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Dd12 & fractions - trying to pinpoint the glitch - UPDATE 09/25


forty-two
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We're on our 4th pass through fractions (SM was our main program, and alongside it we did LoF Fractions, MUS Epsilon, and now we are working through Key to Fractions alongside MM for pre-algebra), and while dd12 can objectively do more on each pass, she still fundamentally feels like she doesn't understand fractions and nothing about that has really changed.  She's a very whole-to-parts thinker, and has to understand the concept before she sees any point in learning the procedure.  Usually this is not a problem, as most of the time she picks up the concept effortlessly.  But there's something fundamental about the fraction concept that she's simply not getting, some misunderstanding that makes all the various fraction operations feel like they are *breaking* the rules she already learnt, instead of extending them.  She's offended by the apparent rule-breaking, thinks that fractions are inherently nonsensical - that there's nothing *to* understand, that my persisting in working on fractions is analogous to me trying to make her accept the legitimacy of 2+2=5 (which she vows never to do) - and mostly she just wants me to quit torturing her with the abominations that are fractions.  She will grudgingly accept that I actually believe that fractions make sense, that it truly is not my intention to make her accept 2+2=5, but she's pretty sure that I'm factually wrong about it being possible for her to ever understand fractions.

With SM, it was me dragging her through the fraction sections, which is why I sought out supplements after the first disastrous foray.  With LoF I had her do it independently, which was in retrospect a horrible plan.  She did great for the first half, and then fell off the train spectacularly, with much weeping and rending of garments.  Next we tried MUS Epsilon.  This helped a lot in some ways - I think she finally got an intuitive sense of what's going on in fraction math, and she was able to do the Epsilon exercises with ease - but it didn't help with her fundamental misunderstanding.  Now she's working through Key to Fractions.  She both hates it and says it is too easy :sigh.  According to her, she hates it *because* it is too easy and thus is pointless.  Usually that means she finds the concept easy but there's something about the execution that is hard.  IDK if it's related to her fundamental misunderstanding - that nothing about this work is helping her connect what she is doing with the fundamental meaning of fractions - or if it's related to something else.  She really presents as 2E in a lot of ways, and so much of math (and school in general) has been working on turning her intuitive understanding into logical/sequential oral/written expression.  (Also a lot of school has been remediating auditory processing weaknesses and working through a reluctance to physically write.)

Anyway, I'm hoping to brainstorm what the fundamental misunderstanding might be, and how to work through it.  I suspect it has to do with not understanding fractions as being division problems, but I'm not sure.  The one concrete thing I've noticed is that she persistently doesn't understand *why* you can go from "1/4 of 40" to "(1/4) *40".  She's improved on every fraction thing that *doesn't* use that premise, but fails on everything that *does* depend on it.  I can get it in her head long enough to get through a chapter, and have done so repeatedly, but it doesn't stick - she fundamentally doesn't understand why you can do that, even as she seems to understand every step of the teaching in MUS on the topic and can do all the MUS problems on it.  I think she understands full well all the reasons people keep saying you can, but she's missing the fundamental premise that *allows* them to do it, that makes those reasons *true* instead of being a convenient fiction.

Edited by forty-two
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She might benefit from seeing it done the Gattegno way.  If you flip to page 38 (it'll say 44-45 down below the book) it starts off with the premise that 1 white block is half of the red (1 is half of 2, or one part of 2).  It continues to use the white blocks to illustrate 1/2 of 4, 8, etc. and then use them to illustrate 1/3rd, 1/4th...and so on and teaching the multiplication of that small block in relation to the big one.  I went through MUS with my oldest and thought it was fabulous for teaching multiplication of fractions, but going through Gattegno for fun with my youngest showed the relationship a different way than using the overlays.  (He later went on to equate that relationship with how MUS teaches decimal multiplication)

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With fractions, we try to draw or use manipulatives for everything. I use graph paper and we draw and slice. Actually, we draw in most math topics.

Not understanding the leap from 1/4 of 40 to 1/4 * 40 can be shown in drawing. I'd let her draw a square, cut it into 40 equal pieces - drawing attention to the fact that she's now made 40 equal groups which is multiplication/division!

Then I'd ask her what we are actually doing when we multiply. Forget fractions, use simple numbers first. What actually happens when we multiply 4*2? We're adding 4 equal groups of two. I'd do this in pictures/manipulatives.

Then, what happens if you have 4* 1/2? We're adding 4 equal groups of a half. I'd draw a circle, cut it in half and ask 'how many equal groups of a half do we have? How many do we need?' 4, so draw one more circle and cut it in half. How many equal groups of a half do we have now? 4! How many whole circles is that? 2! Then I'd follow the same process for 40* 1/2? 40 equal groups of a half. 40* 1/4? 40 equal groups of a quarter. 

For the last one, I'd go back to her square cut into 40 and ask her how she would draw 40 groups of 1/4. I might suggest drawing a cross in each square like a window, then colouring in 40 of the small window squares. If she doesn't see herself, point out that 'wow she's coloured in 10 of the original 40 squares, or 1/4 of the original big square!'

 

Edited by LMD
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Maybe just semantics? 

Let me have two *of* those apples. Two *of* one Apple is two. It's the "quantity", in this case two, *of* the "item", in this case apples. Multiplication is how many of equal groups. 2x1 is two, "quantity", of 1, "item." It is quantity times item. Let's try asking for a quarter of an Apple. May I have 1/4 of an apple? 1/4 is the quantity of an item I want. Quantity times item, same as any multiplication. That's 1/4 x 1. What if the shop has 40 apples and I want one quarter of them? That's 1/4 "quantity" of 40 "item." 1/4 x 40.

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Can she articulate why she doesn't understand why finding a fraction of something makes a multiplication problem? Does she understand that all whole numbers are understood to be "over 1" to convert them to fraction form? For example, you can solve the problem thus:

What is 1/4 of 40?

1/4 * 40/1    We show the understood one to make both terms into fractions

1/4 * 40/1 = 40/4   Multiply across

40/4 = 10   Now we simplify by completing the simple division problem

Some kids need to see that extra step of an understood 1 for a while before they can synthesize automatically as most adults do that you divide the whole number by the denominator of the fraction. If she needs to know why we understand whole numbers to be over one, ask her what 40 divided by 1 is. You still get 40. You haven't changed the number at all. You just made it into a fraction so it is easier to work with when dealing with fractions. It's not that much different than representing 40 as 40.00 when adding and subtracting decimals.

Does she understand that she could also solve the problem this way?

What is 1/4 of 40?

1/4 = .25   When we divide 1 in 4 parts, we get .25

.25 x 40    Now we can simply multiply both terms

.25 x 40 = 10 You still get the same answer as before

 

Does she understand that when you multiply a decimal and a whole number, the whole number gets smaller? Could she understand that 1/4 of a dollar is 25 cents or one quarter? Could she figure out a problem that is worded like this:

4 friends went to an arcade and have 40 tokens to share. If they split them equally, how many tokens does each friend have to spend?

I'm pretty sure she could quite easily see how each friend would get 10 tokens but have her show you the math to prove her answer and then show her it is the exact same problem as what is 1/4 of 40. Help her see that by splitting anything equally, that is the definition of a fraction, something that is split into equal parts. If she needs to, she can make any "what is <fraction> of <number>?" problem into a word problem if it makes it easier to visualize the problem.

As long as the numerator of the fraction is 1:

<insert denominator here> friends shared <insert number here> things equally. How many does each friend get?

What is 1/5 of 110?

5 friends shared 110 things equally. How many does each friend get?

 

If the numerator is not 1:

<insert denominator here> friends shared <insert number here> things equally. <insert numerator here> of the friends decide to donate their things. How many things were donated? 

What is 2/3 of 60?

3 friends shared 60 things equally. 2 of the friends decide to donate their things. How many things were donated?

Yes, it takes longer to do the math that way but if it helps her to understand the big picture to do it this way for a while, she should eventually see that it is easier to skip all these steps and just multiply to get the exact same answer.

Edited by sweet2ndchance
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She needs manipulatives to see it, ideally a few different styles like fraction tiles (my top choice), circles, and towers. We also had a flip board that showed the equivalent fractions, decimals, and percents. 

Lakeshore Learning has a bunch and I personally found it worth buying them:

https://www.lakeshorelearning.com/search/products/page-1/sort-best/num-24?view=grid&amp;Ntt=fractions 

You can also print them on thick paper and laminate (she will handle them a lot). "Printable fraction" should bring up circles and strips easily. You can also find Cuisenare rods and other things. 

I'd buy at least the fraction tower bc I like one 3-d option for comparison. It's also handy to fiddle with in the car. 

 

 

 

 

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I find with my dd I need to do a lot of physical work.  Cut up oranges or slices of cheese or length of string etc 

For what it’s worth although I could work operations on fractions as a kid I didn’t really understand them fully I just accepted them as they were for a long time.  Working through them with my kids has helped them make more sense to me.  

For dd we cut up whole oranges into quarters and then to multiply 3x 1/4 we got 3 of the quarters and put them together to make 3/4 of and orange.  I fully expect to have to repeat this next year though as she will probably forget.  And for my oldest even though we went through and proved how division works several times in the end it just became a case of “multiply by the reciprocal”. We’ve shown why it works, we’ve proven it multiple times but I don’t want to reprove it on every problem.  At this point just do the problems and we’ll readdress it next time it comes up.

probably not an ideal approach but I couldn’t deal with arguing about fractions forever.

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I'm going to agree with everyone else, and also state that for virtually all manipulating of fractions "beyond slices of cake", my mind works MUCH better when considering the fraction bar to be a shortcut to a division symbol plus parentheses.  

That is, 1/4 x 40 is (1x40) div by 4.  However, you don't really get to this interpretation of fractions until Pre-A or A, when you can line things up and start simplifying by crossing out above and below the fraction bar.  You could try teaching it earlier, to see if it helps.

 

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Big picture wise, I like to use money!

Compare decimals and fractions (and eventually percent, but start with decimals and fractions.)

Start with just 1/4, 25%, 25 cents, .25 easy things based on $1.00.

Then, do $2, then $3.

For example, with your 1/4 of $40...

1/4 of $1.00, 1/4 = .25  (1X100/4), Then do 1/4 of $20.  Do as a whole, 1/4 X200, then .25 X 200, then do in parts, 1/4 of $1 + 1/4 of $.

Then, 1/4 of $3, whole of 1/4 X 3.00 then do 1/4 of $1 + 1/4 of $1 + 1/4 of $1.

Then, do for $4.

Smaller numbers and money makes it more tangible.  It also helps to see that even with fractions, multiplication is just repeated addition.

There is also a good Singapore book with all this and bar graphs in one book.  We just moved across the neighborhood, but I can look and see if I can find the book over the next few days/week, it is worth purchasing, they go through a bunch of cases and draw them out.

Either that book or another book also describes two ways of explaining division with fractions, that is also good when you get to that point, if she struggles with that, too.  (Partive and Quotive division)

It might also help to understand that 1/2 is also 1divided by 2, 1/4 is also 1 divided by 4, do a bunch of easy fractions that work out to even money numbers, like 1/2, .50, 1/2 of $1 is 50 cents, do the actual dividing. 2/5 is 2 divided by 5, .4, then do 4/10, work them both out with dividing and with money.

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On November 9, 2018 at 2:08 PM, forty-two said:

She really presents as 2E in a lot of ways, and so much of math (and school in general) has been working on turning her intuitive understanding into logical/sequential oral/written expression.  (Also a lot of school has been remediating auditory processing weaknesses and working through a reluctance to physically write.)

Have you thought about doing evals for this? It sounds like a psych eval and maybe some language testing with an SLP would be in order. You're calling this a math issue, but in some ways it sounds more like a language issue. There's a lot of language involved in math, and in fact dyscalculia affects the part of the brain where language and math intersect, NOT the conceptual understanding. APD and dyslexia are both considered language disorders.

On November 9, 2018 at 2:08 PM, forty-two said:

The one concrete thing I've noticed is that she persistently doesn't understand *why* you can go from "1/4 of 40" to "(1/4) *40".  She's improved on every fraction thing that *doesn't* use that premise, but fails on everything that *does* depend on it. 

That isn't really a fraction issue but how language and math intersect and her flexibility. You could work on that same concept without fractions. I work on flexibility with my ds, because he has autism and is inherently rigid. You might need to back up and make these options for writing (of, parentheses, various signs, rearranging equations, using a variable, etc.) MEAN something by working with them with whole numbers or money or counters. A dc with EF (executive function) deficits will struggle with flexibility, so you might even do other things to work on flexibility like playing FLUX. And when she solves a problem or writes an equation, then say ok what's ANOTHER way we could write this and ANOTHER. 

On November 9, 2018 at 2:08 PM, forty-two said:

she was able to do the Epsilon exercises with ease - but it didn't help with her fundamental misunderstanding. 

Is it possible she basically just memorized one way to see them and understand them and doesn't have the flexibility to apply them to a new setting, a new manipulative, a new way of writing the problems, etc.? 

My ds can memorize things one way and not know how to do it in the next setting. I try to make sure we do a single concept lots of ways, not just with one curriculum, one way. Most kids can do something in a curriculum, do it one way, and "generalize" it to new applications. That's not my ds, so we have to work on all that flexibility upfront. It's an EF (executive function) issue.

On November 9, 2018 at 2:08 PM, forty-two said:

Now she's working through Key to Fractions.  She both hates it and says it is too easy :sigh.  According to her, she hates it *because* it is too easy and thus is pointless.

It may be. Everything you've listed is kind of the same, very b&w. What you might do is find 4 or 5 different fractions systems and have her work through the same concepts in all of them. For instance 

https://www.lakeshorelearning.com/products/math/fractions-decimals-percents/mastering-fractions-hands-on-kit-gr5/p/TT529 there are several grade levels o this kit and the overlays might be a fresh way for her to think about it and put meaning to the language

http://www.ronitbird.com/ebooks/ Ronit Bird uses a bar approach, which is more consistently logically than using circles, and it would be another, fresh take.

https://store.rightstartmath.com/rightstart-fractions-lessons/ RightStart uses their unique fractions puzzle. Dr. Cotter's mantra is that a good manipulative creates confusion. So I'm showing you lots of different ways to approach fractions, and you can do them ALL, doing it with the manip, writing the expressions and equations lots of ways on a whiteboard, till her brain generalizes and goes oh you mean it's always that way no matter how I visualize it, no matter what the manip, no matter what the situation, no matter whether it's whole numbers or fractions? Yes Matilda. 

https://www.amazon.com/Advanced-Pattern-Block-Book-Grades/dp/1583243178 I'm using this right now with my ds (in addition to all this other stuff) and it's fabulous for being a fresh take on fractions.

On November 9, 2018 at 2:08 PM, forty-two said:

all the various fraction operations feel like they are *breaking* the rules she already learnt, instead of extending them. 

I think that goes back to the flexibility thing. Even multiplication/division should have been presented in terms of pre-algebra concepts (equations, missing components, various ways of writing them, etc.). Common Core right now is huge on this, and it's a really strong component of common core. Most of the math in the homeschool community is not CC, and it's missing this. If you went to almost any CC materials for this age for fractions, they'd be doing things lots of ways and working on flexibility. 

My ds has diagnosed disabilities (math, reading, writing) and a language disability and of course his apraxia and ASD and he's a little gifted. For him, I've always said I had to make concepts, terms, numbers MEAN something. I'm using all the things I listed above. No regular math curriculum, just trying to see the same thing 40k ways till it clicks. If you're only picking one thing, try the Ronit Bird stuff just because she's actually meant for dyscalculia (math disability) and super thorough and cheap. At only $10 for an ebook, can't go wrong. With her stuff, I do each step super tediously, till he entirely understands what she was getting at and GETS it and clicks and can think through it lots of ways. I try to go into all the ramifications of it, all the ways it presents. So right now we're using her multiplication ebook (in addition to working on fractions and other things, yes) and we're working on the X4 and X9. So now that he's covered all the X4 (the double of the double), each day I'm asking him lots of ways to write equations for them. So we might say 

4X3=12

12/4=3

1/4 of 12=3

2X2X3=12

(4)3=12

2^2X3=12

Every way I can think of, we're going to write it and explore so I know that that relationship and those numbers got filed under ALL the ways, all the manips, all the applications (measuring and money and time and...) and aren't segregated off as isolated facts. The various ways of writing and using it all have to have meaning and all have to connect to the numbers. Typical kids with no disabilities might be flexible and make those connections and generalize it naturally.

Ideally then what happens is that you use the other structures (pre-algebra notation, fraction bars, etc.) and she discovers why she would want the shortcut of the way we "solve" the problems. You want to do it other ways first till that clicks. With my ds, that click takes a long time. That's why he has a disability, lol. Like we'll do two problems a day for MONTHS, starting with manips and then exploring how it would be written, till finally he goes hey get out of the way I get this. I never go straight to written.

On November 9, 2018 at 2:08 PM, forty-two said:

a reluctance to physically write

Any evals on this? You've got your SLD writing, the working memory (holding thoughts in head while trying to get it out), retained reflexes in the hands that glitch the connections, and well just that writing it herself might be distracting from the thought of the math. I scribe everything for my ds and we work at a whiteboard. Now that he has some things pretty solid, I'm thinking of having him write. I'm really of the Vygotsky school on math. I think they go farther by working with a mentor who pushes their thoughts. You can read about him and the idea of zone of proximal instruction. I don't remember the title of the book I read, sorry, but maybe google a bit. I'm looking on amazon and tons is popping up. 

 

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I found that both of my kids didn't truly understand what was going on with fractions until we got to algebraic fractions.  You might want to try moving on.

Regarding the why for 1/4 of 40 meaning multiplication.  Does she know that "of" means multiplication even if it's not a fraction?  As in: I have two sets *of* three oranges.  Therefore I have two times three, or six, oranges.  2 of 3 is 6.

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1 hour ago, EKS said:

I found that both of my kids didn't truly understand what was going on with fractions until we got to algebraic fractions.  You might want to try moving on.

Regarding the why for 1/4 of 40 meaning multiplication.  Does she know that "of" means multiplication even if it's not a fraction?  As in: I have two sets *of* three oranges.  Therefore I have two times three, or six, oranges.  2 of 3 is 6.


Agreeing. I was wondering exactly this, too.

Long, long ago when I was in public school, we were directly taught that the word "of" meant "multiply" (just as EKS explains above), and it was later on as we did Pre-Algebra / Algebra that the abstract concept and logic of it fell into place. Agreeing with EKS -- if DD is willing, perhaps accept that "of" means "multiply" for now, and move forward, and as the abstract reasoning portions of her brain mature as she gets older, and as she works more with fractions and really *sees* how they work, then the lightbulb may click on for understanding the concept.

And, I know some kids get very stubbornly stuck having to see "why" and really understand the premise before they can move on, so moving forward may not be an option. BEST of luck, forty-two! Warmest regards, Lori D.

Edited by Lori D.
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I am having the same issues with my 12yo Ds. It is a 2E thing for him.

The language of mathematics is every bit as challenging as the calculations in mathematics.

The multiple procedural steps are a hang-up.

We use a lot of cuisenaire rods for bar graphs.

We have cut up a lot of paper this year.

Right now one particular hang-up is in dividing fractions—multiplying by an inverse is blowing his mind even though he “understands” that multiplication and division are related similarly to addition and subtraction as they are inverses of each other.

You know in the movie Hunger Games where they raise their fingers in tribute? Imagine me doing this for you. I get that you are deep down in the trenches. Keep at it, sister!!

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As one of my favorite mathematicians-turned-teacher says, "Fractions are hard!"

Quote

"Fractions are slippery and tricky and, in the end, abstract. It is actually a bit unfair to expect students to have a good grasp of fractions during their middle-school and high-school years. This pamphlet explains why, and offers the means to have an honest conversation with students as to why this is the case. Their confusion and haziness about them is well founded!"
---James Tanton

 

The pamphlet Tanton wrote isn't for students, but for parents and teachers to help us understand some of the reasons our students struggle. Highly recommended!

In the meantime, the most important thing about math is that it's supposed to make sense. If your daughter can't make sense of fractions at this time, perhaps the best thing to do is put them away for awhile and do some other type of math. There are plenty of interesting things to study. No reason to beat your head against a brick wall when you could turn left and discover a fascinating nature trail.

[For example: One of my favorite middle-school math sidetracks is to explore discrete math with Agmath.com's Counting and Probability units.]

And then in a few months, when she comes back to fractions refreshed after the break, she may find that her mind has been working in the subconscious all along and that things make more sense to her. At least, that's how it often worked for my kids.

 

 

 

 

 

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Thank you for all your thoughts and encouragement!  I know I haven't been posting, but I have been mulling over the thread and trying things out.

A week after posting, I went through a lot of the ideas with dd12.  I think she finally made sense of "of = multiply".  (Working through @LMD's ideas was particularly helpful.)

I agree with the people who've said that it's probably something to do with the fraction concept itself.  I've been hoping that hitting it in pre-algebra would help - that seeing it more formally done would help us both. 

That's actually why I did an experiment with AOPS Pre-Algebra - nothing gets into the nitty-gritty whys more.  But I had no idea whether it would work for her: on the one hand, she is generally a strong math student who loves a challenge, and the AoPS pre-test doesn't call for anything more than fraction basics, which she had down.  But on the other, while she intuits most things to do with math, the few that do trip her up, *really* trip her up.  In any case, AoPS was a fail - the detailed proofs weren't making sense to her (found out later that she wasn't able to intuit the jump from elementary math equations to algebra equations, when we did it explicitly in MM - took two days of her working it both ways before she felt solid), she may be all about having an intuitive understanding, but she struggles with explaining it explicitly (and generally doesn't see why she should have to).  On the other hand, it was *great* for bringing misunderstandings to light, and pinpointing exactly where the problem lay.  (Though we didn't work on fractions, I still got a lot of insight into her thinking processes, and it gave me a lot of insight into what might be going on with fractions.)  But it made for long, hard math sessions, when every other day involved being brought to a screeching halt by yet another difficulty being uncovered.  AoPS was hitting too many of her weaknesses - they expected students to intuit a lot of things that she just needed to see explicitly.  So we did some pre-algebra-y MM, and now have started Dolciani.  Originally, Dolciani (which I had on my shelf) looked both boring and intimidating to the both of us, but somehow after AOPS - and me figuring out the basic structure and point of pre-algebra - it looked downright friendly ;).  So far she's doing fine with it - pretty straightforward, no difficulties.


WRT evals, I do wonder.  I've wondered about her for quite a while, but every time I was about to pull the trigger, she'd make a leap.  It's easy to miss her inflexibility on stuff, because she's generally quite flexible - far more so than ds7 (who I'm *really* starting to wonder about, as he's just not outgrowing stuff).  But on the stuff she's inflexible on, she is *really* inflexible on - more politely than ds7, usually, but just as strongly.  But she's generally willing to talk about it, and listen to what I have to say, and I can usually persuade her to give it a trial go.  And it doesn't happen on all that many things (as compared to ds7, who is stubborn and inflexible on what feels like all the things, although he *is* improving, be it ever-so-slow).  And, before puberty, she didn't throw a fit over it, either (also unlike ds7) - we could discuss it rationally; now she cries a lot, although once I calm her down, she's willing to discuss.  I do wonder whether, in our discussions, I'm giving her the words she needs to describe what she's feeling far more than I realize - that it's less a discussion than it is me throwing out words and ideas until something clicks - that I'm doing a lot more of the "putting it into words" work than I realize. 

 

Thanks again for all the thought and effort y'all have put into your replies - I really appreciate it!  I'm working my way through all the links and resources and approaches.

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  • 9 months later...

UPDATE:

God willing, I think we might have solved the fraction problem!  In a ridiculously anti-climatic way, though, which is somewhat disconcerting considering the years of frustration, but I'll take it ;).  I'm cautiously optimistic that it's for real and will stick.

We're doing Dolciani Pre-Alg (1970), and went through the two fraction chapters (one on the nature of fractions and the next on fraction calculations) with great care and attention (on my part), and a miracle occurred!  Somehow, in the first few sections of the nature of fractions chapter, whatever her missing piece was got filled in, and it was clear sailing through both chapters :jawdrop.  I was all geared up, when we started the first fraction chapter, for major weeping-and-gnashing-of-teeth trouble, but somehow it made sense to her this time.

I know a significant part of her problem was how she thought the rules of math changed with fractions - that everything she'd learned in arithmetic suddenly did not apply with fractions.  I talked a lot about how fractions/rational-numbers just expanded the rules, not changed them, and how the expanded rules worked for the whole numbers she already knew, too.  And Dolciani's presentation was right in line with that, and spent a lot of time building up "what a fraction is" before moving to calculating with fractions.  And I liked how they did fraction division - explicitly multiplying by the reciprocal every time, never introducing the "short-cut" of flip and multiply.  And somewhere in there she finally made sense of why multiplying by a fraction can result in a smaller number, not a bigger one.  (She'd previously had the flawed intuition that the nature of multiplication was to make things bigger, while the nature of division was to make things smaller, and fraction multiplication went against that.)

In any case, the pre-alg fraction chapters did just what they were supposed to do: consolidate understanding of fractions, fill in any holes and give a more solid, more accurate, conceptual foundation.  It's by far the most successful fraction experience we've had :).

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On 9/25/2019 at 4:00 PM, forty-two said:

UPDATE:

God willing, I think we might have solved the fraction problem!  In a ridiculously anti-climatic way, though, which is somewhat disconcerting considering the years of frustration, but I'll take it ;).  I'm cautiously optimistic that it's for real and will stick.

We're doing Dolciani Pre-Alg (1970), and went through the two fraction chapters (one on the nature of fractions and the next on fraction calculations) with great care and attention (on my part), and a miracle occurred!  Somehow, in the first few sections of the nature of fractions chapter, whatever her missing piece was got filled in, and it was clear sailing through both chapters :jawdrop.  I was all geared up, when we started the first fraction chapter, for major weeping-and-gnashing-of-teeth trouble, but somehow it made sense to her this time.

I know a significant part of her problem was how she thought the rules of math changed with fractions - that everything she'd learned in arithmetic suddenly did not apply with fractions.  I talked a lot about how fractions/rational-numbers just expanded the rules, not changed them, and how the expanded rules worked for the whole numbers she already knew, too.  And Dolciani's presentation was right in line with that, and spent a lot of time building up "what a fraction is" before moving to calculating with fractions.  And I liked how they did fraction division - explicitly multiplying by the reciprocal every time, never introducing the "short-cut" of flip and multiply.  And somewhere in there she finally made sense of why multiplying by a fraction can result in a smaller number, not a bigger one.  (She'd previously had the flawed intuition that the nature of multiplication was to make things bigger, while the nature of division was to make things smaller, and fraction multiplication went against that.)

In any case, the pre-alg fraction chapters did just what they were supposed to do: consolidate understanding of fractions, fill in any holes and give a more solid, more accurate, conceptual foundation.  It's by far the most successful fraction experience we've had :).

That is awesome!

I love Dolciani!!

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11 minutes ago, ElizabethB said:

I really like both the Dolciani Pre-Algebra and Algebra.  

I have a 70s Dolciani Modern Algebra, plus 60s teacher's editions of Modern Geometry, Algebra with Trig, and Introductory Analysis (aka pre-calc) from my grandpa.  He was a high school math and science teacher, and not only did he teach from original Dolcianis, he was the person responsible for bringing New Math to his school.  I was so chuffed to see those Dolcianis on his shelf!  (He'd had an Algebra 1 book, too, but he'd lent it to my cousin and she unfortunately lost it :sigh.)

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18 minutes ago, forty-two said:

I have a 70s Dolciani Modern Algebra, plus 60s teacher's editions of Modern Geometry, Algebra with Trig, and Introductory Analysis (aka pre-calc) from my grandpa.  He was a high school math and science teacher, and not only did he teach from original Dolcianis, he was the person responsible for bringing New Math to his school.  I was so chuffed to see those Dolcianis on his shelf!  (He'd had an Algebra 1 book, too, but he'd lent it to my cousin and she unfortunately lost it :sigh.)

I have Algebra 1! I have all the others you listed except Introductory Analysis, I didn't realize that's what they called pre-calc, just ordered it, thanks!!  My son sarcastically said that they weren't so modern anymore, LOL. He got "Modern" pre-algebra and algebra with Dolciani.  I don't really like her Geometry that much, though, I am using an actual more modern Geometry with him this year.  I don't love it but it has a fair amount of Algebra review and I like it best of all the Geometry books I have, and I do have a small collection...

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