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progression of learning addition facts


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#1 beth83

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Posted 11 October 2013 - 09:36 AM

Okay, I just need to set some expectations for myself here.  My poor first child.  She is always the guinea pig on teaching me what to expect.

 

Math facts.  What do I expect as far as memorization?  I know this topic is posted a lot, and I read that they don't need to have them solid until the end of 1st or 2nd grade.  But what does the progression look like during the year?

 

We are a couple of week into the Horizons 1 math book.  We are doing flash cards for math facts, and although they recommend doing all the cards that add to 18 right now, I only have her on the +2 cards.  I'm thinking I want these solid, then I will add the +3's, etc.  Good idea, or no?  She cannot remember these to save her life.  It has been 3 weeks now that she has been thumbing through them.  Everyday there is addition review with a number line to use, which is does great, so it hasn't really hurt her work in the book and she is sailing through everything else.

 

When your kids started learning addition facts, did they memorize some right away and add to that?  

 

Or does it take a while to memorize some, even the basic facts?

 

What should her progression look like through the year?

 

Do I know worry about her lack of memorization and just let her play with the cards over and over and over, and play with her c-rods over and over and over, and at some point they will stick?



#2 RootAnn

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Posted 11 October 2013 - 10:33 AM

I'll be honest & say it looks different for each child.

 

My current K'er can tell you the numbers that add to 5, 6, 7, and SOME of the 8s so far because he plays math card games with me. (I modified an Right Start card game to be applicable to other sums.) I was hoping to be to 10s by now, but I'd rather have it solid than move on too quickly. It doesn't have to be drill-and-kill. We also use a Miquon-inspired approach, so I'll say YES to your last question:

 

In 1st grade, I'd let her play with cards over & over, play with c-rods over & over, and through your guided play - they will eventually stick. Don't feel like you have to go at the pace the book gives you. And - Spiral math programs (like Horizons) don't expect mastery right away. 



#3 Mandy in TN

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Posted 11 October 2013 - 11:28 AM

If you roll a die, does she need to count every dot or can she look at it and just tell you the number? If you roll two dice, does she need to count every dot? Can she tell which one has more dots by glancing at the dice (if one has 6 and the other has 2 can she immediately tell that the 6 has more dots)? From there can she grab the 6, say 6, grab the 2, and say 7, 8. 

 

If she has 1 bean in her hand and you ask her how many she will have if you give her 1 more, can she tell you 2? Does she understand that adding one means that she goes to the next number to the right on the number line? Does she still need to use a number line to add 1? When she adds 13+1, can she say 13 one more (the next number) is 14? Does she understand that 13+1 is the same as 1+13?

 

If she has 1 beans in her hand and you ask her how many she will have if you give her 2 more, can she tell you? If she is working on +2, does she understand that she can start at the first number and just count up (13+2 is 13, 14, 15) or does she start at 1 on the number line and count to 13 and then say 14, 15?

 

If she understands the concept, adding 2 shouldn't require memorization at all. IF she understands what she is doing, she should be able to add 2 to a number that you say aloud pretty quickly. At first she may count the numbers aloud. If you say 10, then she says,  "11, 12. The answer is 12." Then, she may count in her head for a little while. Then, it will just become immediate. I wouldn't spent three weeks drilling +2. If she is having that much difficulty, I would back up and make sure she understands that this symbol (2) represents two items (beans, cheerios, or whatever). I would make sure that she could count and understood that 2 is one more item than one and that 3 is one more item than 2.. I would much sure that she could discriminate between more and less (which group has more beans or which dice has more dots). I would have her count and add real objects before we worked with cards that said 3+2. When she begins doing any sort of computation drill, it should be easy- a natural progression of what she already knows.

 

HTH-

Mandy


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#4 sunnyday

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Posted 11 October 2013 - 12:30 PM

Kitchen Table Math has a fun progression that builds intuitively, forming a foundation of counting on and doubles and then filling in the other facts. In Life of Fred the author does everything that adds to 7...then everything that adds to 9...then doubles...then adding to 11, 13, 15, 17, 19. The evens are treated as either doubles or one more than a previous result. Each time a new fact family is introduced, you're meant to drill until it's fairly automatic.

 

For my son, he's now pretty solid on doubles to 9+9 and on numbers within 10. He has all his addition facts to 9+8 some days, but loses them if he doesn't have practice. Right now he is frequently doing addition by making 3s and 5s and 10s, but every time he uses addition he seems to gain a few new automatic responses. Eg. Last week he might have said, "9 + 6 is 10 and one less than 6, 15." But this week he might say "9 and 3 is 12 and 3 more is 15." And next week he might just say "15" and move on, you know? That's what we're moving toward and I feel like a little more drilling and some speed worksheets will get us there by the end of 1st for sure. In the meantime I'm pleased as punch that he has back-up methods that are based on number sense.



#5 beth83

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Posted 11 October 2013 - 01:41 PM

If you roll a die, does she need to count every dot or can she look at it and just tell you the number? If you roll two dice, does she need to count every dot? Can she tell which one has more dots by glancing at the dice (if one has 6 and the other has 2 can she immediately tell that the 6 has more dots)? From there can she grab the 6, say 6, grab the 2, and say 7, 8. 

 

If she has 1 bean in her hand and you ask her how many she will have if you give her 1 more, can she tell you 2? Does she understand that adding one means that she goes to the next number to the right on the number line? Does she still need to use a number line to add 1? When she adds 13+1, can she say 13 one more (the next number) is 14? Does she understand that 13+1 is the same as 1+13?

 

If she has 1 beans in her hand and you ask her how many she will have if you give her 2 more, can she tell you? If she is working on +2, does she understand that she can start at the first number and just count up (13+2 is 13, 14, 15) or does she start at 1 on the number line and count to 13 and then say 14, 15?

 

If she understands the concept, adding 2 shouldn't require memorization at all. IF she understands what she is doing, she should be able to add 2 to a number that you say aloud pretty quickly. At first she may count the numbers aloud. If you say 10, then she says,  "11, 12. The answer is 12." Then, she may count in her head for a little while. Then, it will just become immediate. I wouldn't spent three weeks drilling +2. If she is having that much difficulty, I would back up and make sure she understands that this symbol (2) represents two items (beans, cheerios, or whatever). I would make sure that she could count and understood that 2 is one more item than one and that 3 is one more item than 2.. I would much sure that she could discriminate between more and less (which group has more beans or which dice has more dots). I would have her count and add real objects before we worked with cards that said 3+2. When she begins doing any sort of computation drill, it should be easy- a natural progression of what she already knows.

 

HTH-

Mandy

 

We have been counting real objects FOREVER it feels like.  We have already played with c-rods, beans, mini pumpkins, and every other manipulative I have picked up for the past year.  She full and well understands addition.  She is very quick at the number line.

 

It's not a problem with understanding, as much as quickness.  You can give her any number under 100, and tell her to add 1, and within half a second, she can tell you the answer is one more.  

 

When I give her +2, it takes her two seconds to figure that one out.  She can figure out the problems easily.  I just thought she should have them memorized, without having to figure them out.  Like anytime she sees 2+2, within half a second she says 4.  She has that one down.  She doesn't need to count two more from 2.  She just knows that 2+2=4.  I keep trying focusing on the "quickness" factor.  Should I not be?  



#6 Maela

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Posted 11 October 2013 - 02:19 PM

I do not have a lot of experience, as my oldest is only six.  Just wanted to let you know that we are slowly working our way through the book Two Plus Two Is Not Five and I'm really liking it.  It's all pages of addition/subtraction problems, but introduced in an orderly way.  First you work on +/- 0s, then +/- 1s, then you learn some doubles (2+2, 3+3), etc.  And it teaches little tricks to help learn each type.  We're doing one page each day along with our normal curriculum (Math Mammoth).  I saw Two Plus Two recommended here by (I think) NASDAQ.


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#7 Tanikit

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Posted 11 October 2013 - 02:37 PM

My DD is ending kindergarten and this is how it went for her (if I remember correctly - many of these also happened together) - adding 0 to a number, then adding 1 to a number, then we did numbers that add to 10, then doubles, then adding 2, numbers adding to 5 came in somewhere here as most of them were already covered as were numbers adding to 6. By this stage certain addition facts needed a little bit more work: like 3+5, 3+4 and 4+5 which is where Life of Fred helps since he does numbers adding to 7, 9 and later 13 which seem to cause more sticking than other addition facts since it is easier to go 5+5+1 for eleven (5+6=11) than it is to make a leap to 5+8 =13 simply because 5+3 causes more problems in general. At the moment my DD makes 5+8 =  8+2+3 = 10+3 = 13 (all done mentally, but she will sometimes say what she is doing). 

 

My DD has been through Horizons 1 (to lesson 115 now) and her addition facts below 10 are solid, she knows how to work out addition facts to 20 by making 10s from Singapore Maths and is getting much faster at it and her subtraction is pretty solid though not as fast as her addition yet. Having other ways to work it out is important for me - I want my child to know how to play with numbers and be able to take them apart and build them up with confidence in whichever way works. I have done no drill except for the doubles with my DD - there has been enough in Horizons and in the games we play with math in her general life.



#8 mathmarm

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Posted 11 October 2013 - 02:58 PM

I typically recommend doing the math facts for 10 and 5 first. Explicitly talking about and using the commutative law to consciously reduce the workload.


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#9 Spy Car

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Posted 11 October 2013 - 04:25 PM

Instead of trying to memorize math facts, what about learning the pairs that make "10?"

A fun game for this is RightStart's "Go to the Dump" (instructions on YouTube). Then, after this is mastered, learn to regroup to Tens. So 8+7 is the same as 10+5. And so on. You can do it with the C Rods too.

Regrouping is a skill that "scales" with ever increasing numbers, where "memorization" does not.

Let her do the re-grouping work as the "process" rather than trying to jump an important step in leaning by "memorizing" math facts. Flash-cards are not the way to "learn" math. They may be useful "after" one as learned other strategies for building speed of recall, but can be counterproductive to actual learning.

Bill
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#10 54879525

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Posted 11 October 2013 - 04:27 PM

I did what Spy Car did.

 

When you just cold memorize facts they are easily and quickly forgotten. 

 

 


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#11 PentecostalMom

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Posted 11 October 2013 - 10:17 PM

Instead of trying to memorize math facts, what about learning the pairs that make "10?"

A fun game for this is RightStart's "Go to the Dump" (instructions on YouTube). Then, after this is mastered, learn to regroup to Tens. So 8+7 is the same as 10+5. And so on. You can do it with the C Rods too.

Regrouping is a skill that "scales" with ever increasing numbers, where "memorization" does not.

Let her do the re-grouping work as the "process" rather than trying to jump an important step in leaning by "memorizing" math facts. Flash-cards are not the way to "learn" math. They may be useful "after" one as learned other strategies for building speed of recall, but can be counterproductive to actual learning.

Bill

 

Thanks for the idea! 


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#12 Spy Car

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Posted 11 October 2013 - 10:49 PM

Thanks for the idea!


At the risk of repeating an old story, when my son was young (he likes stories) I made up a thing about "two numbers that were walking in the woods."

The first number was an 8 and he said to his friend 7, oh 7, I so badly want to become a "10."

7 being a kind and giving friend said "what do you need to become a ten" [pause for answer from child]

"You need 2 become a "10," said 7, "I would give you 2."

"Oh would you really?" said 8 excitedly, but what you you become if you give me 2? [pause for answer from child]

So I will be a 10 and you will be a 5. And so it was.

Henceforth these two friends henceforth marched together as partners, 10 and 5, which together makes? [pause for answer from child]

Yes, 15, very good!


I told some version of this story a thousand times :D

Bill
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#13 kiwik

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Posted 12 October 2013 - 03:02 AM

I don't think I ever heard of memorising addition facts before I come here. I suppose I did it when I was a child but it was not a memorisation task like learning my times tables.

#14 mathmarm

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Posted 12 October 2013 - 03:19 AM

I don't think I ever heard of memorising addition facts before I come here. I suppose I did it when I was a child but it was not a memorisation task like learning my times tables.

Honestly, I don't know what the latest obsession with memorizing them is. We explicitly automated 5 and 10 when I was little, the others were just learned through repeated use.

 

At the risk of repeating an old story, when my son was young (he likes stories) I made up a thing about "two numbers that were walking in the woods."

The first number was an 8 and he said to his friend 7, oh 7, I so badly want to become a "10."

7 being a kind and giving friend said "what do you need to become a ten" [pause for answer from child]

"You need 2 become a "10," said 7, "I would give you 2."

"Oh would you really?" said 8 excitedly, but what you you become if you give me 2? [pause for answer from child]

So I will be a 10 and you will be a 5. And so it was.

Henceforth these two friends henceforth marched together as partners, 10 and 5, which together makes? [pause for answer from child]

Yes, 15, very good!


I told some version of this story a thousand times :D

Bill

I just got a warm, fuzzy feeling reading your story. My mama used to do something similar we called them Number-Tales and she'd tell us a few almost every night. As we got older they were the Many Mis(calculated)-adventures of X and Y. :).


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#15 PandaMom

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Posted 12 October 2013 - 10:53 AM

We are almost halfway through Horizon's math 1 right now.  There have been good days and bad days and days where she couldn't tell you 1+1.  At first we started with the facts that added up to 9.  I showed her the flashcards everyday for weeks and then it clicked and now she knows them totally.  I encouraged her to used C Rods for her math exercises.  Eventually I stopped showing her the answers on the cards and if she didn't know it she would use her C Rods to figure it out.  Now she knows her addition facts to 18 very well.  We recently started on subtraction and she's going through the same thing and slowly learning.  She can tell you what 8-1= but not 8-7=.  I just keep encouraging her for she will eventually get it.  I made a chart of all math facts up to 18 and as she mastered them I would check them off.  This gave her a lot of confidence to see what she had mastered and was a challenge for her to master the rest.   DD is PDD-NOS and has cognitive delays which makes it harder for her in many ways.  Math is just one of her likes.

 

Susie

DD(8)



#16 mathmarm

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Posted 12 October 2013 - 05:19 PM

One more thing I thought about. When I was little we automated the 5 and 10 fact families for addition and subtraction and the others we learned through repeated use---but we'd been taught to think of and visualize the relationships at all times from the beginning. I think mama specifically designed/chose to teach us this way so that we could do it that way--but I highly doubt that it was a happy accident or anything. All of my siblings and I did it this way and mama taught her tutoring students who struggled to think of numbers this way too, but it wasn't always successful for older kids who had been 'imprinted upon' by the local school.

 

If we were looking at, say 3 + 4, we usually did something like this:

1) use commutative property to put the larger number first: 4 + 3 = ___?

2) This is obviously smaller than 10

3) How is this related to 5? 4+ 1 is 5, so 4 + 3 is bigger than 5 *light bulb moment*

4) How much bigger than 5 is it well...we can disassociate 3!

5) 4 + 3 = (4 + 1) + 2 = 5 + 2 = 7!

 

I know that seems long and drawn out, but we got it down pat and could do it quickly. We learned math from about ages 3-6 and then started first grade at public school. We were fluent in counting and really understood the basics like place value, skip counting and counting on and back so getting from 5 to 2 by counting took a split second and by the end of first grade, I think we'd all learned the facts for arithmetic. I remember they were a big part of 2nd grade but the teacher gave a sample/placement test the second week of class to see who needed to start where and all of mamas kids always crushed them. :lol:



#17 Rosie_0801

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Posted 12 October 2013 - 05:36 PM

Honestly, I don't know what the latest obsession with memorizing them is. We explicitly automated 5 and 10 when I was little, the others were just learned through repeated use.

 

 

Maybe it is one of those things that worked for special needs students and is now being applied to everyone else too. I'll be teaching maths facts by rote to my dd because of her special needs.



#18 beth83

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Posted 12 October 2013 - 09:28 PM

Honestly, I don't know what the latest obsession with memorizing them is. We explicitly automated 5 and 10 when I was little, the others were just learned through repeated use.


I wouldn't say it is a new obsession. I remember doing "mad minutes" for addition problems in early elementary school. It was a sheet with math facts, we would be timed for a minute, and you raced to complete them all. It was one of my FAVORITE math activities. We were taught the addition concept, then were just drilled after that. I never struggled with math all the way through college, so I have a hard time seeing why there is anything wrong with this.
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#19 mathmarm

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Posted 13 October 2013 - 07:29 AM

I wouldn't say it is a new obsession....so I have a hard time seeing why there is anything wrong with this.

 

I didn't say that there was anything wrong with learning the math facts, and I didn't even mean to imply that there was anything wrong with learning them. I am against the memorization of math facts even thought we did a lot of work with math facts in 1st and/or 2nd grade at my PS also and that was a long time ago.

 

My district is obsessed with math facts in grades K-6 and they have been for the last several years: some kids love it, some kids hate it. The majority seem to do just fine with it, but I will say this: Hundreds of the kids are coming to my classes every semester with practically no number-sense, poor counting skills, shoddy understanding of place-value/base ten and an absence of mastery over 'basic math', so what ever they are doing isn't working very well.

 

**Please keep in mind, this isn't a judgement of you, your kids or your math program at all. I don't know enough about your kids or you and I am not at all familiar with your specific math program, this post is only an explanation of my comment.**

 

So while I don't think that its wrong to learn math facts, or even to devote a significant part of the time to them. I do object to what is the current practice of prioritizing the memorization of math facts over developing number sense, efficient counting and arithmetic skills and trained mathematical thought.

 

My siblings and I were 'the best' math students at school throughout our academic career. Not because our mom was a math teacher, but because she trained us like mathematicians from the first day. For us, doing pages of math facts wasn't about memorizing the answer, it was an exercise in fine tuning our number sense and thinking. We weren't drilling to see just how much we could remember, we were drilling to automate--through training--our thought process (or math voice, as we called it back then). It was like exercising to build muscles or practicing martial arts--you drill it until it is reflexive, automatic and you can get it right, every time, without conscious effort or thought.

 

The steps that I described earlier? The whole: commute, compare, compare, disassociate, count routine that I mentioned that is the sort of thing that we drilled--not exactly the math facts, though through these exercises we definitely learned and committed them to memory, but we didn't set about to memorize the math facts and the point was to get us used to thinking and following logical thought processes. If you did the whole routine on a simple problem than you were wasting time and effort--thus defeating the point. (Do it again!)

 

We learned to think of everything in terms of 5 and 10, so like my '3+4' example from earlier, we looked at it as 5 +2 and we quickly learned that 5+2=7. Mama always taught us that thinking is more powerful than knowledge. Thinking allows you to exploit facts, relationships, rules and rote-knowledge to serve your purposes. (We had a lot of those 'knowledge is power' posters at school and they irked my mom for some reason.) After the 5 and 10 concept was set, we shifted and did a lot of equivalent equations practice also.

 

We did the same the type thing with upper level math not just 'math facts'. It seems a lot more 'long and drawn out' in the early stages but we weren't trying to memorize, we were training the thought processes and trying to master mathematical thought, not just the current topic in the math book.

 

So I guess it could be said that while I am against memorizing math facts, I am not against learning them. I hope that better explains my comment, and again, I am not familiar with your program, kid or you so you guys are probably doing just fine, I'm not trying to snidely imply that your math text is inferior to whatever else is out there. I was just trying to explain my comment a little more and I'm not very eloquent so its hard for me to sum up or be succinct about things without sounding short or blunt. I hope you understand.

 

--mathmarm


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#20 RootAnn

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Posted 13 October 2013 - 12:38 PM

I look at the "memorization" of math facts the way I look at being able to look at a word and know what it is immediately for reading.

 

With phonics, we learn parts of the 'code' and sound out words. After so many repetitions, our brains learn to recognize the pattern and we KNOW what the word is without sounding it out.

Memorizing sight words skips the part about sounding out the word -- understanding the pieces & parts that make it up  -- and just wants you to go immediately to KNOWING the word.

 

With some math programs, they want you to memorize the 'math facts' without understanding them - without 'playing with them' - without manipulating them - without thinking about what they are and what they can be.

Other math programs want you to work with the numbers, think about them, play with them, put them together in different ways (5s, 10s, for example) and the fluency with them comes after repeated exposure so they become automatic without drilling.

 

Just like some kids do better with sight words (or a combo of sight words & phonics), some kids do better with drilling (or a combo of learning/playing & drilling). For some kids, it takes a LOT of practice to 'get' good at recognizing a word immediately or knowing a sum/difference/product immediately.

 

 

I'm an advocate of phonics and of understanding/thinking about the numbers first and foremost.

 

 

If I end up making 'flash cards' of words that we've already studied & analyzed and sounded out over a hundred times so that my kid can get to where he recognizes it right away, I don't think I'm doing a disservice to pure phonics. Same thing with drilling basic math facts if we've been playing with the numbers and thinking about them for a few years without automaticy. Some kids learn differently. Some pick things up quickly. Others need a lot of time and hands-on / practice. I have both kinds (although more of the latter than the former). 

 

I hope the discussion has been helpful for the OP.


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#21 beth83

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Posted 14 October 2013 - 09:34 AM

I look at the "memorization" of math facts the way I look at being able to look at a word and know what it is immediately for reading.

With phonics, we learn parts of the 'code' and sound out words. After so many repetitions, our brains learn to recognize the pattern and we KNOW what the word is without sounding it out.
Memorizing sight words skips the part about sounding out the word -- understanding the pieces & parts that make it up -- and just wants you to go immediately to KNOWING the word.

With some math programs, they want you to memorize the 'math facts' without understanding them - without 'playing with them' - without manipulating them - without thinking about what they are and what they can be.
Other math programs want you to work with the numbers, think about them, play with them, put them together in different ways (5s, 10s, for example) and the fluency with them comes after repeated exposure so they become automatic without drilling.

Just like some kids do better with sight words (or a combo of sight words & phonics), some kids do better with drilling (or a combo of learning/playing & drilling). For some kids, it takes a LOT of practice to 'get' good at recognizing a word immediately or knowing a sum/difference/product immediately.


I'm an advocate of phonics and of understanding/thinking about the numbers first and foremost.


If I end up making 'flash cards' of words that we've already studied & analyzed and sounded out over a hundred times so that my kid can get to where he recognizes it right away, I don't think I'm doing a disservice to pure phonics. Same thing with drilling basic math facts if we've been playing with the numbers and thinking about them for a few years without automaticy. Some kids learn differently. Some pick things up quickly. Others need a lot of time and hands-on / practice. I have both kinds (although more of the latter than the former).

I hope the discussion has been helpful for the OP.

 

Thank you!  I agree with the similarities.  Like I said before, she fully understands the thought behind addition and can do it quickly with manipulatives, etc.  She took off reading and this is the first thing she has really had to try to do.  It has really only been 3 weeks, which isn't long, at all, so I will just give it more time.  

 

If English is turning out to be her strong point, maybe math will always take a little more time.  I just need to be patient!  


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#22 beth83

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Posted 14 October 2013 - 10:27 AM

Oh, I LOVE this book.  I am adding it to my list!

 

I do not have a lot of experience, as my oldest is only six.  Just wanted to let you know that we are slowly working our way through the book Two Plus Two Is Not Five and I'm really liking it.  It's all pages of addition/subtraction problems, but introduced in an orderly way.  First you work on +/- 0s, then +/- 1s, then you learn some doubles (2+2, 3+3), etc.  And it teaches little tricks to help learn each type.  We're doing one page each day along with our normal curriculum (Math Mammoth).  I saw Two Plus Two recommended here by (I think) NASDAQ.

 


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