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How important is review in HS math (pre-calculus)


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My son is (finally) starting pre-calculus. I have Lial's text and Thinkwell pre-calc videos on DVD. 

In looking through the text, I don't see any cumulative review. In all his math to this point, he's used programs that have review built in. I like this for retention.

Is it important? I hate to spend more money, but I want him to retain. 

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20 minutes ago, sbgrace said:

My son is (finally) starting pre-calculus. I have Lial's text and Thinkwell pre-calc videos on DVD. 

In looking through the text, I don't see any cumulative review. In all his math to this point, he's used programs that have review built in. I like this for retention.

Is it important? I hate to spend more money, but I want him to retain. 

Depends on how easily he retains things, I would think. For some kids, it would matter, and for some it wouldn't. 

I don't know the specifics of this program, though, so I can't comment on that. 

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Depending on what you did for Algebra 2, PreCalc might be a review-and-build on it kind of situation for him. Lial's has a lot of problems in it, so you could pull some forward at the end of each chapter/ semester.

There are various worksheet sites and teachers who have worksheets posted if you do some internet searching.

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On 12/2/2020 at 7:09 PM, sbgrace said:

Is it important? I hate to spend more money, but I want him to retain.

You need to ask yourself what specifically you want him to retain and why.

The vast majority of people, even those who did well in precalculus, are not going to remember enough of what they learned to be able to use it even a year later.  This loss is less likely to occur if they have deeply understood the concepts from the ground up so that they can reconstruct whatever it is.  It is unlikely that a typical student using the Lial text will develop this sort of deep understanding.  This is nothing against Lial, it's just that even though the concepts are there, the focus is on procedures--which, frankly, is what continual review will reinforce, sometimes to the detriment of conceptual understanding.

In other words, cumulative review will probably help him retain for the final exam at the end of the year, but that retention won't be permanent if he doesn't continue to use the math.  If you want him to do well on the SAT/ACT, a good prep book or class combined with practice tests in the weeks prior to the exam will suffice.  If he will be taking math that requires his precalculus knowledge in college, a targeted review in the weeks before he needs it would be best.  One good thing is that if the material was learned well the first time, subsequent passes will be much easier and will go far more quickly. 

So this gets back to the original question:  What do you want him to retain and why?  

Edited by EKS
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3 minutes ago, EKS said:

 

So this gets back to the original question:  What do you want him to retain and why?  

I want him to enter college math courses with a solid foundation. I don't really even care if he gets beyond Pre-Calculus. I always felt mastery of Algebra through Pre-calculus was my goal for him. I want him ready for whatever math comes next, after high school.  

Secondarily, he has potential to do really well on the SAT's, but I agree that a targeted review is going to go much further in that area. 

He has always frustrated me a little in that he seems to grasp math concepts well, relatively easily, and then also seems to forget everything equally easily. Maybe I'm expecting too much.

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12 minutes ago, sbgrace said:

I want him ready for whatever math comes next, after high school.

Since most students will have trouble remembering everything about precalculus, calculus courses will generally have some review built in.

14 minutes ago, sbgrace said:

He has always frustrated me a little in that he seems to grasp math concepts well, relatively easily, and then also seems to forget everything equally easily. Maybe I'm expecting too much.

Both of my kids were like this.  I have this theory that bright kids can appear to go from zero to mastery in the blink of an eye, but in doing so, they don't get the practice they need to retain the material long term.  Unfortunately, I was never able to figure out how to time the practice so that it would have optimal impact.  I suspect that the best way to go about it is to make sure they understand the basics and then give them problems that require them to use those basics in creative ways.  But it's difficult to find such problems that are at the right level--if they're too easy, they won't help cement things, and if they're too difficult, they produce frustration.  

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23 minutes ago, EKS said:

I suspect that the best way to go about it is to make sure they understand the basics and then give them problems that require them to use those basics in creative ways.  But it's difficult to find such problems that are at the right level--if they're too easy, they won't help cement things, and if they're too difficult, they produce frustration.  

This is what I do. But I write her questions myself.

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27 minutes ago, Not_a_Number said:

This is what I do. But I write her questions myself.

This is the perfect solution--to be able to tailor the problems to the student in front of you.  Unfortunately, most homeschooling parents don't have the expertise in math to do this well, especially at the high school level.  I am at the point where I can recognize such problems, but I need a repository of them to draw from, and I need to do them myself first.  But at the precalculus level, I can't conjure them out of thin air.  Maybe someday though.

ETA: I think this issue, of needing to do problems requiring creative use of a concept to really remember it is probably the reason that retention of a previous course's material is aided by taking the next course.  

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1 hour ago, EKS said:

This is the perfect solution--to be able to tailor the problems to the student in front of you.  Unfortunately, most homeschooling parents don't have the expertise in math to do this well, especially at the high school level.  I am at the point where I can recognize such problems, but I need a repository of them to draw from, and I need to do them myself first.  But at the precalculus level, I can't conjure them out of thin air.  Maybe someday though.

ETA: I think this issue, of needing to do problems requiring creative use of a concept to really remember it is probably the reason that retention of a previous course's material is aided by taking the next course.  

So, this is my two cents from having monkeyed around with both my kiddo's math and the AoPS precalculus class... what you need is not exactly the creative use of a concept so much as using the part of the brain that actually uses the concept. And the issue with most algorithmic problems is that you quickly train yourself to use the algorithm and your brain no longer retrieves the concept AT ALL. 

For example, lots of classes spend tons and tons of tons of time graphing lines. Especially for mathy kids, after the first 10 lines you graph, you figure out the algorithm, and you NEVER have to again think about the fact that the graph of an equation is all points satisfying that equation for the rest of the class. You have NO IDEA how many times I saw kids in calculus classes who had not retained that essential idea after all their years of schooling. That's because they were overtrained on specific kinds of graphs you can do by rote, and their brain didn't have to even engage with that concept more than a few times, despite all this practice. 

My tack is generally to think of problems that require retrieval of the right kinds of information. For trig, for instance, that has generally meant designing very visual problems using the unit circle. So, like, here's the terminal point for theta, find me the terminal points of -theta, 180 - theta, 2theta and -3theta in the diagram. There's literally NO WAY to do this problem that doesn't require retrieval of the key idea of trigonometry, which is that we figure out things about angles by using their terminal points. My experience has been that this approach is much more successful in getting retention than lots of other approaches, because then when the kids have to do the calculations, their brain automatically retrieves the correct ideas along the way, which means that now instead of pure rote calculations, they actually practice the concept along the way. (I've now done this with a couple of AoPS classes, so can vouch for it.) 

For graphing, I find that the best way to make kids retrieve the IDEA is to actually have them graph a variety of functions from scratch without any shortcuts at all. I am doing this with DD8 now, and I did it to remediate my little sister's mathematics a few years ago, after she had been overtrained on lines without ever really understanding what a graph was. Again, it's not actually creative, but it requires using the correct part of the brain. 

I actually sometimes find that overly creative problems tax one's brain so much that it's hard to put together what you learn from them. So while I like problem solving at later stages of learning (especially for mathier kids), I don't love it as a way to internalize the main ideas. Of course, it does depend on what motivates the kids. I was sufficiently motivated by problem solving that I probably got MORE practice that way, despite the fog surrounding me, lol. But I don't think that's the case for most kids, and I did MANY hours of math a day. 

Edited by Not_a_Number
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24 minutes ago, Not_a_Number said:

And the issue with most algorithmic problems is that you quickly train yourself to use the algorithm and your brain no longer retrieves the concept AT ALL. 

This exactly!

24 minutes ago, Not_a_Number said:

what you need is not exactly the creative use of a concept so much as using the part of the brain that actually uses the concept.

I was using the word "creative" to include this as well, though you are right to make the distinction.  What I mean by creative is really anything beyond simply cranking out the procedures.  I realize that true creativity in mathematics is a different animal.

 

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