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square_25

The nitty-gritty of mathematics education.

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2 hours ago, square_25 said:

 

Interesting. How so? Do they wind up reading as a result? 

My older girl could have probably learned her letters at that age (she learned at 1.5 or so), but it didn't occur to me to teach her to read, so I can't say I've done the experiment! How does it work? 

My mom did the "teach your baby to read" thing with me when I was a toddler.  (In elementary school, I found the box of flashcards she made and used, and she told me a bit about it.)  Pure whole word teaching, but it worked.  Per family stories, I was reading unfamiliar words at 2.5yr.  I remember reading old 60s chapter books when I was 5.  I can't ever remember a time when I couldn't read.  And I read a *lot* - hours and hours a day (still do).  Read everything I could get my hands on, and read under my desk at school.  My reading comprehension was effortlessly good.  (Auditory processing and comprehension was weak, though.)

With my oldest, I liked the idea of reading early, but I wanted to do phonics, not whole word.  I taught her the alphabet - names and letter sounds - when she was a toddler.  She picked them up really quickly, and I thought she'd start reading young, too, but she never made the leap from letters making sounds to sounding out the letters in words.  I did informal sounding out stuff starting around age 4, and that was the year she started recognizing whole words in her environment and getting meaning out of text.  But she still wasn't making the connection between /c/ /a/ /t/ and /cat/, nor between 'c' 'a' 't' and 'cat'.  I started teaching her phonics at 5.5 and it was ridiculously slow going.  She'd known her letters and sounds since before she was 2, but we spent an entire year on CVC words.  I felt so guilty about being so strictly phonics-only, because she was like the poster child for whole language: she resisted phonics, but she naturally used picture clues and context clues and grammar clues.  But when she was 6.5 something clicked and lessons started going faster.  A few months later something else clicked and she made the leap to independent reading.   At the time I thought phonics finally clicked.  Turned out that what clicked was the ability to learn from phonics teaching despite not having the ability to learn to read phonetically (she flunked the Barton pre-screening, which is checking whether a student has the necessary phonemic awareness to learn phonics, as a fluent reader).  She managed to be an almost total sight reader despite strict phonics-only teaching, because she didn't naturally develop the developmental skills needed to learn phonics.  (I worked very hard to remediate her once I realized it.)  But she somehow was a fluent reader anyway. 

Maybe she could have learned to read early if I'd done the whole word approach of teach your baby to read (certainly it would have been earlier).  Or maybe if she'd developed the usual phonemic awareness at the usual times, she'd have been able to make the connection between letters and words earlier.  Or if I'd realized earlier that she *wasn't* developing the usual phonemic awareness at the usual times, and I deliberately remediated it earlier, she'd have read earlier.  But I wasn't really wanting to deliberately and formally work toward early reading.  (And all my kids have needed explicit and extensive phonemic processing work, and I've started later with each child.)

I'm of two minds wrt "does early reading provide a distinct advantage?"  I read early and prolifically and I never slowed up.  How is that *not* an advantage?  But my oldest was 3-4 years behind me, reading-wise, yet once she started reading, she hit the ground running and has never slowed.  She has the same effortless comprehension,  insane speed, and a huge vocabulary and knowledge base.  I see no real differences between our reading experiences at this point.  Reading widely and constantly - and reading fast - seem to be of greater importance than reading early.  (And I'm not sure how to foster those things, either.   Threw me off a lot when my middle loved to read and re-read old favorites, but resisted reading new things.  She's better now, but her knowledge base is a lot less than her sister's at the same age, and I think the lack of reading widely is a huge part.)

Although I do wonder about the intangibles of being an early reader.  Reading is like breathing for me - so effortless and ubiquitous and *necessary*.  I'm not 100% sure my intense relationship with reading is the best of all possible worlds, but it has shaped me so much.  How much is attributable to having pretty much *always been reading*?

Edited by forty-two
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21 hours ago, medawyn said:

I needed to read this exact thing today, because I came perusing this thread to ask for concrete ideas to help teach subtraction.  My oldest is balking at subtraction, and I'm realizing I just need to let him internalize it much more.  I am completely confident that he has internalized place value and addition, but I'm realizing that he needs to spend more time with the relationship between subtraction and addition and just plain more time with subtraction.

Any ideas to help him practice subtraction?  I am wildly fascinated by math because my own education was so lacking, but it makes me insecure in teaching things I probably know better than I feel.

 

Initially teaching subtraction as "finding the difference" or "comparison" rather than "taking away" has really helped my dc understand it conceptually.  For my kindergartner, initial exercises in subtraction involve lots of comparison using cuisinaire rods:  "6-4=?" means "what is the difference between 6 and 4?" or "the difference between 6 and 4 is the same as what?" We then place a green "6" c-rod alongside a shorter purple "4" c-rod and see what rod shows their difference (ie, what rod fits in the space above the 4 rod, to make it as tall as the 6 rod).  A red "2" rod! 

The result is a concrete visual number bond (6 is the whole, 4 and 2 are the parts, and the whole is the same height as the combined parts.) With one c-rod model, you can concretely teach four mathematical sentences and their relationship to each other: 2+4=6; 4+2=6; 6-2=4; and 6-4=2.   

Once my kids were comfortable with comparison, and later with identifying whole vs parts, it was much easier to understand subtraction as "taking away."  Wish I'd known all this when I first started teaching my oldest! 

Check out educationunboxed.com for free videos that probably make a lot more sense that what I just typed.  🙂

Edited by lots of little ducklings
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2 hours ago, lots of little ducklings said:

 

Initially teaching subtraction as "finding the difference" or "comparison" rather than "taking away" has really helped my dc understand it conceptually.  For my kindergartner, initial exercises in subtraction involve lots of comparison using cuisinaire rods:  "6-4=?" means "what is the difference between 6 and 4?" or "the difference between 6 and 4 is the same as what?" We then place a green "6" c-rod alongside a shorter purple "4" c-rod and see what rod shows their difference (ie, what rod fits in the space above the 4 rod, to make it as tall as the 6 rod).  A red "2" rod! 

The result is a concrete visual number bond (6 is the whole, 4 and 2 are the parts, and the whole is the same height as the combined parts.) With one c-rod model, you can concretely teach four mathematical sentences and their relationship to each other: 2+4=6; 4+2=6; 6-2=4; and 6-4=2.   

Once my kids were comfortable with comparison, and later with identifying whole vs parts, it was much easier to understand subtraction as "taking away."  Wish I'd known all this when I first started teaching my oldest! 

Check out educationunboxed.com for free videos that probably make a lot more sense that what I just typed.  🙂

 

I think all of the ways of introducing subtraction have potential conceptual pickles ;-). If you introduce subtraction as "taking away," then you're going to have a harder time with the idea of number bonds. If you introduce subtraction as "the difference," then you're going to need to be careful to make it clear to kids that subtraction has a "direction" -- that is, that 6 - 4 is not the same thing as 4 - 6. If you aren't careful about this, this definition can come back to bite you when you have to learn about negative numbers. 

I think the most important thing is to be mindful of what definition you're using (it should be clear in your own head) and you should know where the conceptual sticking points are. When I was deciding, I thought about it and decided that in "real life," we most often do use subtraction in the context of taking away, and that I wanted to be able to distinguish between 4-6 and 6-4 early. As it happens, my daughter wound up learning about negative numbers at age 5, so our choice worked well for us. But I think both are feasible options! 

 

Edited by square_25
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6 hours ago, drjuliadc said:

It is a very good point that some of “developmental readiness” is prior exposure dependent. I just wanted to point out that a lot of people do teach one year olds how to read. 

It isn't a welcome topic on this board.

I'll go ahead and say something even less PC, which is that the rest of "developmental readiness" is actually intelligence.  Children with higher intelligence are generally "developmentally ready" to do more advanced intellectual tasks earlier.  Not being developmentally ready is a nice way of saying that the student isn't smart enough to learn whatever it is yet (and they may never be--I firmly believe that this is why average achievement levels off in high school).

In other words, if the student has the intelligence to do the thing but seems as though they're not ready, it means that they haven't been educated properly up until that point.

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21 minutes ago, EKS said:

I'll go ahead and say something even less PC, which is that the rest of "developmental readiness" is actually intelligence.  Children with higher intelligence are generally "developmentally ready" to do more advanced intellectual tasks earlier.  Not being developmentally ready is a nice way of saying that the student isn't smart enough to learn whatever it is yet (and they may never be--I firmly believe that this is why average achievement levels off in high school).

In other words, if the student has the intelligence to do the thing but seems as though they're not ready, it means that they haven't been educated properly up until that point.

 

On the one hand, I think you have a point. On the other hand, some kids are just wildly uneven, so "intelligence" isn't a useful way to think of it.

My older girl is just uniformly good academically -- reading came easily, math came easily, advanced thinking is coming easily. She's what I think of a standard high IQ child. My second girl, on the other hand, is so far really asynchronous. She was my precocious talker -- she had 50 words by 15 months, spoke in full sentences by 18 months, and was speaking in complicated and completely grammatically correct sentences by 2. But she was also my late walker -- she walked at 15 months, at the same time as she was really starting to talk. I couldn't put her in a grocery cart without her tipping over until she was older than a year (I have some really amusing pictures of the amazing tipping baby, lol.) So on, so forth... 

My older girl learned all of her letters by age 2. My verbally precocious younger girl, who is now 3.5, is still struggling with the alphabet. She begged to learn to read, because her older sister does, and because she loves listening to stories. Her phonemic awareness is amazing, so we started reading lessons. Overall, she's doing very well... but we've been fighting with recognizing b's and d's for something like 2 months now, and it's is just NOT clicking. She can now do it if she really focuses, but it's not easy. 

This is to say that I'm not at all sure that my younger girl is "less intelligent" than my older, but sometimes asynchronous kids just don't have all the pieces in place at the same time. 

Edited by square_25
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18 minutes ago, square_25 said:

"intelligence" isn't a useful way to think of it.

 

It is not. And not merely because of PC nonsense 🙄 

I have an intellectually gifted child, a child with processing delays, and a straight-down-the-middle. I myself am gifted. I myself struggled with math in school. Womp,womp if only my brains were better 😐. Spoiler alert: gifted kids don't always get everything right away either, as both Square and I have iterated, in so many words.

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2 hours ago, square_25 said:

"intelligence" isn't a useful way to think of it

 

2 hours ago, OKBud said:

It is not.

It isn't useful if you think of intelligence as being fixed or global. 

I have a kid who at age 7 tested as having a below average IQ.  He couldn't read and couldn't count reliably to ten.  After intensive remediation, his IQ (GAI) was approaching the highly gifted range with processing speed at the 2nd percentile.  I get it about asynchronous.  I still say that when a kid "isn't ready" it means that in that moment they aren't smart enough.  It may or may not have anything to do with potential.

Edited by EKS
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This is a really helpful, interesting thread, especially so because I've decided this is my Year of Math in which I will focus on learning to teach elementary mathematics. For moms like me who are trying to find the time to learn or relearn as they go and have several students to keep track of, I wanted to share something that has been working really well for me this year: Instead of having each kid work through a separate math program, I've decided we're all working on the same big topic at once (my students are 4, 6, 9, and 11). This has made it way easier for me to remember what everyone is working on and also go deeper in my own preparation. Rather than prepping a few pages ahead in several different math texts with everyone working on different skills and topics (or, let's be honest, by the end of the term, teaching more reactively as holes in my kids' understanding become apparent), I'm doing a deep dive just on teaching fractions at various stages and from various angles. So I'm doing all my 11yo's problems, working through the fractions chapter in Elementary Mathematics for Teachers, studying the sections on fractions in one of Marilyn Burns' books, reading math teacher blogs, studying Montessori albums on introducing fractions in early childhood, etc, etc. I'm also finding that with just one big topic I'm teaching, it's on my mind more and I think to bring out examples from our "real life" and also just talk more math together at the dinner table and so on. 

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3 hours ago, square_25 said:

 

I think all of the ways of introducing subtraction have potential conceptual pickles ;-). If you introduce subtraction as "taking away," then you're going to have a harder time with the idea of number bonds. If you introduce subtraction as "the difference," then you're going to need to be careful to make it clear to kids that subtraction has a "direction" -- that is, that 6 - 4 is not the same thing as 4 - 6. If you aren't careful about this, this definition can come back to bite you when you have to learn about negative numbers. 

I think the most important thing is to be mindful of what definition you're using (it should be clear in your own head) and you should know where the conceptual sticking points are. When I was deciding, I thought about it and decided that in "real life," we most often do use subtraction in the context of taking away, and that I wanted to be able to distinguish between 4-6 and 6-4 early. As it happens, my daughter wound up learning about negative numbers at age 5, so our choice worked well for us. But I think both are feasible options! 

 

 

Great point; if you opt for a comparison approach, it's very important to help the student establish the concept of whole/parts.  I didn't find this to be any problem at all; the c-rods illustrate this well (much better than bubble-style number bonds, which are still enough of an abstraction that my oldest ds, my dyscalculia kid, stumbled over them until I discovered c-rods).  And of course you do move on to teach the concept of subtraction as "taking away."  But I found that the value in teaching subtraction as finding the difference resides in the number bond created (and the relationship between addition and subtraction that that entails).  This allowed my ds to finally understand multiple digit subtraction with borrowing and tackle word problems successfully.  

"Take-away" was the language with which I was taught, and was never a problem for me as a kid, but I intuitively developed enough number sense to succeed with it.  My ds simply didn't/couldn't.  

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4 minutes ago, lots of little ducklings said:

But I found that the value in teaching subtraction as finding the difference resides in the number bond created (and the relationship between addition and subtraction that that entails).  This allowed my ds to finally understand multiple digit subtraction with borrowing and tackle word problems successfully.  

Interesting! Do you know how it helped? I haven't actually found kids having an easier time with "missing addend" problems than they do with taking away. With my daughter, as I mentioned, we had to work on the idea of number bonds (and I agree that the bubble-style thing is far too abstract!) after we actually figured out how to do multiple digit subtraction with borrowing. The two were basically separate tracks for us :-). So maybe in some sense I did do something like you did, I just didn't give it a symbol... because we did always do number bonds. 

For multiple digit subtraction with borrowing, we simply used manipulatives and took away in whatever order we felt like it. She always wanted to take the tens away first, by the way :-). 

Edited by square_25
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On 1/15/2020 at 10:09 PM, square_25 said:

Interesting! Do you know how it helped? I haven't actually found kids having an easier time with "missing addend" problems than they do with taking away. With my daughter, as I mentioned, we had to work on the idea of number bonds (and I agree that the bubble-style thing is far too abstract!) after we actually figured out how to do multiple digit subtraction with borrowing. The two were basically separate tracks for us :-). So maybe in some sense I did do something like you did, I just didn't give it a symbol... because we did always do number bonds. 

For multiple digit subtraction with borrowing, we simply used manipulatives and took away in whatever order we felt like it. She always wanted to take the tens away first, by the way :-). 

 

I've thought a bit about this, and think it may be that the c-rods presented the clear, visual part/whole relationship that had been missing elsewhere.  

For example, if I had a dozen cookies (or beans, or whatever else was used as a manipulative) and took away 8, DS could see the whole "12", and THEN see the parts "4" and "8" (once we'd done the operation), but he could not visualize the parts side-by-side with the whole AT THE SAME TIME, to recognize that part+part was equal to whole in a subtraction operation, just like it was in addition.  At a very basic level, he was missing that understanding.  Finding the difference (using c-rods, which was a major game-changer for us) helped bridge that gap.  It allowed him to see a part, a whole, and the difference between them at exactly the same time (and in the same bond he'd been using for addition).  

I think, anyway!  LOL. DS is incredibly bright and motivated, but his learning style is the opposite of me in so many ways, and exploring the landscape of his mind has been a fascinating journey of trial and error.  He is also dyslexic, so I often wonder if that process of breaking things into their component parts (whether words into phonemes, or mathematical operations into wholes/parts) is really at the heart of the challenges he faces.  

 

Edited for clarity.

Edited by lots of little ducklings

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1 hour ago, lots of little ducklings said:

 

I've thought a bit about this, and think it may be that the c-rods presented the clear, visual part/whole relationship that had been missing elsewhere.  

For example, if I had a dozen cookies (or beans, or whatever else was used as a manipulative) and took away 8, DS could see the whole "12", and THEN see the parts "4" and "8" (once we'd done the operation), but he could not visualize the parts side-by-side with the whole AT THE SAME TIME, to recognize that part+part was equal to whole in a subtraction operation, just like it was in addition.  At a very basic level, he was missing that understanding.  Finding the difference (using c-rods, which was a major game-changer for us) helped bridge that gap.  It allowed him to see a part, a whole, and the difference between them at exactly the same time (and in the same bond he'd been using for addition).  

I think, anyway!  LOL. DS is incredibly bright and motivated, but his learning style is the opposite of me in so many ways, and exploring the landscape of his mind has been a fascinating journey of trial and error.  He is also dyslexic, so I often wonder if that process of breaking things into their component parts (whether words into phonemes, or mathematical operations into wholes/parts) is really at the heart of the challenges he faces.  

 

Edited for clarity.

 

Oh, yes, DD needed the actual visual to figure out the relationship between addition and subtraction. But DD could calculate differences before that, just not using that relationship. These turned out to be disjoint tools in the toolbox :-). 

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On 1/15/2020 at 12:56 PM, sweet2ndchance said:

Not the person you were asking but I do have a child who learned to read as a toddler. I didn't teach her though, her sister did.

My now-18 year old daughter saw the "teach your baby to read" infomercials for the first time when we returned to the states when she was about 6yo. She was just learning to read herself at the time and made it her mission in life at that time to teach her baby sister, then only about 1.5yo or so. So every day she had her phonics lesson and then she would play school with her baby sister and teach her the same concepts she had learned. Great practice for older dd to solidify her phonics skills by teaching another person. She's also very much the mother hen and loves little kids (and still does). I wasn't really concerned with whether or not younger dd actually learned anything from these phonics lessons her sister gave her at the time. I also never encouraged or discouraged these lessons. I just kept an ear on making sure there weren't any bad habits being taught. Amazingly, older dd was a wonderful and patient reading tutor. Even at 6yo and only teaching concepts she had barely mastered herself.

But learn to read younger dd did. By age two she could read on a kindergarten level (cvc words and such mostly). By the time she was 5 years old and starting kindergarten herself, she could read on a second grade level. By second grade, she maxed out the test at a 7th grade level. Could she have read on a higher level earlier? Maybe but I wasn't interested in pushing her at a young age. I let her go completely at her own pace. If she wanted help sounding something out, I helped her. Other than those "lessons" with her sister as a toddler and preschooler, she never had phonics or reading lessons. She picked up reading on her own after those first lessons from her sister.

Just some observations and other things, younger dd was a very verbal child. By 18 months, she regularly stunned me with her vocabulary and sense of language. Her older brothers and sister were saying things like "want more" and "all gone" and "play outside", 2 and 3 word phrases really, at that age. Ydd was not only speaking in complete sentences but using rather abstract vocabulary correctly. I was holding her in front of a mirror one day, practicing pointing out body parts, and she out of the blue says, " My shirt is very beautiful".  Her verbal prowess only got better as she got older. I think she was predisposed to having no issues learning how to read. Would she have read as early as she did if her sister did not set out to teach her? I cannot say because I don't have another copy of ydd to try out that theory. If I had to guess, based on my experiences with teaching 6 very different kids to read who were all raised in the same environment, my answer would be yes, she probably would have picked up reading on her own, with or without lessons, before kindergarten. She would have been my second child to do so. The lessons from her sister absolutely helped her learn more quickly, but I think she would have picked it up regardless.

 

That's a very cool story!! How amazing that your daughter was a good teacher at age 6 :-). 

I only have two kids, so my sample size is smaller... but I will say that my more verbally precocious child is also the one that's having more trouble learning to read. My older was very normal when it came to speak -- maybe very slightly ahead of schedule. My younger girl, on the other hand, spoke very early and had a 70+ word vocabulary at 15 months (the only reason I know the number that precisely is because I was gobsmacked by it at the time and wrote all of these words down in a text file, and I've recently found this file on my computer.) By 18 months, she was speaking in full sentences. By 2 years, she was speaking in completely grammatically correct sentences. sometimes ones with different clauses. If I remember correctly, her pediatrician said her speech was more than good enough for age 3 at that well check... I also remember that before the pediatrician came in, she said something like "I'll climb down when the doctor gets here" when I asked her to sit by herself ;-). 

On the other hand, as I mentioned upthread, she's my asynchronous child! My older girl learned her letters by age 2. My younger girl is still struggling with distinguishing b's from d's at almost 4, and she shows a strong tendency to see words together instead of breaking them down into letters. This is showing up in a very interesting pattern of strengths and weaknesses when it comes to reading. 

By the way, I would guess that my older girl could have probably learned to read before age 3. We started "100 Easy Lessons" at age 3, and the only trouble we had with it was that she was bored by the early lessons. However, all of the phonics was really easy for her. We sped through the book, and she's been reading ever since. My older girl is my very abstract-minded child: symbols and their interpretations come very easy to her.

Quote

Now, the question of early exposure and the results. I think it still comes down to every child is wired a little bit differently. All 6 of my kids have, more or less, been raised in the exact same literature rich environment. They all saw their parents read for information and pleasure regularly. They were all read to aloud from birth. They were all taught in, more or less, the same manner the basic phonics skills at some point before kindergarten. However, when they could actually read on their own from that exposure and instruction varied widely with no real rhyme or reason that I can discern other than we all have different strengths and weaknesses.

oldest ds was 9yo before he could read on his own

second oldest ds was 4yo when he could read on his own

older dd was 6.5yo when she could read on her own

younger dd 2yo when she could read on her own

younger ds was nearly 7yo when he could read on his own

youngest ds just turned 7yo this month and is teetering on the edge of reading on his own but still not quite there, he's been stuck at this point developmentally for about a year now

My definition of "reading on their own" is the ability to sound out most words on their own and only needing help with phonetic constructs that they have not encountered before. They can fluently blend sounds and decode words but may not know what every word they can read means. However, decoding words is no longer such a strenuous act for them that they cannot derive meaning or think about what they are reading as they read. 

So, all this to say, I think exposure and environment are important. It is important enough that I go out of my way to provide exposure to reading, literature and phonics from infancy. However, in my experience, it is not a determining factor for all children when it comes to how easy or difficult it will be for them to learn how to read. They will still read when they are developmentally ready to understand all the underlying skills for reading. Parroting rules and sounds and the like is not understanding. The light bulb has to click on in their head for true understanding to move them forward. Exposure and environment may help a child who is developmentally ready, but it will not push ahead a child who is just not wired to be developmentally ready to read.

As always, this is just my experience, YMMV.

 

To be fair, I wouldn't expect the same teaching to result in the same outcome for different students :-). I'm currently teaching my second child to read between 3 and 4, and while I would guess that both attempts will be successful, I've needed substantially different methods for the two of them.

For my older girl, we just went straight through "100 Easy Lessons," and everything was a piece of cake. My younger girl, on the other hand, is much weaker with symbol recognition, so she needs a lot more practice with the separate symbols than "100 Easy Lessons" gives -- this book introduces a new letter every couple of lessons, and a new symbol every two days simply wouldn't fly for her (even though we've, of course, been trying to learn letters for years now.) On the other hand, her phonemic awareness is very good for her age -- it's easy for her to break words up into sounds, it's easy for her to rhyme, it's easy for her to replace sounds in words, and I think she could pass the Barton pre-screening if I were able to explain it to a 3 year old :-P. As a result, we've had to really modify the program -- we're only doing "official" reading lessons on the weekend, and in between, we're practicing on words and sentences in books. And she's definitely progressing in a way which makes me feel like she'll be reading a bit after 4! For example, she's picking up words she can read quickly no slower than her older sister, and the fact that she's very orally focused makes it easy for her to string the words together and make sense out of them. For some reason, it's also easy for her to learn new letter combinations. 

Anyway, this is all to say that if we'd done the same thing with DD #2 as we'd done with DD #1, it would have been an unqualified disaster, and yet she's also a verbal and precocious child who's probably able to learn to read early! All of this teaching stuff is so fascinating... I'm so glad I get to homeschool and explore it all. 

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I've been pondering on this thread since it was posted, but today is the first time I've had 15 minutes and some quiet to respond.  My math education was not great - I learned to do the work but I'm confident that I didn't learn it as well as I could have and there was little conceptual understanding unless a light bulb clicked on randomly.  I wanted more for my kids.  My older was shockingly good at math at an early age.  It started with counting blocks as we put them away or counting how many times he could dribble a basketball (100 was always his goal).  Then we'd divided the toys to be put away in 1/2 and we'd each do 1/2, then I'd say 'What if Grammy was here?  How many would each of us put away then?', etc. We worked through oral addition, subtraction, and simple multiplication and division facts.  I introduced variables, first with 'If I needed 11 cookies but had 5, how many more would I need? and then 'So if I said 5 + x = 11...'.  Kiddo grasped negative numbers intuitively, and a friend taught him the idea of perfect squares over Christmas dinner.  All of this was in pre-K, and all of it was oral.  We started with Singapore math in K and did grades 2 and 3, skipping some of the repetition.  The only snag was when we hit  long division - kiddo could not understand the algorithm, which startled me because I'd never really had to 'teach' anything - a few tips, or a look at the example problem and he was off and running.  

I ended up explaining long division by saying what we were actually doing - in 137/4, you put a 3 in the tens spot because it goes 30 times.  So, since you've accounted for 4 x 30, go ahead and subtract 120.  Now, how many times does 4 go into the remaining 17?.  Once that understanding was there, kiddo wrote a problem with several million divided by several hundred, asked if the same rules applied, and then did it.  We stalled for a while at pre-A, because we were using AoPS and, while this kid grasped the math, the attention to detail wasn't there so we kept losing exponents.  But, other than puzzling through some geometry together, this kid mostly teaches himself with occasional help.  He clearly 'sees' the math in a way that I don't.  Sometimes he gets stuck on hard word problems and other times carelessness is still an issue.  But, at no point have I ever read one of those 'trains leaving going at different speeds, what time do they meet' questions and just 'seen' the answer.  I can draw a picture, label it, write 2 expressions and then set them to be equal to solve the problem but I'd never just read it and know the answer.  Right now we use a mix of AoPS and LOF and my kid likes that the same material can be taught so differently and that it's mostly self-teaching.  I have put effort into teaching him how to write what he's just 'seeing' because, well, you need to learn to write math.

My younger is not like this.  She's actually really good at math and loved the way that Singapore teaches adding by forming tens (rather than learning addition facts) and felt the same about subtraction (Singapore teaches 'borrowing' differently than I was taught - if you have 23-9, for instance, and 'borrow' 10, you subtract 9 from the borrowed 10, then add the remaining 1 to the 3).  So, even my non-math-loving kid was better at mental math at 6 than I was after my years in school (where I was told to count on my fingers).  We spent a lot of time with cube blocks early on.  It wasn't done as a fun 'clean up the blocks' think like with older because this kid didn't think it was fun (either the cleaning or the counting).  But, as part of math, she was happy to use them and I think that it really helped her to see the tens, and see that 6x8 was 8x6.  Interestingly, this is also the kid who taught herself the FOIL method of doing multiplication.  We were doing 23 x 35 -type problems with the traditional set-up, and I said 'First you multiply the 5 x 3, then write the 5, put the 1 above the 2 to show that you'll add that 10 later...now multiply 5 x 2, which is really 20 because it's in the tens, then add that other 10.  Now multiply the 3x3, but it's really by 30...etc' and then she said 'So what I'm really doing is adding 15 + 100 + 90 + 600' and she insisted on writing it that way for a while.  I figure at least it will make a bit of Algebra easier.  ☺

But, this kid periodically hits a wall with fractions, particularly dividing by fractions.  Mechanistically she can multiply by the reciprocal, and if asked, when dividing by 3/4, she'll reason through first you multiply by 4 to figure out how many fourths there are, then you divide by 3 to see how many 3/4 there are...and then on the next problem she'll say that she doesn't understand why you do it that way, even though she just explained it.  I figure that time and practice will settle it in her brain eventually...I hope.  

With all of that being said, I've talked about my volunteer tutoring that I've done for the past 5 years.  There are a couple of issues that I see working with these kids. First, some don't have any buy-in.  They don't care if they understand, they just want to be done.  All kids can be that way, but if it's constant and is the norm, I think it can be very hard to get the kids to 'wallow in it' enough to open their brains and let the concepts seep in.  Something that I hadn't considered until I worked with these kids is that, without excellent classroom control, you can't use manipulatives at all.  I love my unit blocks, but with some of the groups they would have become projectiles, or they would have stolen them.  It's not a problem in my home teaching, obviously, but it was an issue that I hadn't considered.  This may be why the kids spend so much time drawing to regroup.  One one hand, I see the point, but I am finding that the kids become dependent on counting and quit thinking.  Like, they'll draw 15 dots and then 16 dots and then count them all, and if they make a mistake they can't find it because the only way to check is to recount and it's tedious.  

I also struggle with how they are being taught several methods to do the same thing.  Singapore math does it too, but it seems to do it differently (although it may be that I was just doing it with my own kids).  Without conceptual understanding, the kids don't realize that they are doing the same thing in a different way and they get really confused.  If you check their work and they see that you've arrived at the answer a different way than how they were taught, even if it's a way that they've done before, they think you're answer is wrong because they don't necessarily understand that you get the same answer no matter how you do it.  There are also a lot of issues, as in every group setting, of moving on before some kids understand the basics.  I can not overstate the awfulness of trying to help kids with long division when they don't fully know their +, -, and x facts.  The number of times I've helped count groups of 6 going into 30something (not skip-counting, counting while holding up a finger every time we get to a multiple of 6)...it's almost like it's 'anti-conceptual' because I think the kids feel like the adults are just pulling this stuff out of nowhere.  

This goes very against the grain of this conversation, but there is a part of me that thinks that some kids would be better off not trying to move much beyond algorithmic arithmetic work, at least for a whil, and maybe ever.  I understand the problems with this, in that they'd be at a disadvantage if they want to move towards more complex math.  On the other hand, they wouldn't be much worse off than I was when I finished school and I made it through calculus.  🙂  I also have concern with how they'd figure out what to do with which kids - I have seen struggling K-2 students have a developmental leap and become adept at disassembling and reassembling numbers as they move in the 3-5th grade range.  But, I've also seen kids bang their heads against a wall for years on end, unable to completely wrap their heads around anything and unable to do simple arithmetic because they don't just learn the facts.  They might be kids who would learn the concept after using the facts for a while, or they might be kids who will never develop the abstract skills to move beyond concrete arithmetic, but either way they'd still be better off being able to do the arithmetic even if they don't understand it.  Some kids seem to be very poorly served by the 'once you understand it conceptually you can do a lot with it' because what seems to result, for them, is 'if you don't understand the conceptual background, all math will seem like magic and you won't be able to get the right answer, ever'.  

 

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1 hour ago, ClemsonDana said:

.....

This goes very against the grain of this conversation, but there is a part of me that thinks that some kids would be better off not trying to move much beyond algorithmic arithmetic work, at least for a while, and maybe ever.  I understand the problems with this, in that they'd be at a disadvantage if they want to move towards more complex math.  On the other hand, they wouldn't be much worse off than I was when I finished school and I made it through calculus.  🙂  I also have concern with how they'd figure out what to do with which kids - I have seen struggling K-2 students have a developmental leap and become adept at disassembling and reassembling numbers as they move in the 3-5th grade range.  But, I've also seen kids bang their heads against a wall for years on end, unable to completely wrap their heads around anything and unable to do simple arithmetic because they don't just learn the facts.  They might be kids who would learn the concept after using the facts for a while, or they might be kids who will never develop the abstract skills to move beyond concrete arithmetic, but either way they'd still be better off being able to do the arithmetic even if they don't understand it.  Some kids seem to be very poorly served by the 'once you understand it conceptually you can do a lot with it' because what seems to result, for them, is 'if you don't understand the conceptual background, all math will seem like magic and you won't be able to get the right answer, ever'.  

 

I just have to quote this^, because *I have two of those kids*.   And I have used more "conceptual" programs (Singapore, Miquon) with them in elementary.

And yes, sometimes all that wonderful, thoughtful, conceptual teaching just muddies the waters.  It becomes one. more. thing. about this whole math mess that they don't get. Again.  It can be very defeating to just not get it. Over and over and over. 

So, in my sample, 2 out of my 6 kids *sometimes in some things* have to learn the algorithm cold, and then get the concept much later.  Only one of mine so far has been truly able to see why dividing fractions is the same as multiplying by the reciprocal. The others did not care at all.  They just wanted to invert, multiply, and move on.  😉 

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Unsurprisingly, I don't agree that some kids just need to do the algorithm because they can't understand conceptual teaching. But I also think there's a misunderstanding of what "conceptual teaching" means. To me, teaching conceptually means teaching the important ideas, like place value, then giving kids tools to use those ideas. Those tools can be manipulatives, or visuals, or mental tricks, or whatever gets them over the hump. I've been using poker chips in my classes and I used visuals with my daughter, and frankly, I don't think it makes any difference (I'm not sold on the tactile component) -- what matters is that the tools you use actually evoke the ideas.

By the way, what I mean by "evoking the ideas" is actually surprisingly tricky, because kids have a tendency to do the easiest thing possible. After you do the first 3 questions asking kids "How many 10s and 1s are there in the following two-digit numbers?", you lose the ability to "evoke ideas," because kids are smart, and they figure out that you want the first digit as the number of 10s, and the second digit as the number 1s. Unfortunately, at that point they stop thinking about 10s and 1s and abstract grouping entirely! That means that after the first few questions, the question is no longer evoking the ideas, and is rather resulting in pattern-matching. To get rich conceptual experience with the ideas, you often need to see them from many sides. 

I also think there's a tendency to equate "teaching conceptually" with teaching algorithms conceptually. The idea seems to be that conceptual teaching is explaining an algorithm REALLY WELL so that the student gets it and thinks of it in a conceptual way. That was the model I used when teaching college kids; that's the AoPS model; that's the model almost all the programs use. And frankly, I think that model is extremely flawed. Kids can only understand algorithms after they have the rich conceptual experiences with the models and symbols and ideas that enables them to connect the algorithm to ideas they are already able to understand. And yes, some kids come in with the ability to see the ideas, or with experience with the ideas, or with the desire to understand the ideas, and that tides them over. But we don't actually provide the useful experiences in the context of classes. 

Anyway, this is to say that I don't teach algorithms until I'm sure we all speak the same language. And by the way, that has very little to do with math facts. There's nothing algorithmic about math facts -- you have to have a way to quickly retrieve addition and multiplication facts, and that's that, and I do work hard to achieve fluency. 

I'm glad you brought up fraction division, because we've been working on fraction division here, too :-). And I'm grateful to Liping Ma for introducing me to how to extend the partitive model of division to fractions. (The partitive model is the one where we say that 63/3 is the answer to "How do we split 63 cookies between 3 people?") Anyway, the way to extend this model is to say that in general, x/y is the answer to "How many cookies does 1 person get if y people get a total of x cookies?" (Yes, we're allowing fractional people here, lol.) If you think about it, that's exactly the "splitting cookies" model, so if your child started out with the partitive model (which my daughter has), it's an excellent way to extend it without causing definition fatigue. 

In any case, my daughter's been working through this definition and finding answers to all sorts of fraction divisions. I haven't managed to stump her yet, and yet she has NO CLUE that dividing by fractions is the same thing as multiplying by the reciprocal. To be honest, she still hasn't generalized her understanding of fractions to realize that 

a/b*c/d = ab/cd,

since she still works calculations like 3/7*6/5 like this:

"3/7*6/5 is... let's see, 1/7 of 6/5th is 6 times as much as 1/7 of 1/5, which is 6 times 1/35, which is 6/35, and we have 3 of these, so the answer is 18/35.”

At some point, she'll be ready for algorithmic methods for calculating operations on fractions, but by then, she'll have had enough rich conceptual experiences that she'll also know other ways to do it. For instance, it's a little silly to do 

1/2 * 4/25 = 4/50 = 2/25, 

since you can just notice that half of 4 of anything is 2 of that thing. 

This is a long-winded way of saying that so far, I've let her work with the ideas until she was ready to generalize herself, and I've been extremely happy with the results. Her understanding of basic arithmetic operations is solid, fluent, and rich. She can explain practically any operation she uses and practically any rule she applies. Frankly, her understanding of arithmetic is miles ahead of most college freshmen I've taught, and I've taught smart kids at fancy schools. And when I finally did teach her the addition, subtraction, and multiplication algorithms, it came ridiculously easy, because we were speaking the same language. (And she now loves the algorithms, lol! She says they are like little calculators making her calculations easier.) 

5 hours ago, ClemsonDana said:

Interestingly, this is also the kid who taught herself the FOIL method of doing multiplication.  We were doing 23 x 35 -type problems with the traditional set-up, and I said 'First you multiply the 5 x 3, then write the 5, put the 1 above the 2 to show that you'll add that 10 later...now multiply 5 x 2, which is really 20 because it's in the tens, then add that other 10.  Now multiply the 3x3, but it's really by 30...etc' and then she said 'So what I'm really doing is adding 15 + 100 + 90 + 600' and she insisted on writing it that way for a while.  I figure at least it will make a bit of Algebra easier.  ☺

 

That's actually an excellent illustration. My daughter did the same thing, but since I was in no rush with the algorithms, I let her do this for a very long time. And as a result, it's so well internalized that I actually didn't explain FOIL to her at all. She immediately wrote down 

(a+b)^2 = a^2 + 2ab + b^2

without any explanation as soon as I started algebra. In general, I've treated all of our arithmetic as a pre-algebra program, and again, I think it's paid off. 

I expect most people will disagree with me, but I've really had an excellent experience trying this method with my daughter. I've been trying out similar tools with my homeschool classes, which has been going well, except that weekly classes aren't quite frequent enough to really get us ahead enough. But I expect the data to be useful... stay tuned ;-). 

Edited by square_25

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I guess what I mean by 'not understanding this conceptually' is that I'm not sure that some of the kids that I've helped understand that 23 is actually 2 tens and 3 ones.  I mean, if they are given the question 'How many tens and how many ones' or 'What is in the tens place' they can answer it, and they can say that 923 is 900 + 20 + 3, but then if you say that 23 x 6 involves multiplying 6 x 3 and then 6 x 20 they'll argue that it's not 20, it's 2, so the answer is 18 + 12 instead of 18 + 120...and then when you go to 23 x 16 its a hot mess.  They don't understand math enough to think through the idea that 6 23s has to be bigger than 18 + 12.  I was helping some with a multiplication approach where they had big boxes labeled 'hundreds, tens, and ones' and when they multiplied by a single digit they drew 3 items in the ones and then 2 items in the tens, and then drew those 6 times each, and then circled groups of ten and drew an arrow showing that they moved one over to the next box when they got to 10.  They became very adept at doing this, but most saw it as a completely different thing from multiplying and carrying over the 10 or 100.  Maybe it's not a problem with teaching concepts but in treating the concept work like another algorithm...if the kids are just going to follow steps in order, they might as well be efficient steps!  I do remember with one kid trying to teach adding by using pennies for ones and dimes for tens, and that did not compute at all.  With no textbooks and no example problem worked at the top of the homework, it's hard to know how to explain it using the exact set-up that the teacher uses, and the kids are adamant that if they do anything different, it will be marked wrong (they are probably right about that).  I have had multiple kids tell me 'You do it your way and I'll do it my way' and if we get different answers that's OK because of course different methods give different answers...like they don't understand that there is one answer for 23 x 16 and multiple ways to get it.  Honestly, having taught my kids and been a homework helper (both officially volunteering and occasionally helping friends' kids), my volunteering with these kids has been the most eye-opening education experience of my life, for many reasons.  

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5 minutes ago, ClemsonDana said:

I guess what I mean by 'not understanding this conceptually' is that I'm not sure that some of the kids that I've helped understand that 23 is actually 2 tens and 3 ones.  I mean, if they are given the question 'How many tens and how many ones' or 'What is in the tens place' they can answer it, and they can say that 923 is 900 + 20 + 3, but then if you say that 23 x 6 involves multiplying 6 x 3 and then 6 x 20 they'll argue that it's not 20, it's 2, so the answer is 18 + 12 instead of 18 + 120...and then when you go to 23 x 16 its a hot mess.  They don't understand math enough to think through the idea that 6 23s has to be bigger than 18 + 12. 

 

I would also be utterly unsurprised if some of your kids don't really know that 6*23 is six 23s! Kids get shockingly minimal instruction in what symbols actually mean. I have a number of kids in my homeschooling class who actually can't describe to me what something like 45 - 34 means in words (or in visuals, or in anything... all they can say "It's 45 minus 34.") This is why I'm so focused on definitions -- before a kid can work with an operation, it's very helpful for them to have a concrete mental model of what it IS. Otherwise, kids aren't reasoning -- they are shuffling symbols.

8 minutes ago, ClemsonDana said:

Maybe it's not a problem with teaching concepts but in treating the concept work like another algorithm...if the kids are just going to follow steps in order, they might as well be efficient steps! 

 

Oh, absolutely. I actually don't think of that kind of homework as conceptual (even though lots of people do), because after you do it once or twice, it becomes completely rote. Plus, most kids didn't have to come up with that way of doing it themselves, anyway, so it's not like they even had to THINK about what multiplication meant to do it. In my opinion, it's best to just slowly work up from "m*n is what you get if you add m copies of n" and work up from there, making sure the model is completely internalized before you provide any shortcuts whatsoever. 

As I mentioned upthread, one thing I learned from my AoPS script rewriting experience is that kids WILL do the thing that requires the least amount of work most of the time. If you want kids to internalize a mental model, you had better provide work that really taxes that model and can't be done with a shortcut. This is why I wind up just doing addition and subtraction with no algorithm -- if you actually have to work with the 10s and 1s (which all of the kids in my classes have been able to so far, by the way), it seriously solidifies your understanding. However, it's also incredibly slow going -- you have no idea how many times I've had kids get "stuck" on having to take away 4 one chips from a single 10 chip! There's a really conceptual leap there, and personally, I try to respect conceptual leaps. 

14 minutes ago, ClemsonDana said:

Honestly, having taught my kids and been a homework helper (both officially volunteering and occasionally helping friends' kids), my volunteering with these kids has been the most eye-opening education experience of my life, for many reasons.  

 

And demoralizing, I'm sure. I've done a lot of classroom teaching (both online and IRL) in the last 12 years or so and that has also been eye-opening and demoralizing. I had extremely smart, articulate, logical kids who had made it to Stanford, and who had NO CLUE how to reason about mathematical operations at all. I don't think they believe mathematics actually involved reasoning. I'm pretty sure they thought it was a set of procedures handed down from above... 

 

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Hah - funny enough, they get that 23 x 6 is 6 23s.  I've never had a problem with them getting that, but then they want to figure it out by counting by 23s, or drawing 6 groups of 23 and then counting them all.  They understand that drawing and counting works, but when its not practical any more....ugh.  

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3 minutes ago, ClemsonDana said:

Hah - funny enough, they get that 23 x 6 is 6 23s.  I've never had a problem with them getting that, but then they want to figure it out by counting by 23s, or drawing 6 groups of 23 and then counting them all.  They understand that drawing and counting works, but when its not practical any more....ugh.  

Ah, cool! That's definitely a good thing :-). That means they are clear on at least one definition, lol. 

I've taught the kids in my classes pretty quickly and successfully to work with 10s and 1s in poker chip form, so I would guess that if you persevered, you could use that approach (perhaps with dimes and pennies, although maybe those are somewhat overloaded symbols and harder to use.) It sounds like they have ingrained counting habits, which can be really hard to break, so I'm sorry about that. I don't mind kids counting for small additions before they remember the facts (and I like counting on as a strategy), but once we get into place value, I'm very firm on not wanting to count large numbers and instead dealing with 10s and 1s separately. 

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1 minute ago, ClemsonDana said:

Well, I can hardly blame them for counting when they're forced to draw so much, for years on end.  


Yeah, I remember your descriptions from another thread — it sounded absurd. I think kids should almost never be drawing things above ten 😕 

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17 minutes ago, square_25 said:

By the way, what I mean by "evoking the ideas" is actually surprisingly tricky, because kids have a tendency to do the easiest thing possible. ...That means that after the first few questions, the question is no longer evoking the ideas, and is rather resulting in pattern-matching. To get rich conceptual experience with the ideas, you often need to see them from many sides.

I've found this pattern matching thing to be a serious problem, and not just in math.  

20 minutes ago, square_25 said:

To me, teaching conceptually means teaching the important ideas, like place value, then giving kids tools to use those ideas.

The problem with teaching conceptually is that the teacher must really understand those concepts themselves and be invested in ensuring that the student gets the tools to use them.  It means that the teacher is able to see when the student is pattern matching and being able to switch gears and go off book to get the student thinking again.  This can be a tall order for a parent who may have struggled with math themselves and who has been avoiding it their entire adult life.  It takes a huge commitment to bring one's understanding up to the "profound" level necessary to do what you're talking about, and this is why so much discussion regarding math education centers on programs rather than methods--people are leaning on the program to do the heavy lifting for them and hoping for the best.  I get it because at one time I was one of these people.

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I'm not sure how to ask this question, so if it comes off wrong, please know I'm not being snarky or trying to stir things up, I'm honestly wondering 🙂

I am one whose own math education was seriously lacking in conceptual understanding. I made A's in math through high school and even college (1 semester each of calculus, algebra, and stats) because I could easily memorize the algorithms. It wasn't until I started hs'ing and I went through MM and Video Text with my kids that I started to understand why the algorithms work.

I really appreciate the thoughts in this thread (and others) that encourage us as hs moms to continue to study and educate ourselves so we can give our best to our kids. I do that, and I hope others do too. I learn a lot from those of you who are more well educated than I am and I'm thankful for your shared wisdom!

But a little bit of the flavor of some responses is quite intimidating to someone like me. Does the fact that I don't know math inside and out mean I'm not qualified to teach my own kids? Some of the responses remind me a little bit of the ps teachers/hs naysayers who look down on hs'ers because we don't have a teaching degree and wonder how could we possibly teach our kids if we're not an expert in every subject. Isn't that one of the things that's so great about hs'ing - that we don't have to be experts but can learn along with our kids and guide them even if we don't know a lot about the topic? Is there something different about math that makes this not true?

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23 minutes ago, Momto6inIN said:

I'm not sure how to ask this question, so if it comes off wrong, please know I'm not being snarky or trying to stir things up, I'm honestly wondering 🙂

I am one whose own math education was seriously lacking in conceptual understanding. I made A's in math through high school and even college (1 semester each of calculus, algebra, and stats) because I could easily memorize the algorithms. It wasn't until I started hs'ing and I went through MM and Video Text with my kids that I started to understand why the algorithms work.

I really appreciate the thoughts in this thread (and others) that encourage us as hs moms to continue to study and educate ourselves so we can give our best to our kids. I do that, and I hope others do too. I learn a lot from those of you who are more well educated than I am and I'm thankful for your shared wisdom!

But a little bit of the flavor of some responses is quite intimidating to someone like me. Does the fact that I don't know math inside and out mean I'm not qualified to teach my own kids? Some of the responses remind me a little bit of the ps teachers/hs naysayers who look down on hs'ers because we don't have a teaching degree and wonder how could we possibly teach our kids if we're not an expert in every subject. Isn't that one of the things that's so great about hs'ing - that we don't have to be experts but can learn along with our kids and guide them even if we don't know a lot about the topic? Is there something different about math that makes this not true?

To encourage you (and others who might be feeling intimidated).....I agree with the sentiments in your last paragraph.  Goodness, I pretty much stink at math about 1/2 way through alg 2.  For alg 2 and precal, I sit with my kids with the solutions manual out while they work through the problems and grade them as they go.  When they get things wrong, I ask them questions and help them think things through.  If that doesn't help, we sit down and work through the problem together and I have them explain to me what is going on at each step. 

Is it the way a math teacher would teach? Nope.  But, it has worked here.  I have managed to raise really strong math students who are way smarter than I am.  🙂 

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11 hours ago, Momto6inIN said:

Does the fact that I don't know math inside and out mean I'm not qualified to teach my own kids? Some of the responses remind me a little bit of the ps teachers/hs naysayers who look down on hs'ers because we don't have a teaching degree and wonder how could we possibly teach our kids if we're not an expert in every subject. Isn't that one of the things that's so great about hs'ing - that we don't have to be experts but can learn along with our kids and guide them even if we don't know a lot about the topic? Is there something different about math that makes this not true?

+1....but! This thread is specifically for people wanting to nerd out on math, people who think about math as having a certain place in life that it doesn't have across the board.  [not me lol]

You have a lot of experience homeschooling. You know what you're doing 🙂 

Edited by OKBud
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8 minutes ago, OKBud said:

+1....but! This thread is specifically for people wanting to nerd out on math, people who think about math as having a certain place in life that it doesn't have across the board.  [not me lol]

Agree!  On both pts!!

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I am enjoying this thread a lot, not because I know much of what any of you are talking about, but because I am loving the idea that even though I don't understand math now I still can and help my kids. I am one who did not get any instruction on math my parents believed that if we wanted to learn we would on our own, they had books upon books and we were supposed to just read them or let them know that we wanted to learn more and even then it was hit or miss if they helped us (either by getting a tutor or a new book even). As a result the most math instruction I had was at 13 and I wanted to know more so I went through 1/2 of Saxon Algebra 1/2 all by myself, it only lasted a few months but I learned a lot in that time. I don't conceptual math but I'm trying everyday to teach something I just don't get and I am having successful results. I have found a math curriculum that is teaching me how to teach and I am pleased with it and I hope to better myself for my last 2.

Anyways keep discussing because I am learning things and I am excited to see where you guys go in this conversation!

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I'm going to quote slightly out of order: I hope that's OK! 

On 1/18/2020 at 9:33 PM, Momto6inIN said:

Some of the responses remind me a little bit of the ps teachers/hs naysayers who look down on hs'ers because we don't have a teaching degree and wonder how could we possibly teach our kids if we're not an expert in every subject. 

 

Honestly, public school teachers can't talk! Most of them aren't teaching anything like conceptually. So I'm not on their side in the least. 

On 1/18/2020 at 9:33 PM, Momto6inIN said:

Isn't that one of the things that's so great about hs'ing - that we don't have to be experts but can learn along with our kids and guide them even if we don't know a lot about the topic? I

 

To be honest, I think the homeschooling community oversells the "You don't need to know it to teach it!" aspect of homeschooling. Yes, I do think it's possible to learn along with our kids and do a good job, but one is still a better teacher for a subject if you a) know it inside and out and b) have practiced teaching it. I'm a very good math teacher because I have a good birds-eye view of high school and college math and because I've experimented with lots of different pedagogical practices. I'm a fine expository writing teacher, because I express my ideas in writing well. However, I'd be a mediocre creative writing teacher, because I'm no good at creative writing, haven't though much how to do it, and have never taught it before. 

On 1/18/2020 at 9:33 PM, Momto6inIN said:

Does the fact that I don't know math inside and out mean I'm not qualified to teach my own kids?

 

I don't think it means that, but I do think you'll do a better job if, as some posters upthread, you teach yourself pretty far ahead. The thing about math that makes it a bit harder than some other topics is that it's so sequential and has so much depth! I think it's useful to know arithmetic inside and out (including how the operations relate) and basic algebra quite well if you're going to teach K-6, because some of what you want to be communicating is a deep understanding and good habits, and that's hard to do if you aren't sure what's coming next. But I don't think that should be an insurmountable obstacle! I think @Lang Syne Boardie had excellent suggestions about how to go about this. I think reading Liping Ma is useful for anyone who wants to see some examples of how knowing the relationships between topics helps you become a better teacher. 

Edited by square_25
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On 1/13/2020 at 10:14 PM, square_25 said:

 

-- The dual meaning of subtraction (it's both "taking away" and "what's left over if we have so many") and the connection with addition. 

 

 

 

For me, and I will add the caveat that this is mostly from working with kids with very significant disabilities, what has worked best is helping my kids see that subtraction is about finding a part, and addition is about finding a total.

So, I might be looking for the part that is left after something is taken away, or the part that is taken away, or the part that I need to catch up, etc . . . but I'm always looking for a part.  On a number line, I might be looking for the number that describes the part I need to travel to jump from here to there.  Or the part that is left after I jump from here to there.  

I'm all about practical applications, given my population, and I find that asking a kid whether they're thinking about groups that are all the same, and whether they're looking for a part or a total is the most effective way for kids to be able to figure out what operation to use.  

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2 minutes ago, CuriousMomof3 said:

 

For me, and I will add the caveat that this is mostly from working with kids with very significant disabilities, what has worked best is helping my kids see that subtraction is about finding a part, and addition is about finding a total.

So, I might be looking for the part that is left after something is taken away, or the part that is taken away, or the part that I need to catch up, etc . . . but I'm always looking for a part.  On a number line, I might be looking for the number that describes the part I need to travel to jump from here to there.  Or the part that is left after I jump from here to there.  

I'm all about practical applications, given my population, and I find that asking a kid whether they're thinking about groups that are all the same, and whether they're looking for a part or a total is the most effective way for kids to be able to figure out what operation to use.  

We talk about putting together versus taking away. I think it’s the same idea, though!! The visual matters.

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On 1/18/2020 at 6:44 PM, ClemsonDana said:

Hah - funny enough, they get that 23 x 6 is 6 23s.  I've never had a problem with them getting that, but then they want to figure it out by counting by 23s, or drawing 6 groups of 23 and then counting them all.  They understand that drawing and counting works, but when its not practical any more....ugh.  

 

I tell my students that math is the one area in life when the goal is to be lazy.  That counting is a strategy that solves all sorts of problems, if you have enough time, but realistically no one has the time to solve problems that way, so mathematicians have developed strategies to get the job done with a lot less work.  For example, if someone asked me how much further is it to Saturn than to the moon, I could count out 1,200,000,000,000 poker chips one for each meter, and then take away 384,400,000 of them and then count what's left and then they could visit me in the "old lady home" to find out the answer.  OR, I could learn how to subtract multidigit numbers today and tell them tomorrow. See how lazy that is?   Then when they fall back on wanting to count, I just tell them they aren't being lazy enough, which usually makes them laugh. 

Edited by CuriousMomof3
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