Barack Obama’s election to the presidency was a watershed moment for black Americans. And if Hillary Clinton becomes president, it will be celebrated as an equally historic moment for women. But what of Sen. Bernie Sanders, who recently became the first Jewish-American presidential candidate to win a primary? He’s closer to reaching the U.S. presidency than any other Jewish person in history, yet his achievements aren’t exactly being celebrated by the American Jewish community as a milestone.

The ambivalence of American Jews toward Sanders can be explained partly by Sanders’ own seemingly lukewarm embrace of his “Jewishness,” experts say. When Sanders won the New Hampshire primary last week, he delivered a stirring speech that highlighted his family history: “I am the son of a Polish immigrant,” Sanders told the crowd. It was an odd choice of words that didn’t go unnoticed by many in the American Jewish community.

“It’s a bit unusual for a Jew — no less a 74-year-old Jew — to refer to himself as the children of Polish immigrants,” Michael Cohen wrote in the Tablet, an online magazine about Jewish culture. “The historical Jewish experience in Poland is not what one would exactly call ‘rosy.’ Being a Polish Jew is not necessarily a point of pride; it’s just very odd for any Jew to invoke their Polish heritage rather than their Jewish heritage.”

Indeed, the recognition of Jewish heritage is important to American Jews — according to a 2013 Pew Research study, 62 percent of U.S. Jews said that being Jewish is mainly a matter of ancestry and culture, more so than religious practice itself. And ignoring that heritage is troubling to some.

“Everybody knows Bernie Sanders is Jewish, but he’s had a lot of trouble talking about it. And some, not all, within the Jewish community are uncomfortable with that,” said Jonathan Sarna, author of “American Judaism: A History,” and professor of American Jewish history at Brandeis University.

“The great irony of the story is that Bernie Sanders does not have Jewish grandchildren — but Mr. Trump does. That perhaps explains why Bernie Sanders has not been greeted by the Jewish community quite the same way that, say, Joe Lieberman was when he ran for vice president in 2000,” added Sarna, noting that Sanders’ wife is not Jewish, and their children and grandchildren don’t seem to identify at all with Jewish heritage.

“The secular Jewish community is still deeply worried that their grandchildren won’t be Jewish, just like Sanders’ grandchildren aren’t,” said Sarna.

To some, the term “secular Jewish community” may sound like an oxymoron, but in fact a full 22 percent of those who identify themselves as Jews in the United States describe themselves as having no religion. Rather, they celebrate their heritage by enjoying some truly excellent bagels, a well-placed Yiddishism or maybe a Woody Allen film. Sanders likely falls into that category — but even to many who fall in this group, a sense of preserving Jewish heritage is important.

“Judaism is very much about communal factors. To be a good Jew is not to be religiously observant or knowledgeable, it’s to be part of the community,” said Ken Wald, professor of political science and American Jewish culture at the University of Florida. “Sanders doesn’t invoke a sense of ‘he’s one of us’ among American Jews. For some people it’s a bit troubling that he will be seen by non-Jews as representative of Jews.”

Of course, in many ways, Sanders’ Jewish background is impossible to miss, says Deborah Dash Moore, director of the Frankel Center for Judaic Studies at the University of Michigan.

“Bernie Sanders, however long he has lived in Vermont, is a Brooklyn Jew, with many of the negative old-time stereotypes associated with such Jews: brash, nonreligious, socialist (or, more generally for the stereotype, radical troublemaker), working-class background, second generation where Jewishness is largely unconscious — that is, it is not performed through religion but rather lived through who you are,” Moore wrote in an email to International Business Times. “He represents a kind of ethnic Jewishness that is out of style these days among American Jews.”

That could be because Pew’s 2013 survey suggested that the American Jewish identity, partly as a result of increasing intermarriage, is becoming muddled. When asked what it meant to be Jewish, 69 percent of Pew respondents answered with “leading an ethical life,” while just 19 percent said it meant “observing Jewish law.”

“That Pew study opened a lot of fears,” said Sarna. “Everybody recognizes it, but at the same time, the hope is that somehow, through culture, there will be a maintenance of continuity.”

And while many secular Jews will be attracted to Sanders precisely because of his liberal leanings, his unclear stance on Israel is a turnoff. According to Pew, 69 percent of American Jews are either very or somewhat attached to Israel.

“I couldn’t tell you what position Sanders takes on the Middle East. Sure, he did the requisite kibbutz volunteer work as a kid, But other than that, he hasn’t shown any particular commitment to Israel,” said Wald. “Hillary Clinton is seen as somebody who is pro-Israel. Bernie Sanders is kind of a blank slate on that issue.”

The lack of emphasis on Sanders' Jewish identity, however, is also a sign of progress, said Sarna.

“It demonstrates that Jews are well integrated into America,” he said. “Many Jews don’t feel they have to vote on the basis of kinship, but rather on the basis of whose policies they support."

It is notable that “someone who looks as Jewish as Bernie Sanders can run for the presidency, win a primary and the Jewish issue is practically not talked about at all, in much the same way we can have three Jewish Supreme Court justices,” said Sarna. “That speaks of enormous progress in America.”