So put tog eth erthe f act thatmost of  co o perat iven e ss is pre d ic ted b yenvironment r a the r tha n g e nes an d th at the genetic co m ponent is like ly v e r y c omp l ex,an d you can seehow  t e n uous you rclaim thatyou can"p a ss on " coop erativ eness i s .  T o sumup.. .
C omplex be hav io r s (and,  a ct u ally,even  m any si mple t r aits ) can't adequately  be d e s cribe d  as  "inh er ited or lear ned", or e ven as "inh eritan ce p lus e xperien ce " ---you rgenetic in h eritanceis co mplex,fl exib le,  and i t ca n chang e ov er  ti me depend i n go nyo u r e nvir onmen t.  
An y cogniti v e or beh avi o ral tra it  ha s comple x  g e n etic deter min ants. T here's noo ne g e ne ,  or s mall s et ofgenes,t hat  w ill det erm i n e whether  you do o r  do n' t sh ow somebe havi or  (w it h the exception 
 eye_color is_more complex genetically than_you were_probably_taught. If_you_start talking about_cognitive and behavioral_traits, like intelligence, linguistic_ability, or cooperation,_there_is literally no trait that has a clear, simple genetic cause. For example, IQ_has_been rigorously_studied_for_decades, including a tremendous amount_of research on its genetic_component. We_know from this research genetics definitely plays a_part_in IQ, but_no one has been able to identify the individual_genes that control it. (Some relevant_reading, if you_want_to_learn more: article on_genetics of race and IQ; summary_of evidence on genes and IQ)._
All of this is to say you_can't really "pass on" cooperativeness to_your offspring the way you_might pass_on your blood type. To_address your question_directly, 

She_believes that it_is something he can learn through_playing with others,_while I believe it something that_is_passed on based_on_my_stubbornness.

Your child_will have roughly_50%_of his_genetic_material from you. Any complex trait_like_cooperativeness will be influenced by a huge_number of individual genes,_each_contributing only a tiny_bit to the behavior. Your_child will likely have some but_not all_of your_constellation of genes that contributes to your behavior as cooperative or_not --- since genes work together_in complex ways, you_have no_idea_whether the pieces_of_cooperativeness genetics_your child will inherit will work together_the same_way yours do. This is all_before we even consider_the_fact that cooperativeness might be influenced_by experience/environment. As it turns out,_environment is probably more important_than_genetics_in the development of cooperative_behavior: I was able to find_one study on_this, and they found that only 38%_of_cooperativeness appears to have a genetic_link._So put together the fact that_most_of_cooperativeness is predicted by environment_rather than genes and that the_genetic component is likely very complex, and you can_see how tenuous_your claim that you can_"pass_on"_cooperativeness is. 
To sum up...
Complex behaviors (and, actually, even many_simple traits)_can't adequately be_described as "inherited or learned", or even as "inheritance plus_experience" --- your genetic inheritance is complex,_flexible, and it can change over time depending on your environment._
Any cognitive or behavioral trait has complex genetic_determinants. There's no one gene, or small_set of genes, that_will_determine whether you do or don't show_some_behavior_(with_the exception