One of the great joys--as well as debates--of this play is trying to determine who is taming whom. Petruchio is overt about his intentions to nearly everyone, including the audience, and there is no doubt he sees himself on the role of falcon-trainer. However, Katherine is a bright woman (which is why she suffered no fools as suitors) and once she catches on to her husband's plan, she masterfully plays him like an instrument. Perhaps, then, they are each training the other. In the end, the audience is more hopeful for and interested in this pair than in either of the other two married couples because of their complex relationship.
I think this is an attempt at humor. I say attempt, because I don't really find it funny. It would have been uprorious in Shakespeare's day though. Shakespeare had a way of using the relationship and power struggle between men and women humorously. Ok, I admit it's a little funny.
I read this question a little differently. You asked if Shakespeare presented his opinion of women. In the particular example you gave of training a woman as one would train a falcon, I don't think is Shakespeare's opinion. I think he is showing how society views women so he can create a contrast that does reveal his own opinion. Shakespeare shows how Petruchio believes he can tame Katherine; this would be in line with Elizabethan society ideas. But, he turns around and shows Katherine to be a strong, independent woman. This would not have been desirable at this time period. Shakespeare then shows, as post 5 stated, that Katherine also trains Petruchio. I think Shakespeare shows us his own view juxtaposed to that of society at the time.
I think the image of training a falcon is one that we can find in various places throughout Shakspeare's plays, and I partly agree with the analysis given above. However, is not the fact that it is Petruchio who has to "train" Katharina indicative of a certain sense of male superiority? What evidence is there in the play that Katharina returns the favour and gets to "train" her husband? At the end of the day the analogy of a falconer with his bird always leaves the falconer having more power, whatever the kind of relationship that he has with his bird. I like to think that this play is actually being very subversive in terms of its discussion of gender relations, but at the same time I think that the message of male dominance and supremacy is hard to avoid.
I would also add that the ending of the play can have two subtly different interpretations. Is Katherine truly tamed? or is Petruchio only training her enough to "play the game" better in public display? I have always thought that the second reading was more interesting. Petruchio may actually enjoy the wit and fire of Katherine and doesn't want her to be rid of those qualities, but instead, use those qualities in more appropriate ways. I read the last scene where Petruchio and Katherine are with all of the people to be almost a private joke between the two -- that she is saying and doing all the right things because it is what she needs to do for Petruchio. I like to think there is a certain glint in their eyes as they deliver their lines.
Okay, my response is going to pale in comparison to the awesome response before it, but I just had to say that even though we could never tell for sure what Shakespeare is thinking, my opinion is a big fat "Yes"!
Leave it to Shakespeare to literally "shake" things up in Elizabethan times. Such a rogue! Women were generally not considered equal to men during Shakespeare's time and some of his stronger women (Portia, Viola, Katarina, Lady Macbeth, even Juliet comes to mind) broke the Elizabethan mold, thank goodness.
If you want to see a set of TV directors and producers who think the same thing and made it applicable to our time (or, okay, at least to the 1980s), watch the episode of Moonlighting that presents a parody of The Taming of the Shrew. It is absolutely not to be missed.
In Shakespeare's The Taming of the Shrew, Petruchio is a pretty savvy guy. He knows what he wants, and he is well-aware of the challenge before him to not only win Kate's heart, but to "train" her as well so that their marriage will be satisfying to both of them.
I don't think that Shakespeare shows any sense of superiority of women in having Petruchio handle Kate as if she were a falcon. Petruchio is simply applying what he knows will work with a falcon, on a woman—much as he sees it work with birds, insinuating that this is simply an example of logic and good reasoning. If it works with birds who are very smart, but obedient when trained, the relationship between the two will be strong.
Thus have I politicly begun my reign,
And 'tis my hope to end successfully.
My falcon now is sharp and passing empty;
And till she stoop she must not be full-gorged,(175)
For then she never looks upon her lure.
Another way I have to man my haggard,
To make her come and know her keeper's call,
That is, to watch her, as we watch these kites
That bate and beat and will not be obedient.(180)
Shakespeare is not in the habit of putting down women. There are women in several of his plays that are full of surprises, and able to obtain what they want either through subterfuge or determination. And not all of them are "good" people, either.
For instance, in Shakespeare's Macbeth, Act Two, scene two, Lady Macbeth is much stronger than her husband; killing is not something she shies away from if it will give her what she wants: to be Queen. She speaks harshly and cruelly to motivate her husband to kill the King. When he returns with the murder weapons, she takes them back and smears Duncan's blood on his servants to implicate them in the murder. At this point of the play, she feels to wash her hands will clear them of the guilt of what they have done. Macbeth is slow to follow her lead, at least at this point, but Lady Macbeth is seen as a strong, no-nonsense woman.
You do unbend your noble strength, to think
So brainsickly of things. Go, get some water
And wash this filthy witness from your hand. (60)
Why did you bring these daggers from the place?
They must lie there. Go carry them, and smear
The sleepy grooms with blood.
I'll go no more:
I am afraid to think what I have done; (65)
Look on't again I dare not.
Infirm of purpose!
Give me the daggers. The sleeping and the dead
Are but as pictures; ’tis the eye of childhood
That fears a painted devil. If he do bleed, (70)
I'll gild the faces of the grooms withal,
For it must seem their guilt.
I do not believe that there is any disrespect intended in Shakespeare's portrayal of Petruchio and his plan. I think that Petruchio is using common sense where he has seen it work before— and remember, he is able to deal with Kate in ways no one else has ever been able to.