We were on the beach when we spotted the brown bear. My wife Laura and I, with three of our children, were conducting a survey of shore birds for a local science project near our home in Alaska. The bear was perhaps a quarter of a mile away. Looking through our birdwatching telescope, I saw she was thin and wandering aimlessly. It was a cold April day and I guessed she was behaving oddly due to hunger.

Inland, beyond a bank of dunes, there was forest and the bear moved in that direction. After a few minutes, I told the others I was going to climb the nearest dune, just to check she had gone. I hadn't taken the first step when my daughter Grace said, "Dad, it's right behind us!" There she was, no more than 60ft away.

I work for the US Fish & Wildlife Service and always carry a gun or pepper spray in bear country, but I'd never seen so much as a paw print on that beach. There are established rules for bear encounters and my family are well drilled. Bears rarely attack people unless provoked and I was confident we'd see this one off by letting it know we were human. "Hey, bear," I shouted. "Get out of here!" We bunched together to make ourselves look more imposing and clapped and yelled, expecting her to back off. She bounded towards us.

"Get behind me," I said. "Stay with me and don't run, no matter what." An aggressive bear will usually make a "bluff charge" to frighten away an adversary, growling and roaring, but pulling up short at the last minute. This one was silent, ears pinned back – the sign of an animal that's going in for the kill.

Gripping my telescope tripod in both hands, I held it in front of me to form a barrier as the bear cannoned into me. Her huge head was level with my chest and shoulders, and the tripod lodged crossways in her mouth. She bit down and I found myself supporting her weight. I'm a big man, 6ft 3in and 280lb, but the bear was at least half as heavy again. I knew I wouldn't be able to hold her for long.

Even so, this was a fight I had to win: I was all that stood between the bear and my family. Grace was 11, Damien eight, and she could have snatched either of them up in her jaws and made off with them. Laura had seven-month-old Camille on her back, and the world's fastest sprinter would stand little chance of outrunning a brown bear.

The bear swiped at the telescope, shearing it clean off the tripod. I raised my left forearm to protect my face; the beast's jaws clamped down on it and her paw slammed into my ribs. It felt as if my arm was in a vice, and I sensed my bones were going to break.

Drawing back my free hand, I struck the bear as hard as I could, five or six times, aiming for the eyes and nose. She opened her mouth and I grabbed at her fur, trying to push her away; Laura tells me I was actually wrestling with the bear at this point. Then, as abruptly as it had begun, the fight ended. The bear retreated back up the dune, before doubling back to perform a bluff charge – the first time I felt genuine fear. The initial attack can't have lasted longer than a minute, and I'd reacted instinctively, but I knew that if the bear was really determined to have the last word, there was nothing I could do to stop her.

Apparently satisfied that we posed no further threat, she moved off, swiping at a fence post as she went. Laura called 911 and by the time the police arrived, she was nursing Camille – who had slept through the incident – in the back of our van. There were no tears and no one was shaking, just as there had been no screaming during the attack. Rather than feeling traumatised, we were all glowing, buzzing at our good fortune. Shots rang out in the distance: the troopers had been warning some walkers farther up the beach when the bear had appeared suddenly and charged them. They returned in a pick-up truck with the bear carcass in the back.

When we got home, Grace and Damien excitedly relayed the story to their seven disbelieving siblings – one day, we'll tell Camille what she slept through, too. My arm was bruised, but the outcome for us could hardly have been better. I'm proud that my family remained clear-headed when panic could have led to a very different outcome.

• As told to Chris Broughton.

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• This article was edited on 1 July 2013. In the original, an editing error put Toby Burke's home in Arizona when it is thousands of miles farther north, in Alaska. This has been corrected.