A year or two for math facts, money math, fractions / decimals etc? Maybe we have a vastly different understanding of what needs to be achieved in those areas? I can't imagine this taking more than a couple of months, at most.
How in depth and to what level of exposure are you talking about? Brief introduction? Or in depth mastery at higher levels? I could briefly introduce all of those subjects to a student who has never had any formal exposure in a couple of months or so. Unless they are really, really, really good at mastering math concepts quickly or have had significant exposure prior they will not have MASTERED these concepts or internalized them to the point of complete automaticity.
I am looking at the Math on the Level list of concepts needing to be mastered by the end of 8th grade and it is 140 items long. Many of these things build on each other so it isn't 140 separate unrelated concepts but it is a lot of concepts and each one of those 140 items may take days or weeks to really understand and master. Even after mastery it may take additional review for the knowledge to stick long term so besides learning and mastering the concept older concepts need to be reviewed as you move forward. For some kids they can learn with minimal on-going review. For others they will need daily or at least weekly review of anything that has already been introduced. Add in any deficits (such as low working memory, low processing speed, ADHD, dyscalculia, dyslexia, etc) and it will probably take longer, maybe much much longer to master these concepts.
For instance, fractions can be exceedingly difficult for a child to master. Doesn't mean they can't grasp the concept pretty quickly. But it may take quite a while to be able to convert between fractions with different denominators or to multiply fractions or change mixed numbers to improper fractions and vice a versa, or reduce fractions to their simplest form, etc. And then to be able to apply that knowledge in multiple settings in real life and in word problems and on tests will take time. They need exposure, then mastery, then automaticity. That usually does not happen overnight at all. That takes time.
When I was in school concepts were introduced and then we moved on. There was very little review. The assumption, I guess, was that once a concept was introduced and a student showed they understood it then it was not only mastered but internalized and no further review was necessary since there would be additional exposure the next year. For many students this approach left them with a very weak foundation in math and struggling mightily in later classes. They not only hadn't mastered the material it had not been internalized. There was no automaticity. They were using all their brain resources trying to remember these processes instead of learning the new material being introduced the next year.
I hit High School with a very weak foundation in fractions, decimals, percents, etc. even though I had exposure every year. There is a difference between exposure and mastery. There is a difference between master as in showing you fully understand a concept and automaticity as in internalizing the material to the point you no longer need much or any review to be able to do the math cold after having been away from it for weeks/months/years.
Most students can grasp fractions, decimals, money, etc. at an exposure level pretty quickly, especially as an older student (barring potential complications from LDs). Those same students will probably need a LOT more than just exposure to master and internalize those concepts at a more in depth level and to be able to perform those tasks smoothly without help in different scenarios in real life and on different types of tests.
Just ask all the math teachers and tutors out there trying to help High School and College students get through higher level math classes. As far as I can tell the bulk of the issue usually centers on a weak foundation in fractions, decimals and percents. These are kids that were exposed repeatedly to those concepts over the years. What they weren't able to do with just exposure was master and internalize these concepts at a deeper level.
Edited by OneStepAtATime, 20 April 2017 - 07:17 AM.