When I first started coming "out", I also started learning about my own misconceptions of what being transgender means. Now that I've been out for a while, I can say that this ignorance was common to the larger cisgender society we live in. And I've come to believe that being transgender is being on a spectrum of gender development issues that involve gender identity, gender preference, gender presentation, and gender manifestation.
To start, for most people, gender identity is a simple concept: you are either male or female. But what happens when the mind develops in a way that this concept doesn't fit? Like many transgender people, one can reject the genitalia that one is born with and look to have both surgeries and hormone treatments to correct this physical anomaly. However, many transgender people do not suffer severe gender dysphoria and choose paths which may only involve hormone therapy and other, less radical body modifications. To some cisgender people, this is beyond comprehension, and it is an abomination. But why? A baby's gender is the first thing people want to know when a child is born, as if defines much of the path that person's life will follow from birth to death. Anyone who challenges that basic path might be a threat to social unity, and that has to be stamped out before the threat becomes a reality.
Next comes gender preference, a concept with which people are only now starting become comfortable. Like gender identity, this is considered a threat to society by many cisgender people. However, many cisgender people have grown to understand this concept and accept that people with non-traditional gender preferences should be treated with respect and accepted by society.
However, gender expression (presentation) is not as understood by the general populace. Why do some lesbians present with a more "masculine" appearance? Why do some "straight" (in gender preference only) prefer to present as females? Many in cisgender society have problems processing who and what a person is when a person has an androgynous (or non-conforming) gender image that resists categorization. This may trigger cisgender society's deepest fears, as people tend to fear most what resists categorization and being understood.
Lastly, gender manifestation (or, I should say, being intersex) is something that is either hidden or "corrected" at an early age. For example, many babies born with ambiguous genitalia have been "fixed" to look like "normal" females. But this potentially creates a problem for these children as they grow up into adults, as their gender identity may be in conflict with their "corrected" gender manifestation. As a result, many intersex people are demanding that no "corrective" surgeries be performed until the child knows enough about its own gender identity and can provide input into the process of gender identification.
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With all of the above being said, I have to focus on the misconceptions people have about transgender people. For example, not all of us want to have surgeries to convert our genitals to that of the sex of which we identify. One person I know has had hormone therapy, but chooses to retain her masculine sex organs. (Who wants to live a life without being able to have an orgasm again? This person needs to preserve her ability to self-pleasure.) Others need to have genitalia which resembles that of their gender identities. And then, there are issues related to how many gender related physical traits that transgender people feel they need to fix before they feel they are of their identified sex. (Hair transplants, Voice Adjustments, etc.) As they used to say in old commercials, your mileage may vary.
Another misconception is that many cisgender people have about us transgender people is that we want different genitalia, so that we can have sex with our "former" sex. (I knew a LCSW who believed this, and I am glad she is not in my life.) For the most part, one's gender preference does not change after one has gender corrective surgery (GCS).
What I found surprising is how many cisgender people react when they interact with non-op transgender people (like me) who travel with legal IDs which conflict with their gender presentation. Some, like my Texas friends, are amazed that I will often travel in female mode while holding male ID. Others will (in bad taste) wonder aloud whether a transgender person is a male or female. If I had heard that while boarding a ship on one cruise, I'd have asked the couple: "Why don't we go back to your room, and the three of us find out? It could be some great fun." That would have thrown them for a loop.
Yet, when it comes to sports, more of the general population doesn't want transgender people to compete against people not belonging to their natal sex. But what happens when a person's natal sex is ambiguous, or when hormone levels are that of the opposite sex? There is a female runner who has been disqualified from running against other females because her naturally occurring levels of testosterone are that of a male. How should we accommodate people like her? There is a transgender female swimmer who is now disqualified from being in competitions against other females - even though many of her peers support her inclusion in these competitions. I feel that if a transgender person's body has not undergone the puberty of their (at birth) assigned sex, and has only gone through the puberty of their identified sex, they should be allowed to compete against others of their identified sex as they will have no advantage from the wrong puberty. And even this is an issue that perplexes many in society.
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I feel that we have a long way to go before society understands and accepts transgender people. At best, we are considered anomalies whose identities are not fully accepted because they are ambiguous. At worse, they trigger irrational fears in others that will trigger them to cause us harm. Hopefully, things will change for the better for us soon.