Image caption Author Michael Pearl says sales have been steady

A child-raising book that advocates whipping with branches and belts has sold hundreds of thousands of copies to evangelical Christians. But the deaths of three children whose parents appear to have been influenced by the authors' teachings have provoked a growing backlash.

The implements can vary. For a child under one year old, a willowy branch or a 1ft (30cm) ruler is recommended. For older children, a larger branch or a belt.

But the objective of the "spanking" described in Michael and Debi Pearl's To Train Up a Child is the same - making children surrender completely to their parents' will.

"Training is the conditioning of the child's mind before the crisis arises; it is preparation for future, instant, unquestioning obedience," reads a passage from the book's first chapter.

The "training" is meant to start early and pre-empt the need for punishment. But if the child is already rebellious, parents are told to "use whatever force is necessary to bring him to bay".

"If you have to sit on him to spank him then do not hesitate. And hold him there until he is surrendered... Defeat him totally."

Hannah (not her real name) grew up in a community of Independent Fundamental Baptists in north-western Florida. Her parents obtained copies of books by the Pearls when she was about nine and her sister seven.

The spanking began shortly afterwards and continued for at least eight years. In the first five years, it usually happened several times daily.

One day, when she was 14 or 15, her father heard a story about Hannah getting into a fight with a boy at church.

"I'm still not sure honestly what I was being accused of, but my dad just completely flipped out because whatever he heard was just atrocious," she says.

There's no way that a person who reads the book could be led to violence Michael Pearl

He used wooden rulers, or yardsticks, to spank her, snapping about five in the course of the beating - her mother kept a dozen in the house because they broke so often.

"When I couldn't sit down a couple of days later he was like: 'Stop being so melodramatic, what's wrong with you?' Then he had mother look and [my coccyx] was incredibly bruised and swollen."

Hannah, now in her mid-20s, says her father was "horrified" and never spanked her again. But her mother continued, using a plastic blind handle that she thought was less likely to leave marks on her children's skin.

Like other people who have witnessed Michael Pearl's advice being put into practice, Hannah says her parents were seduced by the idea of a simple formula that would make their children compliant.

"The problem is that [Pearl] tells you you have to break your children," she says. "And to get there you have to be completely ruthless."

To Train Up a Child is widely seen as the most extreme of the publications produced by conservative Christians in the US who advocate corporal punishment.

It is produced by the Pearls' organisation, No Greater Joy Ministries, which is attached to the church where Michael Pearl is a pastor in Pleasantville, Tennessee. First published in 1994, the book soon became popular among fundamentalist, non-denominational groups outside mainstream Christian culture.

Within these tight-knit communities, many families educate their children at home, viewing schools as having a harmful social environment and being insufficiently religious. The Pearls started homeschooling their children in the 1970s, when the practice was still novel.

Homeschooling in the US Began spreading in late 1970s and early 1980s

Early supporters included progressive left-wingers and Christian fundamentalists

Restriction and regulation of homeschooling has been gradually relaxed

An estimated 1.8 million children homeschooled in 2011

64% said desire for religious instruction was an important motive Sources: US National Center for Education Statistics; Rachel Coleman

Books, magazine and videos produced by the Pearls are passed around between families, or given as gifts for new-born babies and couples getting married.

Elizabeth Esther, a blogger who grew up in a conservative Christian community in California and describes herself as a "recovering fundamentalist", says that in her church the Pearls were "basically held up as the sterling example of how to raise your children before God".

The group said its revenue for the 2012-2013 tax year was $1.5m, 60% of which came from sales. Some products are donated to prisoners and military families, and boxes of books have been sent to US troops in Iraq and Afghanistan.

To Train Up a Child has sold more than 800,000 copies, according to Michael Pearl. Sales have remained steady in recent years and are only boosted by attacks, he says. "We have several million very happy and cheerful parents and kids who've seen great, wonderful fruit from that book and other things we've written."

Extracts For the under one year old, a little, ten- to twelve-inch long, willowy branch (striped of any knots that might break the skin) about one-eighth inch diameter is sufficient. Sometimes alternatives have to be sought. A one-foot ruler, or its equivalent in a paddle, is a sufficient alternative. For the larger child, a belt or larger tree branch is effective.

The parents who put off training until the child is old enough to discuss issues or receive explanations find their child a terror long before he understands the meaning of the word. A newborn soon needs training

PARENTS MUST ASSUME THAT PART OF THE CHILD'S MORAL DUTY WHICH IS NOT YET FULLY DEVELOPED. The parents' role is not that of a policeman, but more like that of the Holy Spirit

When the time comes to apply the rod, take a deep breath, relax, and pray, "Lord, make this a valuable learning session. Cleanse my child of ill-temper and rebellion. May I properly represent your cause in this matter."

Make it a point never to use your hand for spanking. The hand swatting is a release of the parent's own frustration. Furthermore, where the child is concerned, the hand is for loving, not martial arts.

Matthew (whose name has also been changed) grew up in a homeschooling family in the mid-West that expected just such positive outcomes from To Train Up a Child.

Spanking with wooden rods and branches started at a very early age for Matthew and his two younger siblings. In the first 10-12 years it happened daily to weekly, he says, before becoming less frequent but more severe.

He says there were no serious injuries. But there have been cases ending in tragedy.

Three child deaths Hana Williams, 13, hypothermia, malnutrition (2011)

Lydia Schatz, 7, massive tissue damage (2010)

Sean Paddock, 4, suffocated in blankets (2006)

In 2010, Lydia Schatz died after being beaten, three years after arriving in California from Liberia. The following year, another adoptee, 13-year-old Hana Williams, died from hypothermia and malnutrition after being left in the back yard in a small town in Washington state.

The Schatz parents are serving long prison sentences after pleading guilty to charges including second-degree murder, torture, voluntary manslaughter and unlawful corporal punishment.

The Williams parents were sentenced in October to decades in prison for manslaughter.

Investigators said both sets of parents had followed advice from To Train Up a Child, a copy of which was reportedly found in both homes.

Michael Ramsey, a district attorney who prosecuted the Schatzes, said he was planning to mention the book as a contributing factor if the case had come to trial.

Though he did not want to detract from the parents' responsibility in causing Lydia's death, he said the book's ideas were "wholeheartedly embraced by the Schatzes", and "the entire philosophy of the book is intended to lead someone down that slope".

Image caption Carri Williams, adoptive mother of Hana, was sentenced in October to nearly 37 years in prison

In a third case Sean Paddock, a four-year-old whose adoptive mother in North Carolina had turned to the Pearls' teachings, died from suffocation after being wrapped tightly in a blanket.

All three of the children who died were reportedly beaten with a plastic plumbing tube similar to one that Michael Pearl had mentioned as a possible spanking tool.

But Pearl himself denies that To Train Up a Child can trigger abuse, pointing out that the book cautions against "brutality" and using spanking as a vent for anger.

"No court, judge, police or child protection service has ever accused us of doing anything that was an endangerment to children," he said.

"There's no way that a person who reads the book could be led to violence. That may not prevent violence if that's part of their nature, but it's not going to lead them to do something that's contrary to their own set of values."

The book's critics, who include Christian groups, atheists, parenting activists and academics, disagree.

To Train Up a Child is "quite singular in its orientation toward punitiveness toward children in general but also infants", says George Holden, a professor of psychology at the Southern Methodist University in Dallas.

"It preaches a number of very dangerous views, that could very easily result in physical child abuse if one follows what they advocate."

Holden has backed one of the three main online petitions demanding that Amazon and other book suppliers stop selling To Train Up a Child.

The petitions gained momentum in recent weeks after a fresh push by UK bloggers. They have registered more than 200,000 signatures in total, though some people may have signed up to more than one of the campaigns.

British bookseller Waterstones says it is not stocking the book and Foyles, also based in the UK, said it had removed the book from its website.

Amazon, however, has shown no sign of changing its policy. The company said in a statement: "This book has been widely debated in the media, and on Amazon, for many years, and anyone who wishes to express their views about this title is free to do so on its product page on our website."

Corporal punishment 34 nations prohibit corporal punishment in all settings

nations prohibit corporal punishment in all settings 23 of them are in Europe

of them are in Europe 48 have expressed a commitment to prohibit all corporal punishment

have expressed a commitment to prohibit all corporal punishment 114 have no clear commitment to reform, including the US and UK Source: Global Initiative to End all Corporal Punishment of Children

People involved in the campaign say they are not seeking to have the book banned. Any legal challenge would be highly unlikely to succeed, at least in the US, where the first amendment of the constitution offers authors a high level of protection.

But they point out that increasing numbers of conservative Christian families have been turning against the Pearls' teachings, partly in reaction to the three deaths.

One American woman who blogs under the name Hermana Linda and runs a website for Christians seeking information to counter the Pearls' mindset says she has seen many people reacting with disgust to their teachings.

A concept called "gentle parenting", which opposes spanking, has been slowly gaining ground, she says. Others see a similar trend worldwide.

"National studies have been showing a steady decline in support for parental corporal punishment," says Robert Fathman, a clinical psychologist and campaigner against physical punishment for children.

"The US has a long way to go but we are moving in the right direction."

Follow @BBCNewsMagazine on Twitter and on Facebook