The yearlong siege of Gaza may soon end with the new cease-fire there, marking the eclipse of one more American-backed Israeli policy that backfired by strengthening extremists.

Here in Gaza, sulfurous with fumes from cars burning cooking grease because the siege has made gasoline scarce, the entire last year of the blockade feels not only morally bankrupt — a case of collective punishment — but also counterproductive. The fragile new truce between Hamas and Israel just might create a new opportunity to stabilize the Palestinian territories, but only if we absorb the lessons of what has gone wrong.

Consider Adham Sharif, a 26-year-old man whose only child, a baby girl named Mariam, had a tiny hole in her heart and needed surgery to repair it. Gaza hospitals were unable to perform such an operation, but doctors said that surgeons in Israel or in neighboring countries could save her.

In theory, there was an exception to the siege to let people out of Gaza in medical emergencies. But Mr. Sharif could not get the Israeli permit for Mariam to leave, and she died in November. “It’s so hard,” he told me. “You see your child dying, and you can’t save her.”

Does Mr. Sharif blame Hamas as the cause of the blockade that cost his daughter’s life? “Of course not,” he said. “I blame the ones who closed the border: Israel. And America, its ally.”

Now when he hears of extremists firing rockets at southern Israeli towns like Sderot, Mr. Sharif has a warm feeling all over.

When Hamas won democratic elections in Gaza and then seized full power a year ago, there were no good choices for Israel and America. Hamas includes terrorists, Islamic fundamentalists and ideologues, and it has cultivated ties with Iran. It has decent governance by the region’s devalued standards — it is not particularly corrupt; it delivers social services efficiently, and the streets are safe — but it runs a police state and alarms all its neighbors.

Of all the bad choices, Israel chose perhaps the worst. Punishing everyone in Gaza radicalized the population, cast Hamas as a victim, gave its officials an excuse for economic failures and undermined the moderates who are the best hope of both Israel and the Arab world.

If the U.S. and Israel had formed a Joint Commission to Support Hamas Extremists and Bolster Iranian Influence, they could hardly have done a better job. The episode is the latest evidence that hard-liners in Israel, Palestine and America all reinforce each other. Arab terrorism led to the rise of Israeli hawks and to two invasions of Lebanon. The first Israeli invasion helped give birth to Hezbollah, and then the Israeli assaults on Palestinian police helped nurture Hamas.

So while Israelis denounce Hezbollah and Hamas, they helped create them. And while Palestinians denounce the separation barrier, their suicide bombings built it.

“Extremists need each other, support each other,” noted Eyad el-Sarraj, a prominent psychiatrist in Gaza. He laments that the siege of Gaza has discredited pro-American voices: “Whoever is not going along with the U.S. is a hero, even the crazy ones.”

The U.S. and Israel devoted their energies to punishing Hamas and didn’t work to make a success of our preferred interlocutors. So moderates like Mahmoud Abbas, the Palestinian president, now come across as weak, irrelevant and ineffective, while Hamas emerges as the victor.

We should talk to Hamas, not because negotiations will necessarily get anywhere, but because a failure to negotiate will necessarily get nowhere.

Israel’s decision to block Gazans from studying abroad was particularly shortsighted. Educating Gazans might help build a contingent of moderates, but Israel has continued to block three Fulbright scholars from leaving for the U.S.

“For Israel to have a better future, it should want neighbors with better education,” Zohair Abu Shaban, one of the Fulbright students, noted reproachfully.

So far, Hamas has outmaneuvered Israel and the United States. Opinion polls this year show Hamas gaining over all in the West Bank and Gaza. And, when we help Hamas, we inadvertently boost its backer, Iran.

Perhaps most depressing, large Palestinian majorities — more than before — now favor terror attacks. A university student in Gaza, Rajaa Batrikhi, 20, told me she has suffered so much from the siege that she relishes the rocket attacks from Gaza on Israeli towns.

“I think it’s good when we hit them with rockets,” she said defiantly. “Our rockets are nothing like the rockets they hit us with. At least they feel the fear that we feel every day.”

It’s a credit to Israel that it was willing to negotiate indirectly with Hamas, and with the truce, we now have a chance to break this downward spiral. Let’s stop bolstering Hamas.