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He asked for TV, and we refused as he took too long a time to get washed (for everything we ask, he skips and starts playing or watching books).
Some may say it is wise to be specific in managing expectations. For example, a parent might say: " You may have TV if you get washed and come straight here, without playing or looking at books. But if you choose not to get washed and come straight here, you may not have TV. The choice is yours." This sets clear expectations and the child will know you can be trusted to follow through. From reading the post, the child may have the perception that parents agreed to TV and then changed their minds arbitrarily, leaving the child feeling disempowered. The child may not understand the role of his own choices as leading to the consequences.

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When he decided to come back he discovered that instead of the grated carrots (the starter), he now had smashed patatoes ... So he screamed again .... And so on.
Announcing, "the carrots will be served for 5 minutes, then it is time for potatoes" can be used to manage the child's expectations.

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Several tantrums a day... They can stop just in a blink of an eye, and then he becomes a normal kid, just as if nothing happened.
Some have said that "tantrums" are purposeful attempts by a child in control of himself to manipulate a situation; "meltdowns" are cognitive overload by a child overwhelmed and therefore not in control of himself. Effective parenting may respond differently to tantrums than to meltdowns.

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This morning, as another tantrum occured (this time because he was looking at a book on the sealife instead of having his breakfast), she said to DW that she's not sure to be able to go on with homeschooling
Some may say this sounds arbitrary and punitive... why punish a child for looking at a book, even during breakfast time? Isn't this type of flexibility a benefit of homeschooling? If there is truly a need to stay on schedule, might a parent advise, "No books until after breakfast. That's a house rule. If you choose to break the rule, I will mark that on the calendar. 5 broken house rules in a month will indicate you have chosen to no longer homeschool, but rather have chosen to return to school."

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For instance, following the yesterday event, DW wants to punish him for what he did by not allowing him to watch TV today
Were expectations proactively managed regarding the result of behavior choices? For example, "A tantrum today will result in loss of TV privileges tomorrow."

From the child's point of view, it may be that he never knows what to expect. It may appear that adults cannot be trusted, but rather make up arbitrary rules on the fly, according to their moods.