Let's talk about mastery (math)

[Chrysalis Academy]

Can we talk about mastery?  How you decide you have it?  What to do when you had it but it goes away?  :rolleyes:

 

So, I expect to see 90% or above, with the errors silly or calculation-type errors, rather than conceptual errors, before we consider a topic "mastered" and move on.  This is achieved by my dd rather easily.  She always learns what is in front of her without too much trouble - she gets stumped and has to work hard on the more difficult word problems, but she seems to learn to do the basic calculations quickly and easily.

 

But then she forgets stuff . . . yesterday it was how to multiply postive & negative integers (are the products pos or neg?).  Before that it was dividing fractions - which number do you flip, again?

 

On the one had, this seems perfectly normal:  whenever you are learning something new you need multiple exposures/repeated practice, I think.  If it was a piano piece, I wouldn't expect her to practice till she could play it right once, then walk away and come back 4 months later and play it perfectly.  Right? She might need to practice again to get back to the same level of mastery.

 

On the other hand, I worry a little bit that these types of forgetting mean that she doesn't really understand at a deep, conceptual level?  Maybe she's like me, a good pattern matcher/good memory, and she's fooling me by doing well on the test, without really learning the material deeply?

 

How the heck do I tell?  And what do I do when she shows this kind of forgetting?  Usually when I remind her, she says , "Oh, yeah" and then can do the problems just fine.  Should I have her do review problems/pages on the forgotten concept?

 

Do I have something to worry about here, or is this kind of forgetting normal?

[Penguin]

I have a "forgetter." Someone else can speak about normalcy :)

 

Most of the time we do 45 minutes on the main lesson (AOPS Prealgebra) and then 15 minutes of review/drill.

 

But maybe it is genetic...LOL.  When I was in high school, I did not take math my senior year (I had already taken everything that my school offered and dual enrollment was not an option)...and I was a mechanical engineering major.  When I started my freshman year, I had some serious "forgetting" to get over - fast!  Everything did come back quickly, but it was a painful lesson that has stayed with me.

 

 

ETA:  To be more accurate, I should have said that the review/drill really is not every day.  More like two or three times per week, and the "math hour" stretches to about 70 minutes.

 

 

 

[8filltheheart]

I'm not sure what the answer is. Do you do number of problems based on how well you perceive her understanding? Does she progress through all materials regardless of whether she seems to understand the concepts quickly or not?

 

I don't mark out problems bc my kids understand the problems already. I figure if it is easy, then they simply do the easy problems. But, my kids have all used spiral problems through elementary school so review is always built in.

 

But division of fractions and asking which one you flip and the rules for pos/neg sound more like procedural implementation vs solid conceptual understanding bc of the way you phrased the questions. But, it is really hard to say from a brief description as to what she is thinking /not thinking.

[JoJosMom]

FWIW, my 11 year old DD is the exact same.  She did Saxon 7/6 last year in 5th grade at a private Christian school, but we had to do a mop-up operation with MM6 last semester when she came home.  We have recently started pre-algebra.  During the pre-test, despite having completed the MM6 year-end review with a better than 90% average, she had a minor meltdown over one of the questions.  The issue?  She claimed that she had never learned how to do decimal division.  Umm, yeah, ya did. Sigh.  I have been assured by moms with way more experience than I have that it's the age.  I'm going with that.

 

 

(BTW, this is why we're doing pre-algebra.  I know some just go on to algebra, but we need the growing up time here.)

[Chrysalis Academy]

Right - I will need to probe for that (what she is thinking/not thinking) more deeply.  But that was exactly how she phrased the question: "What do I get when a multiply a positive and a negative number again? Is it negative or positive?" which is why I was taken aback . . . . The question implies that she just understands it as something she has memorized, and forgotten, rather than something she understands the *why* of.

 

Same with fraction division: we've gone through the whys, she's done lots of problems and gotten them right.  But she still has trouble remembering "which number you flip" despite the fact that when she asks that, I take her through the whole discussion of what fraction division *is* and explain it conceptually, I don't just give her the answer.

 

I, and her math books, explain these thing conceptually.  But I can't tell if she's learning them conceptually.  She never seems to have trouble with the work I assign her.  And we aren't talking easy-peasy work here, she's doing MM6 for practice on algorithms, and Jousting Armadillos (which is based on Jacobs) for learning new concepts.  She likes both of these, and does very well with them . . . but then she asks these questions that make me think, "Where have you been????" and "How can you do all your homework correctly, and score above 90% on your tests, and still not know the answer to these questions?????"

 

 

[TracyP]

Well if mastery = not forgetting, then my dd has mastered nothing. I'm kidding... sorta. I know some kids don't need it, but my dd needs constant review. If I tried to nail it down, I would say that my dd needs weekly review of a new concept for a year. At that point, she won't lose it.

 

I like the phrase "use it or lose it" much better than "if you truly understand you won't forget". Of course, understanding is important, but practice and repetition are how most people will retain info.

 

We have finished MM, so we are using CLE alongside Challenge Math for the rest of the year. I'm not sure if dd needs the spiral review as much as she used to at this point, but I'm not taking any chances. I may even add CLE to our line up for next year.

[8filltheheart]

Rose, maybe this approach will help. I have a motto (bc it applies to me for anything beyon simple alg 2!! So I know how true it is!). Simply bc you can understand how to a problem was solved when it has been explained is not the same as your being able to do the problem.

 

Rather than you explaining to her, step back and have her explain to you. Can she teach you how to solve a problem? In order to teach you, she is going to have to explain WHY she is doing certain things.

[kiana]

Just because you're explaining it conceptually doesn't mean she's listening to your explanation. She may just be going 'ladidadida' until you get to the 'how to do it' part and then turning on her ears.

[Chrysalis Academy]

She must be growing again - this morning her brain was pretty dysfunctional! :lol:  We're talking about a kid who has grown more than 7 inches in the last year.  That takes a lot of energy.

 

But she did all the integer problems in Zaccaro Real World Algebra without any trouble with the integers - although she did temporarily forget how to do long division!  :rolleyes:  So I'm pretty sure she gets that concept.  I'm going to have her do all the Fraction word problems in Zaccaro challenge math tomorrow, and explain to me how to solve them.

 

I think maybe that is what I"m needing: a method to make sure she really gets it (because just getting the problems right isn't enough, of course  :glare: ).  If she can explain to me how to do it, maybe that will make us both more confident that she is really mastering the underlying concept.

 

I think I have two issues:  the relation of mastery and practice (as in, how much continued practice is needed for previously mastered topics? what does the need for period practice say about the level of mastery?)  And, second:  can a kid get all the problems right (procedural mastery?) but not really understand the underlying concept?  This is a bigger worry for me.  My child typically does not have a very hard time with the numeric operations side of math.  What she does find difficult are word problems.  I'm the same way.  Trying to figure out what this means, exactly, and how to tell when it's time to move on to the next thing.

[Chrysalis Academy]

Just because you're explaining it conceptually doesn't mean she's listening to your explanation. She may just be going 'ladidadida' until you get to the 'how to do it' part and then turning on her ears.

 

Um, yes!!!!!!!!  I think this is what is happening sometimes.  Making her explain to me how and why she's doing what she's doing might help me catch this.  Any other ideas?

[kiana]

I think I have two issues:  the relation of mastery and practice (as in, how much continued practice is needed for previously mastered topics? what does the need for period practice say about the level of mastery?)  And, second:  can a kid get all the problems right (procedural mastery?) but not really understand the underlying concept?  This is a bigger worry for me.  My child typically does not have a very hard time with the numeric operations side of math.  What she does find difficult are word problems.  I'm the same way.  Trying to figure out what this means, exactly, and how to tell when it's time to move on to the next thing.

 

Yes. Yes, they can.

 

I think having her explain back to you what you just explained may help. Be prepared for a lot of uncomfortable silence and 'ummm' the first time you try it.

 

Difficulty with word problems is often caused by not understanding what the operation means. I have a lot of students who can easily compute 1 and 3/4 divided by 2 if I tell them to compute it, but if I say "A cake recipe requires 1 and 3/4 cups of sugar. How much is required for half a recipe?" will get flustered and start doing crazy stuff with the numbers. I don't have any real remedy other than more practice.

[Woodland Mist Academy]

Rather than you explaining to her, step back and have her explain to you. Can she teach you how to solve a problem? In order to teach you, she is going to have to explain WHY she is doing certain things.

 

This approach has been helpful here.

[regentrude]

What you describe (not "remembering" which fraction to flip, or whether the product of a negative and positive number is negative or positive) would, to me, indicate that she has tried to memorize, but not conceptually understood, what is going on.

 

Something that has been understood on a  fundamental level will not be forgotten.

I clearly distinguish this from careless mistakes such as overlooking a minus sign and forgetting to distribute it; a student who is told "there is something wrong here" will catch this mistake -whereas a student who can not "remember" how something was done is missing conceptual understanding,because there is nothing to memorize in math.

 

In order to determine whether she works by pattern matching and memorizing, you might want to have her narrate her solutions and explain the problems to you. I use this technique with my students all the time, and it is the best way to find out what the student simply recalls from having seen before, and what he has understood, i.e. mastered. I would use this technique throughout the math work and only move on if she is able to teach you the concept, not just be reiterating the procedure, but also by answering all your "why" questions.

 

ETA: To answer the question in a later post: yes, t is perfectly possible with some curricula for the student to get all problems correct because she has drilled a procedure, while not understanding the concept. This is where a well designed curriculum plays a big role; with some curricula, it is impossible to get the problems correct when you have not understood the concept, because the problems are constructed to require thinking how the concept has to be applied, as opposed to just applying a drilled algorithm.

[Chrysalis Academy]

That's what I'm worried about . . . along with Kiana's suggestion that even though I explain the concept, she's not really listening.  She's just fastening on to "what do I need to know to answer the questions."

 

She's smart, and fast, and has a good memory, so she is definitely trying to skate by on memory and pattern matching.  Ask me how I know this? This is exactly how I got through math, up through calculus.  

 

Well, I'm not going to let her get away with it.   :toetap05:   But I can see that I will definitely have to change things up:  more word problems and *making her explain why she is doing what she is doing*  

 

I have been guilty of accepting high scores at face value.  So either we have to use a curriculum that is difficult enough that she can't skate and get high scores,  or I will have to force her to articulate her conceptual understanding much more.  Or both, of course.

 

 

[regentrude]

She's smart, and fast, and has a good memory, so she is definitely trying to skate by on memory and pattern matching.  Ask me how I know this? This is exactly how I got through math, up through calculus.  

 

Well, I'm not going to let her get away with it.   :toetap05:   But I can see that I will definitely have to change things up:  more word problems and *making her explain why she is doing what she is doing*  

 

I have been guilty of accepting high scores at face value.  So either we have to use a curriculum that is difficult enough that she can't skate and get high scores,  or I will have to force her to articulate her conceptual understanding much more.  Or both, of course.

 

You may have mentioned elsewhere, but I have forgotten: have you looked into AoPS? That would completely eliminate the problem ;-)

[Chrysalis Academy]

:lol:  Yep, that's what I'm thinking.  I've got AoPS sitting right here.  I was worried it was going to be too challenging, but I'm now thinking she's just snowing me, making me think she knows what she's doing . . . I think maybe it's time to put the proverbial feet to the proverbial fire, and hope that the proverbial s*&% doesn't hit the proverbial fan . . . 

[RootAnn]

Almost all my kids are clearly 'forgetters' - but my dd#2 clearly doesn't understand concepts until she's been through things about six (thousand) times. She's so focused on 'just getting it done' that she does not care what she's doing. Eventually, it dawns on her that there is something else going on and then she pays enough attention to figure out what it is. That's when the light bulb goes on. (How do I know this? Daily observance & much experience. This is the child that did 'calendar' activities with me for two solid years and then asked one day, when she was about eight years old, "What's that little square on the calendar mean?  :confused1: ) I also admit that this is what I'm like. I can "solve" a Rubik's cube by following an algorithm, but cannot explain why a set of moves does a certain thing or what to do when given a certain set-up without following the algorithm. I haven't bothered to figure out the 'why' - and dd#2 does this with just about everything in life. DH is not at all like this and boggles at the fact that I am. :leaving:  

 

DD#3 still forgets what she's previously memorized, but she understands the *concept* behind most of her math so far, so she just re-figures out how to find the answer and moves on.

 

All of them, so far, have benefited from spiral math programs in elementary.

[FriedClams]

I am essebtially taking Algebra 1 alongside my DD (you can read a post on how I do middle school math on my blog - posted today). There are days I forget stuff. It's mastery not perfection. That's why (IMHO) Saxon puts the lesson numbers beside the problems - so students can switch back and review a concept, especially one that hasn't shown up in a while. If test grades are 85% or better I wouldn't worry. The test material is more systemically reviewed before the test. I remind myself - they're learning it, practicing, trying - not doing it perfectly every time.

[TracyP]

As far as I can tell, my dd will completely understand a concept  - I make her verbalize how she came up with the answer, I have her teach me, she solves word problems that demonstrate mastery - but she still has two issues. 1) She will completely forget if she doesn't review. 2) She will always revert to procedural understanding at the end of the day, i.e. remember the procedure but not the concept. I feel like my dd is doing really well in math, but I also don't want her to just skate...

[Chrysalis Academy]

As far as I can tell, my dd will completely understand a concept  - I make her verbalize how she came up with the answer, I have her teach me, she solves word problems that demonstrate mastery - but she still has two issues. 1) She will completely forget if she doesn't review. 2) She will always revert to procedural understanding at the end of the day, i.e. remember the procedure but not the concept. I feel like my dd is doing really well in math, but I also don't want her to just skate...

 

This is how I feel, too.  I was being playfully harsh above. I don't think she's *just* skating, or that she's intentionally snowing me.  I do feel like she demonstrates mastery via her performance on her daily work as well as her test results -both of which include word problems.  And we do talk about the math, and it does seem like she understands the concept.  She can explain why she did something right after she learns it . . . 

 

But.  But she forgets stuff sometimes, and in the case of the two examples I started with, they are thing that make me think she either didn't learn or has forgotten the concept.  

 

Regentrude says that if you really understand the math, there is no memorizing.  So the fact that she forgets (i.e. is searching for a memory, rather than looking at the problem and figuring it out) makes me wonder if she ever understood it in the first place.  But it really seems like she did understand it, she just forgets.  And when reminded, she can then apply it correctly again, even to word problems.

 

So is she understanding, or isn't she? Is she mastering the material or isn't she?  I really don't know.  I thought so, but now I'm wondering.  

 

So that's why I started this thread, really, to talk about mastery: how they show it, how you know it's for real, etc.  I've gleaned a a couple of great suggestions: make sure she can explain why she's doing what she's doing, and make sure the math is hard enough that she is challenged and can't just pattern match.

 

But I still have trouble deciding when she's "mastered" something, if it doesn't stick long-term.  My sense is that all these basic operations should be mastered before starting Algebra.  But mastered when? When you finish studying them, or right before you start studying the Algebra??

 

Can you even talk about having mastered something you've forgotten? That doesn't make any sense to me whatsoever.

[regentrude]

 

But I still have trouble deciding when she's "mastered" something, if it doesn't stick long-term.  My sense is that all these basic operations should be mastered before starting Algebra.  But mastered when? When you finish studying them, or right before you start studying the Algebra??

 

Can you even talk about having mastered something you've forgotten? That doesn't make any sense to me whatsoever.

 

Again, it depends WHAT you forget, and how you handle it.

I completely agree that the basic operations must be mastered before algebra. And I do think they should be mastered when you finished studying them.  I see two different components here: mastering the concept, and mastering a specific algorithm. Ideally, I would want the student to develop the conceptual understanding as well as computational proficiency with an algorithm through practice.

Now, it is still entirely possible the student forgets the standard algorithm for multiplication or long division if she has not used it in two years. That does not mean the concept has not been mastered. But the student who understands what she is doing will be able to compensate for this by coming up with a way to perform the multiplication or division based on her conceptual understanding, maybe re-deriving the standard algorithm, or coming up with an alternative way. I would not be worried if my 9th grader had forgotten long division but could, if he had to, divide number.  In my experience, reviving a forgotten, but previously mastered, procedure takes only a few practice problems (heck, *I* did not remember long division when my kids needed it and had to painstakingly recall how that was written down)

So, yes, one can master a drilled procedure and still forget it; but I do not believe one can master a concept and forget it.

[lewelma]

Rose, I have not read all the responses, but thought I should tell you that I think it perfectly normal. I do think she can conceptually understand it and then still forget it with disuse. I am currently relearning physics and I can absolutely get it, really really get it, and if I take two weeks off, it is gone. Poof. You just have to keep reviewing it.

 

My son about four months ago, asked me how to multiply decimals. This was the older son who just got into the Olympiad camp! He had just forgotten. Sure he could have reasoned it out, but he just wanted me to remind him as he had more pressing math to get through. He also thinks he has pretty much forgotten all of the combinatorics he has learned, because he is not using it enough. This boy only learns conceptually. So it is not that he did not really understand it.

 

So don't worry, just review. A lot.

[TracyP]

 

Regentrude says that if you really understand the math, there is no memorizing.  So the fact that she forgets (i.e. is searching for a memory, rather than looking at the problem and figuring it out) makes me wonder if she ever understood it in the first place.  But it really seems like she did understand it, she just forgets.  And when reminded, she can then apply it correctly again, even to word problems.

 

So is she understanding, or isn't she? Is she mastering the material or isn't she?  I really don't know.  I thought so, but now I'm wondering.  

 

 

I don't know, either (with my own dd). What you mention about not thinking just searching for a memory sounds like my dd. My dd doesn't like to *think*. She expects the answer to miraculously appear in front of her. I have tried to keep her challenged to avoid this, but it hasn't worked. :tongue_smilie: Right now we are taking a break from moving forward to work on problem solving. We work buddy style so I can make her slow down and think. When we talk through it and she uses that thingy between her ears, lol, I feel like she does know what she is doing. But then I wonder if I am doing too much hand holding. Maybe I am walking her through it, but she isn't owning it.

 

But I still have trouble deciding when she's "mastered" something, if it doesn't stick long-term.  My sense is that all these basic operations should be mastered before starting Algebra.  But mastered when? When you finish studying them, or right before you start studying the Algebra??

 

Can you even talk about having mastered something you've forgotten? That doesn't make any sense to me whatsoever.

 

 

I don't know. I agree with you on the one hand, but on the other I think of ALL the things I forget. More than I would like to admit. I'm just not sure. This is a great topic.

[Chrysalis Academy]

Rose, I have not read all the responses, but thought I should tell you that I think it perfectly normal. I do think she can conceptually understand it and then still forget it with disuse. I am currently relearning physics and I can absolutely get it, really really get it, and if I take two weeks off, it is gone. Poof. You just have to keep reviewing it.

 

My son about four months ago, asked me how to multiply fractions. This was the older son who just got into the Olympiad camp! He had just forgotten. Sure he could have reasoned it out, but he just wanted me to remind him as he had more pressing math to get through. He also thinks he has pretty much forgotten all of the combinatorics he has learned, because he is not using it enough. This boy only learns conceptually. So it is not that he did not really understand it.

 

So don't worry, just review. A lot.

 

This post made me crack up, as well as feel better!  Thanks!

 

I actually think that you have hit on something:  call it having more pressing math to get through, or call it being lazy, but I do think that sometimes she asks because she just happened to forget the algorithm at the moment, and she needs to use it in a word problem and doesn't want to stop and think about it because it will break her train of thought on the problem.  I can totally see that!  It's like, long division is a tool, I just need the tool, just gimme the tool, darn it, don't make me stop in the middle of my job and re-invent it!

[Chrysalis Academy]

I don't know, either (with my own dd). What you mention about not thinking just searching for a memory sounds like my dd. My dd doesn't like to *think*. She expects the answer to miraculously appear in front of her. I have tried to keep her challenged to avoid this, but it hasn't worked. :tongue_smilie: Right now we are taking a break from moving forward to work on problem solving. We work buddy style so I can make her slow down and think. When we talk through it and she uses that thingy between her ears, lol, I feel like she does know what she is doing. But then I wonder if I am doing too much hand holding. Maybe I am walking her through it, but she isn't owning it.

 

 

I don't know. I agree with you on the one hand, but on the other I think of ALL the things I forget. More than I would like to admit. I'm just not sure. This is a great topic.

 

Man this sounds exactly like what happens at my house!!!  Right down to the me worrying I'm doing too much hand-holding as I walk her through things.

 

And yes, one of the big bonuses to me working through math just ahead of dd is that I have more sympathy with her mistakes.  I am far from flawless, either in memory or in procedure!!!  I just want her to be *better* than I am:  better at math, better at being persistent, better at really learning stuff and not settling for coasting . . . 

[TracyP]

Man this sounds exactly like what happens at my house!!!  Right down to the me worrying I'm doing too much hand-holding as I walk her through things.

 

And yes, one of the big bonuses to me working through math just ahead of dd is that I have more sympathy with her mistakes.  I am far from flawless, either in memory or in procedure!!!  I just want her to be *better* than I am:  better at math, better at being persistent, better at really learning stuff and not settling for coasting . . . 

 

Hear, hear. :cheers2:

 

[kiana]

As far as "remembering which number to flip", here is what I would do if I had a habit of forgetting.

I do know that 4 / 2 = 2.

So that means that if I'm going to do (4/1) / (2/1) I should definitely get 2.

If I flip the first one, I have (1/4)(2/1) = 2/4 = 1/2. Uh oh!

If I flip the second one, I have (4/1)/(1/2) = 4/2 = 2. Oh, that must be the one I flip.

 

I didn't have an issue remembering this, but there were other places where I continually forgot and had to do a quick check with something where I *knew* the answer to remind myself I was doing the problem right. For example, does f(x-c) translate to the left or the right (for c > 0)? Well, ... y = x^2 has a 0 at x=0. y = (x-1)^2 has a 0 at x=1. Oh, it must move to the right.

[WinsomeCreek]

I didn't read the other responses yet, so this may be redundant. I chalk what you're describing up to 'kids are weird.' Really, they are. In a case like that I'd just jog the memory and a few days later ask to have it explained back. We all have brain farts. I mean, yes it could mean more, but just as likely not.

[Chrysalis Academy]

Yes. Yes, they can.

 

I think having her explain back to you what you just explained may help. Be prepared for a lot of uncomfortable silence and 'ummm' the first time you try it.

 

Difficulty with word problems is often caused by not understanding what the operation means. I have a lot of students who can easily compute 1 and 3/4 divided by 2 if I tell them to compute it, but if I say "A cake recipe requires 1 and 3/4 cups of sugar. How much is required for half a recipe?" will get flustered and start doing crazy stuff with the numbers. I don't have any real remedy other than more practice.

 

And this is why I like Zaccaro so much.  The word problems at the end of the Fraction chapter really check for this: if you don't understand what you are doing you won't be able to solve those problems.  I've got her working through those and she's doing fine so far.  So I think  maybe she does actually understand integers and fraction division, maybe?????

[Chrysalis Academy]

I always tell ds to plug in easier numbers (ones where he already knows the answer) if he's not sure what to do. That usually helps him figure it out for himself. Does he do it when I'm not there? Probably not. My hope is that someday he'll hear me in his head saying that during a test or something. 

 

:lol: Exactly!!  I tell dd "replace the numbers with 2 and 10 and figure out what you'd do to them."  We'll see if she remembers today while she's working on math while I'm not here!

[wapiti]

I can give you a silly example of forgetting.  My dd worked through the exponent chapter (2) in AoPS Prealgebra, completely independently, two and a half years ago.  It just so happens that there haven't been exponents much involved in what she's done in her algebra class due to the order of the Prentice Hall text, but I believe a chapter on them is coming up for her class.  I don't think she remembers anything!  She could figure it out if she thought about it long enough (as I do when I forget), or, if she's being lazy, I could review with her for a few minutes and then she'd probably be able to pass the chapter test.  She's been pressed for time lately, so panicky about understanding quickly, even though I thought we had conquered that when she was at home...

 

Then there was the moment last summer when ds, working in Intro to Algebra, had a brief "duh" when he forgot how to subtract large numbers with regrouping, something he was proficient in during preschool/K.  He literally had tears in his eyes.  About five minutes later, he remembered, but it was a scary five minutes.

 

So, I agree about reviewing!  There are lots of ways to do that.

 

Rose, if you are wanting to be sure she understands, and have evidence in the form of problems solved, I think what you're looking for is a non-plug-and-chug set of exercises, such that she has to think her way through each problem rather than match the pattern.  Obviously that's what AoPS is all about, but IIRC MM has some decent word problems too.  I like to take problems from a variety of resources, especially for review - I have enough math books laying around  :tongue_smilie: and I believe so do you :).  Also, don't forget about Alcumus!!

[Chrysalis Academy]

Thanks, wapiti!  It helps to hear other people's stories!  I agree with you completely: the last thing this child needs is a page of identical problems to "review" - she would fly through it in a minute, and it would tell me nothing about mastery.  I am taking the approach you suggest, and yes!!  I do have lots of math books on the shelf!

 

MM's word problems are great.  Our usual pattern with MM is that she flies through the numerical problems perfectly or with only silly fogetting type mistakes, and then has to sloooooowwww down to do the word problems.  This is the main reason we've kept going with MM this year:  it's a good mix of review/practice of the algorithms plus some great word problems.  Zaccaro has great word problems too, and the end of chapter word problems really mix it all up, so they have to choose which operation is appropriate for each problem. 

 

My main issue with AoPS has been being intimidated by it myself.  Also, i just do not love the wordiness and convolution of the book.  But, I'm about to pass the PreA section of Alcumus!  :party:  and I agree that it's a great way to check for mastery.  There is no way you can fake it or pattern match on those problems! I am planning to introduce her to Alcumus soon.

[JoJosMom]

Chrysalis Academy:

 

• THANK YOU. 
• This has been an enormously helpful thread. 
• AOPS Pre-Algebra is now on my Amazon wish list.
• I think we have the same child.  (I think some of the issues being discussed are age-related.  Mine is growing like a weed, as well.  With that much energy being expended on growth, it only makes sense that the brain is getting short-changed a bit.  Once again I am reminded, I need to practice patience.)

 

 

[Chrysalis Academy]

You are welcome!  It helped me a lot, too.  She did the Fractions word problems in Zaccaro Challenge Math without too much trouble, so I think she gets the concepts.  I think she just needs practice to make remembering them automatic.  That's what review is for!  

 

It's a good reminder to me to pause and check for retention, and review as needed.  I can get so gung-ho about charging ahead sometimes!  So I have to put together the "go slow to go fast/review" with the "more challenging problems" thing in some way that makes sense. 

[Chrysalis Academy]

Can I just say that I think Alcumus is brilliant?  It's really giving me a whole new perspective on mastery.  This weekend I "passed" (green bars) the PreAlgebra level.  I thought I'd be moving on to Algebra . . . but no.  It's sending me through the whole thing again, to "master" (blue bars) the content.  At first I was kind of peeved - ready to move on! But I realized that it was exactly what I needed: revisiting the topics, most of them I am flying through, and I have so much more insight into how to approach them than I did the first time.  A couple of topics, I just passed by the skin of my teeth the first time, and I'm having to work a little harder.

 

It's given me two insights, though:  first, it really highlighted my besetting sin: always wanting to move on to the next thing, ready or not.  It made me really stop and think, am I sometimes doing this to my kids? Rushing them on to the next thing when really, they could use another pass through to cement some of the basics?  I'm planning to be much more mindful of this in future.

 

The second thing is insight into mastery.  Mastering topics on Alcumus can't really be faked, the way it can on Khan Academy and some of the other online "adaptive" programs I've seen, where you can basically pattern match your way to correct solutions.  This is a really good thing.  But I feel like that when you master a level on Alcumus, what you are *really* mastering is problem solving.  I've had to use different techniques to get through Alcumus - watching the videos, reading the book, walking away and letting a problem "cook" then coming back to it.  They have all been valuable and useful.  And I realize, again, that *this* is what I want for my kids - less a perfect mastery of math operations, and more a mastery of the ability to solve problems, to look for help from multiple sources, to walk away and come back, to manage frustration, and to persist until something is really mastered.  

 

ETA:  As 8 pointed out, I phrased this badly:  I don't mean that I don't want them to perfectly master math operations!  Just that I want them to *also* master problem-solving.  And I think you go about mastering these things differently, maybe?  Meaning that a kid can do pages and pages of problems, perfectly, but still be completely helpless and flustered when faced with word problems on a topic they "know" - like Kiana was describing above.  Problem-solving does need to be worked on explicitly.  Thanks, 8, for helping me clarify what I meant!

[Professormom]

Can I just say that I think Alcumus is brilliant? It's really giving me a whole new perspective on mastery. This weekend I "passed" (green bars) the PreAlgebra level. I thought I'd be moving on to Algebra . . . but no. It's sending me through the whole thing again, to "master" (blue bars) the content. At first I was kind of peeved - ready to move on! But I realized that it was exactly what I needed: revisiting the topics, most of them I am flying through, and I have so much more insight into how to approach them than I did the first time. A couple of topics, I just passed by the skin of my teeth the first time, and I'm having to work a little harder.

 

It's given me two insights, though: first, it really highlighted my besetting sin: always wanting to move on to the next thing, ready or not. It made me really stop and think, am I sometimes doing this to my kids? Rushing them on to the next thing when really, they could use another pass through to cement some of the basics? I'm planning to be much more mindful of this in future.

 

The second thing is insight into mastery. Mastering topics on Alcumus can't really be faked, the way it can on Khan Academy and some of the other online "adaptive" programs I've seen, where you can basically pattern match your way to correct solutions. This is a really good thing. But I feel like that when you master a level on Alcumus, what you are *really* mastering is problem solving. I've had to use different techniques to get through Alcumus - watching the videos, reading the book, walking away and letting a problem "cook" then coming back to it. They have all been valuable and useful. And I realize, again, that *this* is what I want for my kids - less a perfect mastery of math operations, and more a mastery of the ability to solve problems, to look for help from multiple sources, to walk away and come back, to manage frustration, and to persist until something is really mastered.

Thanks for this Rose! Ds is starting the aops pre-alg in about a month and I have been waffling with running through it myself first. You helped me make up my mind:-)

[8filltheheart]

And I realize, again, that *this* is what I want for my kids - less a perfect mastery of math operations, and more a mastery of the ability to solve problems, to look for help from multiple sources, to walk away and come back, to manage frustration, and to persist until something is really mastered.

I agree with your post except this part. Kids do need to master math operations in order to master problem solving. Mastery of problem solving without mastery of basic operations is an oxymoron. If a student doesn't understand order of operations, fractions, decimals, etc they will be hampered in successful problem-solving.

 

But, yeah, I don't understand the rush forward in any subject, not just math.

[Chrysalis Academy]

I agree with your post except this part. Kids do need to master math operations in order to master problem solving. Mastery of problem solving without mastery of basic operations is an oxymoron. If a student doesn't understand order of operations, fractions, decimals, etc they will be hampered in successful problem-solving.

 

But, yeah, I don't understand the rush forward in any subject, not just math.

 

You are right, of course! I posted this morning before I had my coffee.   :rolleyes:  I didn't mean to imply that I don't want them to master operations, too . . . just that it isn't enough.  I want them to master *both* operations *and* problem-solving.  It's just . . . it's a lot easier to figure out how to teach the first than the second, sometimes?

[Acadie]

I wonder, after reading this thread, if I've been looking at the same issue too superficially, but I'd been thinking dd11 needs a short review sheet of math through 6th grade to keep it all in mind at one time.  

 

I do think she understands the "why" as we go through each section of MM, and we slow down and supplement as needed, but a few months later she often forgets part of the procedure.  I was thinking a brief review sheet--let's say around 10 pages for all math through 6th grade--might be something I could point to when she forgets something, and visually recalling a familiar review sheet might help it stick.

 

Do you think this would be short-circuiting conceptual understanding?  

 

Amy 

 

 

[regentrude]

I do think she understands the "why" as we go through each section of MM, and we slow down and supplement as needed, but a few months later she often forgets part of the procedure.  I was thinking a brief review sheet--let's say around 10 pages for all math through 6th grade--might be something I could point to when she forgets something, and visually recalling a familiar review sheet might help it stick.

 

Do you think this would be short-circuiting conceptual understanding?  

 

Yes, I think it would. I would be concerned that her "forgetting" a procedure in such a short amount of time (you say a few months; we are not talking about several years) means that she never really understood what she is doing conceptually and also that she never mastered the algorithm through enough practice to move the skill into long term memory.

If she understood the concept, she should be able to reconstruct the algorithm by examining what she knows about the concept.

If she really practiced the algorithm to mastery, it should not be forgotten after a few  months.

 

The brief review sheets can only have overviews over algorithms; they can't remedy the lack of conceptual understanding. They are a crutch. I would go more slowly in make sure that both the conceptual understanding is cemented and the procedural skill practiced until it becomes automatic. Only then I would move to the next topic.

 

ETA: A well designed math curriculum should have continuous review built in, without making it "extra". Skills learned in the earlier years should be required for solving the problems in the later years, since math builds on itself. I have a hard time envisioning a math program that is so compartmentalized that students won't continuously use the earlier skills.

[Pen]

We are 'use it or lose it' types, I guess. I forgot a whole (almost entirely) language due to lack of use. And math in a certain sense is a language. Sure, a line is 180 deg. but now, are two angles making up that 180 deg called "supplementary?"...well, probably, but I always have had to review that sort of thing if I am about to take an exam that will ask for it. And it has more to do with language than math. I think even memory of the "math facts" have a lot to do with language. 

 

I also have been struggling to figure out how much review is needed. I think the answer may be that in general, the more times ___  has been reviewed, the longer between reviews are then needed to keep ___ fresh. But also certain subjects, or sub-subjects, for certain people may need more or less review.

 

Even when I know for sure I have understood a concept in the past, I often find that without use I forget it. And I forget procedures even more easily. And even more easily than that the words for things. I can do long division fine and know the answer is a "quotient," but recalling which is the dividend and which the divisor with the similar sounding names nearly always requires me to look it up yet again.

 

My own forgetting may turn out to be helpful to my son. I have a feeling that when my ds gets to where I no longer remember my math, that it may actually help him to be learning it without me being able to teach him, especially if I let him help with my own struggles to remember. I do not know this yet, but it is a theory.  He was on the phone at one point with an older relative who had forgotten which fraction gets turned into its reciprocal for fraction division, and so he explained it. That real explanation to someone who was trying to figure it out, not trying to test him, was very helpful.

[kiana]

We are 'use it or lose it' types, I guess. I forgot a whole (almost entirely) language due to lack of use. And math in a certain sense is a language. Sure, a line is 180 deg. but now, are two angles making up that 180 deg called "supplementary?"...well, probably, but I always have had to review that sort of thing if I am about to take an exam that will ask for it. And it has more to do with language than math. I think even memory of the "math facts" have a lot to do with language.

 

Frankly, I wouldn't consider forgetting the name 'supplementary' to be lack of understanding. I *would* consider forgetting that a line was two right angles to be lack of understanding. I wouldn't even consider lack of memory of math facts to be lack of understanding.

 

Of course, I would consider a student (of a certain age) who can understand that 6x8 is 6 groups of 8 or 8 groups of 6 but can't figure out 6x8 without drawing 6 groups of 8 and counting the groups to be handicapped by an inadequate math education. But this problem is not lack of understanding but lack of computational fluency. And frankly, for any sort of genuine math education, we need both. We need students to understand multiplication in multiple ways, as 'x groups of y', as scaling ('6 8-foot boards is how long?'), as area ('a fence 6 feet by 8 feet will fence in how many square feet?' but we also need them to be able to come up with 48 in a non-time-consuming manner.

[regentrude]

We are 'use it or lose it' types, I guess. I forgot a whole (almost entirely) language due to lack of use. And math in a certain sense is a language. Sure, a line is 180 deg. but now, are two angles making up that 180 deg called "supplementary?"...well, probably, but I always have had to review that sort of thing if I am about to take an exam that will ask for it. And it has more to do with language than math. I think even memory of the "math facts" have a lot to do with language.

 

 

s. I can do long division fine and know the answer is a "quotient," but recalling which is the dividend and which the divisor with the similar sounding names nearly always requires me to look it up yet again.

 

But neither of this is MATH. And knowing the names has absolutely nothing to do with mathematical understanding.

I would expect a student to KNOW that two angles that make up a straight line add to 180 degrees, or how to perform a division.

Knowing the terms is nice to communicate about math, but it is not math. Reciting definitions perfectly does not mean the student has understanding. And really: terminology that is used is retained, and terminology that is not used is probably pointless in the first place.

(I have no idea how all those angles are called, but I can use their properties and prove tricky geometry stuff. Way more important)

 

It has been my experience with school curricula that a disproportionate amount of emphasis is put on vocabulary. If anything, it's worse in science. Many "science" tests or worksheets are almost exclusively about terminology, but not about scientific content. What many educators do not understand is that knowing the names of things does not teach the slightest bit of actual concept. You could call the dividend and the divisor anything you want, it would not change the concept of division. Just as memorizing Newton's laws verbatim does not mean the student has any understanding about physics.

[Chrysalis Academy]

This is a great point.  I've been realizing this as I work through Alcumus problems:  I haven't thought about geometry much since the 9th grade (other than figuring out the square feet of farm fields and stuff like that) but when I got to that section of alcumus, by gum I remembered all the properties of parallel lines and triangles!

 

OTOH, I must have been sick the day (week? year?) that they covered LCM & GCF in what - 6th grade?  7th grade?  I had a vague memory of what they were, but I swear I had never, ever, ever seen prime factorization used to calculate them until I started reteaching myself math a couple of years ago.  I had to (re)learn all that stuff from scratch.  It made me wonder - is it possible that they never taught us that? Or that it happened while I was on vacation or something? And how is it that I managed to do math through calculus without really understanding factors?  

 

I kind of flipped when Shannon started using the "Birthday Cake Method" she read in Math Doesn't Suck to do GCF & LCM problems.  It looked all snazzy, but she clearly didn't have a clue why she was doing what she was doing.  And hence remembered nothing about LCM from one day to the next - once the pattern wasn't in front of her, it was gone.

 

Well.  :toetap05:  needless to say that book is going back to the library, and we covered it again, from the beginning, with FEEEELING.  I think she gets it now.  We'll see tomorrow!  :smilielol5:

[Pen]

But neither of this is MATH. And knowing the names has absolutely nothing to do with mathematical understanding.

I would expect a student to KNOW that two angles that make up a straight line add to 180 degrees, or how to perform a division.

Knowing the terms is nice to communicate about math, but it is not math. Reciting definitions perfectly does not mean the student has understanding. And really: terminology that is used is retained, and terminology that is not used is probably pointless in the first place.

(I have no idea how all those angles are called, but I can use their properties and prove tricky geometry stuff. Way more important)

 

It has been my experience with school curricula that a disproportionate amount of emphasis is put on vocabulary. If anything, it's worse in science. Many "science" tests or worksheets are almost exclusively about terminology, but not about scientific content. What many educators do not understand is that knowing the names of things does not teach the slightest bit of actual concept. You could call the dividend and the divisor anything you want, it would not change the concept of division. Just as memorizing Newton's laws verbatim does not mean the student has any understanding about physics.

 

I agree with your points above. But I also think there is a language aspect to math that affects memory and also understanding.

 

Philosophically speaking there may be a MATH that exists independent of language and independent even of human beings. Sort of a Platonic absolute paradigm of math.

 

There may be mathematical understanding that can take place in someone who has no language at all, neither in the sense of English, German or Chinese, nor in the sense of numerical symbols. For example, there is at least apparently a sense of quantity that exists in babies and also animals. A sense of more than and less than, and probably a sense of shape, size, distance and speed which may be understood without language or math symbols.

 

And yet at another level, Math and Language are related. At one level there is the--and I agree least important aspect that involves words like dividend or divisor--and then moving to terms like numerals, and operations, it starts to get more important as it becomes a numeric language.   For example, take :  10 X 11 = 110 and II x  III = VI  which could both represent, in our usual English explanation, three sets of a pair of items, depending on the numerical system being used. And what procedures we can do, and how conveniently we can do them, can depend on the numerical language system. So it might be different for someone dealing with base 2, or Roman numerals, or Mayan knots, though the concept of multiplication might be at some conceptual level the same. But if you looked at the first example and were picturing one hundred ten sheep or whatever, in your mind, I, as I wrote it was only picturing six of them. I took a multiplication problem that I can easily picture in my mind as objects in groups and then translated that back into some symbols.