I hesitate to respond to your message, in case I get drawn into an /Alice in Wonderland/ tunnel... However I'll giver a try
>... And if students don?t yet know the concept of *average* then they can?t possibly be thinking about why it is important.
Perhaps they will benefit from an example of what an average is, prior to a statement of what the concept 'is'. I can imagine a teacher explaining to a class that they could put all their lunch moneys into a pile & then redistribute it equally so that each student has the same amount.... <Caution: if you are a teacher, be aware of the possibility of some involved parent taking umbrage at "income redistribution"!>
>These are 6th grade students. You don?t need to waste yet another day with math avoidance, just walk into the classroom and ask them ?How do we (teachers) >determine your final grade?? and go from there.
But wouldn't the more astute little tykes say that teachers' pets get the higher marks & the trouble-makers lower?
>Start with the average of two scores, then 3, 4 etc. Ask them why it is useful in the end to give an average score rather than a list of 30 scores.
Students may not see the need for a single number, rather than two or five. The class might consider a student who has written 5 math tests. He can either tell his poor old man: "my average on the last 5 math tests was 64% - or - he could say "I got 80% on the 1st four and 0% on yesterday's." Maybe his mom is trying to quit smoking: "I have had 7 cigarettes in the last week: 1 day on an average" vs "I had 7 cigs a week ago but none since"
So: Students: when would a single number be good enough? monthly class absenteeism? maybe to compare all the grade 6 classes. Or tardiness?
>And there isn?t a *rule* and I don?t know why you keep using that term. There is a generalized definition, the sum of the values over their count, but that has >to be interpreted over many contexts. With a straight list. With something like 3 kids scored 100 and 2 scored 50, what was the average score? To something >like, the average of two numbers is 30 and one of the numbers is 50, what is the other number? It isn?t that I don?t find it important to make mathematics >applicable to the student. I do this more than any activity you have yet to post an example of.
These are the 'mathy' topics that should indeed be mastered. But if there is some context (why we might want to use averages) I would hope the success ratio might be higher, on average;)
> 40 years ago you would be teaching math for daily living. Somewhere in those 40 years the political forces decided that you should instead pretend to teach >advanced courses like algebra to very unprepared students.