Gifted Issues Discussion homepage Forums General Discussion How to debug/correct basic math problems

I've been been partial homeschooling my DD (7yrs) for about 6 weeks now. We work primarily on math and literacy for 2 hrs a day together. We started the homeschooling because I felt she wasn't working at an appropriate level in literacy, nor at an appropriate speed and in the appropriate manner for math (ie the level was not necessarily wrong, so much as approach and pace).

A few weeks in I was commenting to DH, and probably here, that her conceptual ability was well ahead of her basic arithmetic/facts. However, watching her work and also being reminded by a recent post of HKs about WHY I wanted to home-school math in the first place, has me wondering if she's not so much struggling to learn math facts as completely confused about basic principles. One of the concerns I was trying to explain to her school was I felt like coming at a vast array of concrete representations of the same idea again and again and again was confusing her (ie "Didn't we just do this? So this must be new... So there must be some trick here... Ummm...."). They want the kids to "discover for themselves" "authentically" the meaning of every possible concrete representation of an idea, like it's new and amazing...

She's struggling incredibly with basic addition and subtraction and watching her I can't quite tell if she's

a) using multiple methods in a conflicting manner

b) suffering from a working memory or processing deficit (ie the method is fine but she keeps losing her place and miscounting). She does have ADHD, this is feasible BUT she is on medication, which does work, and her WMI on the SB5 was 140+ BEFORE medication....

Can anyone suggest the right approach to figuring out if this is a method problem or a maturity of mental math ability problem? I am assuming I would address fixing these causes differently.

While we are speaking of her IQ testing - quantitative reasoning was her weakest area on the SB5 (96th), the tester noted it was too early to tell if she had an actual relative weakness or simply hadn't developed the skills yet through lack of interest (she was 5.25yrs). She's not crazy out there in math by any stretch (holding her own whole grade skipped, but no more than that in math), but she should surely be well able to manage subtraction and addition within 20, or at least within 10.

Thanks Portia, she's definitely completely sound on quantity (if you mean correspondence) but like your son something seems to be going horribly wrong with a partially on the fingers and partially in the head approach to addition and subtraction. I see her waggling her fingers or touching her knuckles and re-doing and re-ding it because she's gotten muddled. I am not sure if it's a problem with her ability to think or a problem with using a completely erroneous method (or weird amalgam of approaches that were never supposed to be smerged).

She's got a pretty good attitude and has not given up because the work is too easy - she's been at vaguely the right level for the last 18 months at least, but becoming increasingly confused in some ways. I am pretty sure her mental math was the same (or better) at 4.5yrs old when compared to right now.

Struggling how, exactly?

And in what SORTS of representations?

Standard over-under notation? Side-by-side, equation-style?

Manipulatives or a drawn representation of them?

First thing that I would try is to have her "teach" a stuffed animal or pet-- or you, if she will-- how to do the problems she is working on. Let her do a few of those, maybe even pipe up with questions if a 'gap' isn't being explained-- that will let you know whether or not she's "smushing" methods somehow and adding 2+2=5 as a result.


Then you'll know what you need to remediate, if there's a problem in something she's learned (or erroneously applied/synthesized).

Perfect suggestion, thank you!

Her understanding of what the question is asking seems correct in side by side OR over under, as is her approach to multiple forms of writing out subtractiong with re-grouping. But when she solves the smaller parts of the question (11-7, 8-5, etc) in her head she takes an obscenely long time and is wrong at least half the time. Zero problems doing it with beads, though we've not done it a lot. she seems annoyed by the beads and to want to do it in her head but then to go horribly wrong.

Edited to add that she clearly understands what is happening in a drawn representative of 83-29, but will still choose to work out the ones in her head and get them wrong.

Well, have you asked her HOW she's doing the problem in her head?

Like-- is she visualizing beads? Is the problem that she can't "hold" the image? Maybe it would help to arrange things in rather fixed "patterns" so that she can recognize "shapes/patterns" (like the number of symbols on playing cards, for example-- you don't have to "count" an 8 to know how many hearts are there, right?) and speed things up that way.

I tend to work basic math quite visually, myself. I'm terrible at rote memorization, so fluency was a VERY slow go for me personally.

Is this difficulty in the context of two-digit subtraction? Is she also having trouble with addition, or ONLY with subtraction?

I tended toward the latter-- for some reason, that is more complex to do mentally if you're working problems visually. I think it requires an extra "holding/manipulation" step over addition.

I was just talking with DH about this, she's a very private little person (partly this is extreme perfectionism, can't bear to find out she's wrong), so she's mubbling and waggling her fingers enough that I can tell she's doing SOMETHING but not what exactly. I will try the "teach teddy" approach tomorrow and see what that reveals.

I think it is worse with subtraction (need to pay more attention), so I have been teaching her to add up to the number she needs to subtract from (ie calculate 13-7 like this: 7 + 3 [to get to ten] + 3 [to get to thirteen] = 7+6 SO 13-7=6). She has no problem following this concept at all, perhaps she will start applying it herself.

She's equally strong verbally/visually on IQ test, but it is entirely possible that she does think about this sort of thing visually, which I don't AT ALL. Edited to add: There is no way I could visualise numbers based on a deck of cards for example.

Forgive me if this is a stupid question, but maybe she needs to do some rote facts work? Or is the concern that somehow the basics aren't there--like, there is an inability to conceptualize 8-5?

I feel like with some kids, they just need to get the facts cemented before moving on or they will get hung up? No? Feel free to shoot me down.

Ultramarina - I am just trying to figure out if its nothing more than needing some rote learning, or if there is some other issue.

Today we did a bunch of exercises around counting 1s/10s with manipulatives and manipulating ideas about those numbers verbally, and it was interesting the odd gaps she has. No trouble conceiving that 12 blocks could represent 12 ones OR 12 tens, could count the 12 "tens" and say "12 tens" could count them by 10s and say "120", could write "12 10s = 120", but verbally struggled with converting back and forth between 12 10s and 120. It seems much easier for her to count the objects by hand but to think about the concepts seperately from the manipulatives (or maybe just on paper) does that make any sense at all? It seems when looking at the numbers it's easy to understand that 12 groups of 10 equals 120, by when saying it outloud she gets confused about where to put the zero? Maybe this is a visual thinker issue?

I think the adding/subtracting/counting difficulties may be age and/or ADHD related. She was having trouble accurately counting larger groups of beads today as she'd loose her place or miss one. Like wise when asked to add or subtract without beads she wants to use her fingers but the gets muddled.

Maybe she just hasn't found the method that works for her yet?

Just bumping this to ask for more help re math facts to 20... I've given up on timez attack as DD is too unfocused to even notice and read the sums reliably before the time is up, let alone answer them. We've been trying this one today:

http://www.harcourtschool.com/activity/thats_a_fact/english_K_3.html

I have no memory of mastering these facts myself. Or even of my first child working through this to be honest (we had MUCH bigger problems with literacy than addition/subtraction facts, I guess, so this just happened in the background and was never poor enough to be anyone's primary concern). So for something so basic I feel rather out in the dark. I can tell that addition/subtraction facts (not concepts) within 20 is a problem for my DD but I am not clear what the ideal way to fix it is.

Doing the sums on the link above she is slow but accurate within 12, very slow and less accurate within 20 (but struggling to attend at all). Note that she is in general slow with most things. Slow to talk, slow to respond... Presumably this is ADHD related. My husband is a bit like that, his sister is actively difficult to talk to she is so slow to reply... So slow might not mean too much.

She's using her fingers a lot or just staring into space and I feel like maybe I should be teaching her better methods, but I am not sure what they would be. When I am faced with sums within 20 I either "just know" the relationship, or if I can't remember (my memory is impaired by a health problem), then I will very rapidly break it down using the gaps either side of 10... But I am not sure if this is what I should be teaching her? Or how?

Does that make any sense at all?

Also worth noting that with the timez attack retention she can't get past she'll struggle with different sums each time and get a different set of problems to practice each time. So it's not been a case of she just didn't learn 2 or 3 groupings. One pass through she'll be fast with certain facts and then not get them the next pass...

Quoting you...One of the concerns I was trying to explain to her school was I felt like coming at a vast array of concrete representations of the same idea again and again and again was confusing her (ie "Didn't we just do this? So this must be new... So there must be some trick here... Ummm....")....

When I read this, I just thought that perhaps a video would help if this is her thought process. All kids know a video is the same thing again and again, right?

My son (first grade) has learned a lot over the last two years using the Mathtacular series of videos, and he still has a lot more to learn from them. I like to watch them, too. They do little skits that are silly, but get the word across. (Much less dry than Khan Academy, imo.)

(He's in the 99th percentile for the nation in math according to the MAP testing this fall. I think videos help tremendously. Especially for a visual learner.)

Thanks Anerteine, i will look at that link!

Portia we have different currency to you guys, no 1s or 2s, without small coins I can't see how money would help. She has no trouble using beads to demonstrate. But she just isn't fast or accurate at all with facts that really are needed for double digit subtraction with regrouping. Subtraction wihin 11-20 is fairly fundamental to subtraction like 73-17, which she understands conceptually but makes computational errors in the small steps (ie 13-7).

I have to say I don't get your thing about the names making 11-20 harder at all, I'm assuming a personality thing, for me 25-9 is not easier than 15-9, it's dependent on 15-9... But maybe it will make more sense to DD.

I'm thinking of googling for number bond worksheets. Maybe that's what she needs to memorize...

Have you tried giving her a calculator to just play with math as she sees fit?

Thinking maybe she just needs to absorb in her own way and something freeform like a calculator would let her do that. I'd try to look at where her strengths are and use those as a conduit to the learning. Is there something she learns seemingly effortlessly?

Reading came easily, math concepts came easily. Rote learning does not. I'll think some more about visual representations I think. I feel like she does better with abstract discussion than concrete materials. Like she needs to be told (probably with a visualization) how to think about these problems, not to spend endless time manipulating beads or some such.

Do you guys still have 5 cent pieces - it is amazing how doing money sums is simplified when you only have 10, 20 and 50 cent pieces. Although I did have to explain rounding to to ds6 the other day and the difference between the rounding most of the shops use and what you would use in maths - so it has added to the complications in that way. And I would really like to be able to use coins for place value explanations.

Then I would try the play with numbers approach and pattern concepts. You might like some of the ideas here: http://www.gifteddevelopment.com/Articles/instruction/i90.pdf

My DS just told me today that he realized adding even numbers always gets even numbers, adding odd numbers also get an even number, but adding an even and an odd yields an odd.

He also likes the idea that adding the digits of a number with nine as a factor ends up with nine. And that you can move up the squares by adding the next odd number to the previous square:
1 + 3 = 4
4 + 5 = 9
9 + 7 = 16

I can't even imagine memorizing any addition past 10 when a process works fine.

Puffin she has no trouble at all with things like 10-5, 50-5, 20-10, 50-20, 20+50+5, etc anything I could demonstrate with our coinage I am very confident that even if she was slow she would be accurate (though its hard to say if having the coins in hand would slow her down or speed her up, compared to being asked verbally I could time it, I think she'd be faster verbally). Her problem is NOT with the concept of addition or subtraction.

I am particulalry sure about the coins given her approach to 50-20 would be "thats 5-2, so 30". I think the problem is fundamentally that she's no doing it in her head using memory or patterns like that when regrouping is required, because regrouping requires that you know subtraction for 11-20 or are fast and reliable at working it out. And she's got neither. Like zen scanner said I wouldn't have thought you need to know above 10, but working with her ive realise I do just KNOW the bond between 5, 7 and 12, or 8,8 and 16, etc, I also use tricks around 1s, 5s 9s and 10s. She's not got any of that and I'm not sure why (or why it's hard).

I'm hoping zen scanner has hit on it with the last post. I need to read that link on a real computer.

Maybe I should try teaching times tables and all the patterns and tricks for learning them, Imaybe that's what she needs to get this.

Your DD sounds quite similar to mine. She loves playing with beads but if I try to get her to use them to do tedious calculation, she'd probably run away.

With rote learning, she's very selective. If she isn't interested then she won't remember. It's really extreme.

For her, DreamBox was quite helpful in improving the math fluency. They go over number bonds visually using different tools and things started clicking for her eventually. I use this abacus to help her get unstuck when she gets confused:

http://www.learningresources.com/product/2-color+desktop+abacus.do

This.

Fluency comes with practice and meaningful application-- or not at all-- for DD and I both.

I cannot simply MEMORIZE anything just because I want to. Certainly not numbers. Not my driver's license, not a phone number, nada.

I never pushed this on DD. She's completely fluent now, and has been for years-- but she definitely wasn't when she was 7-9yo.

One helpful tip to add to the thoughts above--

Do PeeChee folders still have a times-table on the inside? That's a good way to "see" the times tables in a visual format.

Another idea-- try using a 100's chart to actually visualize the PATTERN involved in addition and subtraction.

Number lines are helpful to me, personally. This is actually a useful skill, by the way, when they get into functions and translations/inversions later on.

One idea for homeschooling (since you are doing so) is to use Math Games-- file folder games.... there's a book with reproducibles in it that has a BUNCH of basic math facts drills organized as simple board games. DD found those much more tolerable than any other kind of drill at this age.

File Folder Games for Math

You could also play a regular board game with a pair of D-20's from a local gaming store. That way you could practice the subtraction facts that she doesn't know.

Thank you all!

Zen Scanner I read that article you linked to a few years ago when researching about my elder DD, who fits the "visual spatial learner" description much more typically. And I used a similar article, but spelling related, with great great success for improving her spelling, but she'd already nearly mastered times tables with timez attack by the time I read this article so we stuck with what we were doing.

This quote from that article is my eldest to a t:

"The right hemisphere cannot process series of nonmeaningful symbols,"

She was my most mathematically advanced toddler - but took MONTHS, long after she was doing addition and subtraction past 10, to learn to recognise the digits 1-9. She just could NOT map the concept of a number to symbols at ALL. And reading, OMG reading... Just reliably having the alphabet took us nearly 3 years.

I think the reason I am puzzled by this issue with my 2nd DD is that she's not appreciably sequentially impaired, I don't think of her as a visual spatial learner. Certainly she's V/S gifted, but her Verbal IQ is equally high as her NV, I think of her as quite balanced. She learned to read easily (and phonetically), she learns spelling with ease (not one look and never forget, but easily and reliably and transfers it to her writing). This issue with math seems odd to me.

BUT that article is exactly what I had been thinking of using when it came to teaching her times tables, I have just been holding off because of this addition/subtraction fact issue. AND today it has hit me that I have been stupidly doing exactly what annoyed me about her school teachers - failing to give her a big enough picture. Possibly she NEEDS to sit and look at a number chart and see addition, subtraction, multiplication and division patterns all together to really start seeing the patterns and thus learn them. She's absolutely all about patterns, DH and I have described her as "highly pattern seeking" on more than one occasion.

I didn't mean could she do it. I was just curious about the currency thing. Math mammoth etc do quite a bit on money and so did we when I was young and we had 1,2 and 5's but it seems pointless now.

I hope you get a solution soon.

Do you have 5s? We do. I have to say I never think to work with money with my kids because we use cash so rarely. Even if I send them into a shop alone, I will sometimes give them cash but often they'll take a credit card now that you can just wave the card at the machine for small purchases. Talk about lost skills!

How is shop rounding different for you? Our prices are now either in increments we actually have (5s or 10s), or are .99, which would round up by any method yes?

You've got that right!

Ah, the education value of going to the sweet shop to buy candy on the walk to school at so many a penny etc.

No wonder we never had trouble with simple arithmetic and even fractions were easy in those days LOL

I haven't tried an abacus no. We might try the plastic money or an abacus if my current plan doesn't work. Which is to work on some number grid and number bond stuff. I really appreciate all the ideas.

Cuisinaire rods or unit blocks can help

Especially with the half p coin.

Hmmmmm.... maybe.


Honestly, manipulatives like this were of NO help to DD or myself either one. Truthfully, the manipulative was just, like-- well, one MORE thing to try to manage. It just didn't "map" mentally at all.

Even with an abacus-- I'd encourage free play and then use the CONCEPT to discuss a way of approaching the math, not necessarily mandate that she use it to work math. Does that make sense?

DD sounds exactly like this-- she's a purely phonetic reader, symbols she has ZERO trouble with, it's just a numbers thing for her. She seems to prefer the context and a MENTAL construct to work within, or it just doesn't stick. She's not quite so extreme that way as I am, but my DH is also this way to some extent. I have to have it in my head to use it. He can (sometimes) use what is on paper.

Seriously-- both of us got labeled "probably slow" because of arithmetic troubles in elementary school. Uhh- yeah, just, no. We both have terminal degrees in the physical sciences, in math-heavy disciplines.

My prediction is that people like this don't develop fluency until they NEED that fluency for some other purpose. If you place load on the skill, it will come.

I should have elaborated....I was given the cusinaire rods to play with in 5thgrade by my teacher. Everyone had an individual set. There were some activities and games but mostly free exploration to find patterns and discover with them. I actually made several connections to patterns that I had never noticed with just numbers or other manipulatives. I never had to use them for any particular assignment, they were just available.

My kids have a set here at the house and pattern blocks and a balance with gram cubes and fraction bars...all that stuff we have played with together and then I just let them explore it on their own.

YMMV

hee - that's funny! i started with money as a way IN to arithmetic with DD5. i think it really helped for her to see the connections between 1s/10s/100s so multi-digit addition/subtraction was a snap. i did have a hard time tracking down pennies since Canada recently retired the 1¢ piece, but it was really worth the trouble. fractions and multiplication/division have also been well served by money counting - oh, and analog clock problems worked, too! those have also been great for reinforcing number sense in a fun, visual way.

but i bet DD5 would enjoy some Sweet Shop Math even more!

My mom used to joke that I could solve any problem if it had a dollar sign in front of it.

I'll be curious to see how HK's experiment goes, but I wouldn't hesitate to give her pocket money to calculate purchases when you're out. The motivation of the purchase may be just enough "load" to induce full interest/effort if that's the weak link.

What's the square root of $1?

We don't have 5s. 9.95 should and usually is rounded down to 9.90, 9.99 to $10. But of course in maths. 1.55 is rounded to 1.6 (2 do).

I did a lot of work with DS in the early days with short cutting computation- we broke things up into manageable parts, we did a lot of mental math on car rides, we checked out the Sir Cumference book on how to organize to accurately count large numbers. The main "gist" was mental organization to compute quickly. I was a budget manager for years, so learned some skills (for most people, it is a skill) with quick calculations.
It does sound like she may need some help with "math facts" and tips for quick calculations. For instance, once I was sure that my son understood multiplication and struggled through a good deal of mutiplying on his fingers and counting groups on paper, I did successfully use flash cards and the iPad to facilitate memorizing the table (same with division).
Sometimes it takes a bit for things to click- I was like that as a kid with math- not always quick but when I got it, I got it better than most.

Better still if I'm short that much... (-$1)

Part of that is anything that I want it to be.

I like negative square roots and money. I'm going to build an island with my proceeds.

This sounds a lot like my DD, too. But she's really good at memorization and is begging for some "new big idea" math so I told her we would do a little push on math facts adding/subtracting 1-20 for a month (as folks have recommended here before). Rods have helped for sure. But she'll touch fingers and mutter and know and forget facts... And she definitely thrives on being shown approaches to problems instead of "discovering" (seriously the math at her school is going to be too much for me... But there are other kids in the same boat so I hope with a bit of math fact hot housing we can join up and ask for a real accelerated math group). Showing her column math and regrouping made things exciting enough for her to really want to work on facts. She's just aching for new ideas, challenge... She intuited negative numbers and certain patterns in graphs and fractions etc etc at an early age... But yeah those little facts!

Thanks everyone! I'll report back late next week I guess, we don homeschool Friday mornings, she just goes straight to school on Fridays. And obviously nothin is likely to happen instantly. But I think Monday I will goring out some blank number bond temates and get her to figure out ever number bond to 20, maybe not the ones 12-18 where one is over 10, because those are covered by knowing 1-10.

Forget imaginary numbers and imaginary money - pah!

How about imaginary gifted education?

I've only skim read so forgive me if this has already been suggested but I found the most helpful learning tool for numeracy facts teaching to a gifted child was to explain the system. We use the decimal number system and so I explained what that meant. I demonstrated how it differed from other number systems (ones they were familiar with like roman numerals and others unfamiliar to them) but essential explained the system, ensured it was logical and there was nothing weird or complicated about it. The best visual tools for this are the base ten blocks (which come in handy for decimals and much more later on too)

A toy cash register, some toy or real money and some stock for a child to set up shop with can be a fun way to practice addition and subtraction.

I also made some 'rulers' that were ten evenly spaced (marked and numbered) units long out of paper strips placed side by side into 'columns' to demonstrate the 'overflow' into the column to the left for addition to account for the full addend and the 'drawing off' from column to the left for subtraction if the current column runs out before the subtractend has been fully accounted for.

My DD found this super easy to understand and she got the idea in minutes if not seconds. But like me she gets pictures and diagrams quickly - not sure if that makes us 'visual learners' or not - I call it mechanical aptitude but I am not a psychologist.

YMMV

Update. We've been discussing this chart:

http://www.eduplace.com/math/mw/background/2/01/te_2_01_overview.html

and filling out all the number bonds up to 20 on these blank sheets:

http://fivejs.com/wp-content/uploads/2011/12/number-bonds.pdf

And talking as we go about the relationship between the patterns on the chart and how many number bonds there are per number, it seems to be going exceptionally well.

Great update and resource find. Funny how many resources there are to try to train kids as top-down thinkers, but how sparse when they come to the table with a top-down skillset.

Indeed. And I feel stupid for pulling her to homeschool and then making the se errors myself... Never mind.