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Teaching properties of numbers.

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1 hour ago, square_25 said:

Do you ever find that presenting multiple methods ever confuses kids? I ran into this issue at some point a few year ago with subtraction: I wanted my daughter to think about questions like 73 - 1 and 73 - 69 in different ways, in the sense that one is easier as "what's left over?" and the other is easier as "what do we take away?" and I found her starting to apply algorithms in a rather confused way. I wound up backtracking and just teaching a single method (basically, take away the second number from the first, end of story, you may need to draw a picture and regroup) and she did a lot better conceptually than she did when I tried doing multiple methods from the get-go. Of course, sometimes it was less efficient, but it just made more sense to her. 

By now she can fluently do both, but she had to have a solid definition that she always worked with first, and it surprised me how long it took to get the definition fully absorbed. Of course, she was quite little at the time. On the other hand, I've seen very similar issues in middle school and high school students, where we don't allow them enough time to play around with the basic idea before we provide the tricks, and that makes them more confused as opposed to less. 

 

My daughter has dyscalculia and multiple methods didn't confuse her because she was used to me presenting multiple methods. It gave her greater mental flexibility, which she sure needed. I think the understanding that the same concepts can show up in different configurations is incredibly important. Her understanding of that helped rather than hindered because she's a bright girl who would get bored with the overfamiliarity of one method and would be completing assigned tasks by rote before she had actually learned the maths. 

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38 minutes ago, Rosie_0801 said:

 

My daughter has dyscalculia and multiple methods didn't confuse her because she was used to me presenting multiple methods. It gave her greater mental flexibility, which she sure needed. I think the understanding that the same concepts can show up in different configurations is incredibly important. Her understanding of that helped rather than hindered because she's a bright girl who would get bored with the overfamiliarity of one method and would be completing assigned tasks by rote before she had actually learned the maths. 

Interesting! We definitely do a variety of problems in which subtraction shows up... but somehow it helped for my daughter to have a solid understanding of what the operation means before starting to play with how to use it. I guess when we defined it, we just did a bunch of calculations with it, some of which required regrouping and some didn't. So maybe not so much variety at first. I suppose our only "algorithm" was drawing the picture and crossing out the number that we're taking away. 

I have no experience with dyscalculia, though. How does hers manifest? Does that mean she has trouble understanding numbers?  

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Sometimes we tend to overcomplicate the entire "what curriculum to use" scenario. Curriculum can be used as a guiding tool that you just teach according to the needs of the child sitting in front of you. I dont use anything structured at all for any subjects at the elementary age other than their math textbooks and then teaching math is discussing what is going on.  It is equally fun to sit back and let them explain things to you. 

If you have the ability to teach vs needing a scripted curriculum bc you have no idea what you are doing, use the curriculum as your guide for problems and target skills and just teach concepts in a way that connects with your child. It is actually a far better approach than relying on a textbook to teach.

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2 hours ago, square_25 said:

Do you ever find that presenting multiple methods ever confuses kids? I ran into this issue at some point a few year ago with subtraction: I wanted my daughter to think about questions like 73 - 1 and 73 - 69 in different ways, in the sense that one is easier as "what's left over?" and the other is easier as "what do we take away?" and I found her starting to apply algorithms in a rather confused way. I wound up backtracking and just teaching a single method (basically, take away the second number from the first, end of story, you may need to draw a picture and regroup) and she did a lot better conceptually than she did when I tried doing multiple methods from the get-go. Of course, sometimes it was less efficient, but it just made more sense to her. 

By now she can fluently do both, but she had to have a solid definition that she always worked with first, and it surprised me how long it took to get the definition fully absorbed. Of course, she was quite little at the time. On the other hand, I've seen very similar issues in middle school and high school students, where we don't allow them enough time to play around with the basic idea before we provide the tricks, and that makes them more confused as opposed to less. 

I have not found that to be the case with my kids.  We teach multiple strategies, practice a specific one, then practice mixed issuing a strategy of choice.  I think in your example of whether to see the difference (73-1=?) or look at it like a missing subtrahend (eg 73 -?=69) is easily resolved by teaching number bonds/part-while models from the beginning.  I can say to my 1st header “are we looking for a part or a whole? Yes, and what’s the missing part?”  Or as easily she could think of subtraction as a model of comparison and see there’s only 4 difference between the two. No regrouping needed yet because 4 is a difference you can count on/back easily.

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2 hours ago, square_25 said:

 

Interesting!! That makes a lot of sense. What kinds of things were the teachers having trouble understanding, if you don't mind giving me examples? I don't have any firsthand Common Core experience.  

For the elementary student it was nearly everything outside the standard algorithm. I am not blaming teachers - most elem teachers don’t have to take conceptual math courses, they have new curriculum thrown at them regularly and with short notice, they have more than math (and too many students) to focus on, and after teaching math the X way for 15 years it can be hard to transition to the Y way.

For my high school student it was similar - teacher was a computational teacher and dd had been in years of conceptual math.  She made calculation errors at time (ADHD partly to blame) but she could explain why something didn’t turn out tightbonbthe board while teacher was lecturing (that didn’t go over well), or explain in clearer terms to her peers than the teacher did, because she understood the way the math worked, the concepts behind it.

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7 minutes ago, Targhee said:

I have not found that to be the case with my kids.  We teach multiple strategies, practice a specific one, then practice mixed issuing a strategy of choice.  I think in your example of whether to see the difference (73-1=?) or look at it like a missing subtrahend (eg 73 -?=69) is easily resolved by teaching number bonds/part-while models from the beginning.  I can say to my 1st header “are we looking for a part or a whole? Yes, and what’s the missing part?”  Or as easily she could think of subtraction as a model of comparison and see there’s only 4 difference between the two. No regrouping needed yet because 4 is a difference you can count on/back easily.

Would you teach multiple strategies before definitions, though? I like strategies after the basic definition is down. I suppose if you're talking number bonds, then you could define a -b to be "the number which added to b gives a" which is totally how I've taught vector addition to middle school and high school kids. But I've seen struggles with two-step definitions in that case: I have to remind the kids to check that the addition actually works a lot. 

I guess (and this is also based on teaching older kids) I worry about not having a firm definition first. The way I think about math, you have definitions, then you show that different things are really the same. Like, for example, what is a*b? You have to define it as a copies of b, I think. You can't just say it's also b copies of a, because that's not a definition. But you can say it's the same because here's why. 

I've gotten in the habit of giving a firm definition and letting students "play" with it. I find that sometimes the students find the strategies themselves, which can be more rewarding. But I might be overgeneralizing from fairly mathy kids. On the other hand, those same mathy kids had real trouble if they weren't given enough time to get a feel for a concept... 

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5 minutes ago, Targhee said:

For the elementary student it was nearly everything outside the standard algorithm. I am not blaming teachers - most elem teachers don’t have to take conceptual math courses, they have new curriculum thrown at them regularly and with short notice, they have more than math (and too many students) to focus on, and after teaching math the X way for 15 years it can be hard to transition to the Y way.

For my high school student it was similar - teacher was a computational teacher and dd had been in years of conceptual math.  She made calculation errors at time (ADHD partly to blame) but she could explain why something didn’t turn out tightbonbthe board while teacher was lecturing (that didn’t go over well), or explain in clearer terms to her peers than the teacher did, because she understood the way the math worked, the concepts behind it.

Ah, it sounds rough to be taught things outside the standard algorithm by someone who doesn't really get it. And some of the Common Core methods I've seen are not as straightforward as they might be, too... I imagine they become gobbledygook if presented by someone who's even a bit confused. 

Yeah, I imagine it's a tough job, having to relearn things all the time, especially since in my experience elementary school teachers tend to like kids a lot but don't tend to like math much, on average. Maybe some conceptual math classes would be a good idea. I have to say I find adults tougher to teach than kid: they are much more math phobic and set in their ways. 

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30 minutes ago, 8FillTheHeart said:

Sometimes we tend to overcomplicate the entire "what curriculum to use" scenario. Curriculum can be used as a guiding tool that you just teach according to the needs of the child sitting in front of you. I dont use anything structured at all for any subjects at the elementary age other than their math textbooks and then teaching math is discussing what is going on.  It is equally fun to sit back and let them explain things to you. 

If you have the ability to teach vs needing a scripted curriculum bc you have no idea what you are doing, use the curriculum as your guide for problems and target skills and just teach concepts in a way that connects with your child. It is actually a far better approach than relying on a textbook to teach.

Honestly, I don't expect to find a curriculum that suits fully, given how opinionated I am ;-). I've taught math for quite a while, and I tend to have very specific ways I work, and I'm fine with that. 

On the other hand, having some compatible materials is nice on the days when I want to take it easy, and it's also good to be taken outside my bubble: I have a pretty good idea of where I'm going with the concepts I'm teaching, since I've taught higher math, but I can certainly forget about skills or types of problems that can be handy. For instance, we won't stick with Beast Academy too much, I think, but I liked being reminded about the puzzlier kinds of problems and plan to both use the ones in the books and make up some of my own.

So, basically, I plan to stick with what I'm doing, but I very much appreciate the suggestions on this thread. They won't become our main curriculum (I'm going to continue making it up, like I have been doing) but I'm pretty likely to incorporate them and also just read them over for inspiration :-). It's always helpful to have things to consult! 

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19 minutes ago, square_25 said:

Would you teach multiple strategies before definitions, though? I like strategies after the basic definition is down. I suppose if you're talking number bonds, then you could define a -b to be "the number which added to b gives a" which is totally how I've taught vector addition to middle school and high school kids. But I've seen struggles with two-step definitions in that case: I have to remind the kids to check that the addition actually works a lot. 

I guess (and this is also based on teaching older kids) I worry about not having a firm definition first. The way I think about math, you have definitions, then you show that different things are really the same. Like, for example, what is a*b? You have to define it as a copies of b, I think. You can't just say it's also b copies of a, because that's not a definition. But you can say it's the same because here's why. 

I've gotten in the habit of giving a firm definition and letting students "play" with it. I find that sometimes the students find the strategies themselves, which can be more rewarding. But I might be overgeneralizing from fairly mathy kids. On the other hand, those same mathy kids had real trouble if they weren't given enough time to get a feel for a concept... 

It isn’t only “a copies of b.”  It’s also a sets of B, a scaled by b, an array of a by b, it’s combinatorics, it’s skip counting, it’s many things... When children are young, and for some learning styles even when they are older, a verbal definition is insufficient and even developmentally inappropriate.  A concrete model should be used. They can infer relationships much better than with a verbal definition. The verbal definition will come, but the concrete idea (and higher reasoning skills) must be there first. The concrete definition comes first. And yes, they discover relationships and strategies for themselves as well.

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3 hours ago, Targhee said:

It isn’t only “a copies of b.”  It’s also a sets of B, a scaled by b, an array of a by b, it’s combinatorics, it’s skip counting, it’s many things... When children are young, and for some learning styles even when they are older, a verbal definition is insufficient and even developmentally inappropriate.  A concrete model should be used. They can infer relationships much better than with a verbal definition. The verbal definition will come, but the concrete idea (and higher reasoning skills) must be there first. The concrete definition comes first. And yes, they discover relationships and strategies for themselves as well.

Right, it’s all of those. My point is that you have to define it as one of those first. The fact that it’s also other things is not obvious. Those are things you have to explain to be consistent with your original definition.

I defined a*b as adding a copies of b. It wasn’t really a verbal definition, since we then practiced actually doing it. How do you define it?

Edited by square_25

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1 hour ago, square_25 said:

I have no experience with dyscalculia, though. How does hers manifest? Does that mean she has trouble understanding numbers?  

 

Dyscalculia is basically an absence of number and place value sense. With a neurotypical kid, you're teaching them the correct words and how to use something they are primed for. With dyscalculia, it just isn't there. You have to teach them to have a number sense and place value sense. (Whether this is possible when dealing with very severe dyscalculia is debatable.) With more severe forms than my dd has, there can be other stuff like an inability to feel time passing and other such inconveniences.

When my dd was preschool aged, she really got into the idea of jigsaw puzzles, but had absolutely no ability to do them. I had to teach her to think of matching the colours, shapes, patterns etc, and despite thinking this was all terribly cool, it was a real struggle for her to learn to think of those things. When I was that age, I was doing jigsaw puzzles picture side down because there wasn't enough challenge in doing them the right way up. As she got older, the real struggle was keeping it in her head. She forgot how to tell time on the last page of the Kumon workbook. And yes, she needed the drill of a workbook, she was never going to learn to tell time on an analog clock without it. So we had to start right over again. Now she's been in public school for two years, I'm seeing that not only will the maths fall out of her head like that, but the teaching methods will too, if they are not reinforced. She seems to have retained her number sense, but after two years without the necessary methods (the Papy minicomputers from CSMP in particular) her place value sense is slipping. We had "stealthed" her dyscalculia so it no longer showed up on academic testing, but I think it would show again in another year or two.

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4 hours ago, HomeAgain said:

Square_25,
Have you read Liping Ma's Knowing And Teaching Elementary Mathematics?  You might really enjoy it.

I haven't! Thanks for the recommendation. 

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3 hours ago, Rosie_0801 said:

 

Dyscalculia is basically an absence of number and place value sense. With a neurotypical kid, you're teaching them the correct words and how to use something they are primed for. With dyscalculia, it just isn't there. You have to teach them to have a number sense and place value sense. (Whether this is possible when dealing with very severe dyscalculia is debatable.) With more severe forms than my dd has, there can be other stuff like an inability to feel time passing and other such inconveniences.

When my dd was preschool aged, she really got into the idea of jigsaw puzzles, but had absolutely no ability to do them. I had to teach her to think of matching the colours, shapes, patterns etc, and despite thinking this was all terribly cool, it was a real struggle for her to learn to think of those things. When I was that age, I was doing jigsaw puzzles picture side down because there wasn't enough challenge in doing them the right way up. As she got older, the real struggle was keeping it in her head. She forgot how to tell time on the last page of the Kumon workbook. And yes, she needed the drill of a workbook, she was never going to learn to tell time on an analog clock without it. So we had to start right over again. Now she's been in public school for two years, I'm seeing that not only will the maths fall out of her head like that, but the teaching methods will too, if they are not reinforced. She seems to have retained her number sense, but after two years without the necessary methods (the Papy minicomputers from CSMP in particular) her place value sense is slipping. We had "stealthed" her dyscalculia so it no longer showed up on academic testing, but I think it would show again in another year or two.

 

Ah, got it. That sounds like a challenge. Why did she go back to public school, if you don't mind me asking? 

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1 hour ago, square_25 said:

 

Ah, got it. That sounds like a challenge. Why did she go back to public school, if you don't mind me asking? 

 

Her father made her.

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7 hours ago, square_25 said:

 

Right, it’s all of those. My point is that you have to define it as one of those first. The fact that it’s also other things is not obvious. Those are things you have to explain to be consistent with your original definition.

I defined a*b as adding a copies of b. It wasn’t really a verbal definition, since we then practiced actually doing it. How do you define it?

You don't have to explain everything first. One of the benefits of concrete manipulatives is the opportunity to discover math concepts on their own. They are not necessary for that, but add additional opportunities. There have been many, many times my kid has said he figured out something in a way I never taught him, or utilized a concept I never taught him. Half the time I feel like I'm just playing catch up trying to teach him the words to use to discuss these things. I'm not sure how unusual my kid is in that regard, his younger brother seems to be following the same way. 

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For us, cuisinaire rods have been FANTASTIC for internalizing basic properties of numbers.  My six year old knows backwards and forwards (haha) that 2+5 and 5+2 are equal, and can prove it with the rods.  Similarly, once my kids start multiplication, they know that a 2x4 rectangle will fit perfectly over a 4x2 rectangle, so whether they use 4 2-rods or 2 4-rods, we end up with the same rectangle.  Perfect squares or cubes are very concrete when you hand the kid a bunch of 1x1x1 cubes and ask them which quantities can be made into a perfect square or cube, and which cannot.  Prime numbers can only be made into a 1xA rectangle, no other options.  Composite numbers can be made into at least one other rectangle.  And so on, and so on.  

My 4yo, through math play, now know that things like 6+1 are 7, but it is also 5+2 or 3+4.  This is just from playing.  We do so much play with the rods before starting formal school that many concepts are just understood.  

When my oldest daughter was 7 or so, I was teaching cubes to my oldest son.  I asked him if 1000 was a perfect cube, and he was finding it mathematically, by doing prime factorization.  She kept shouting from the other room- "it's a perfect CUBE!  A CUUUUUBE!!!"  I finally asked her what made her say that, and she just took the 1000 cube out of our box of manipulatives and said, "Look at it!"- she had internalized the lesson that perfect cubes, are, well, perfect cubes.  

Check out educationunboxed.com to get started with them. 

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Oh, I didn't really finish my train of thought.  By playing with the mainipulatives, your kid will internalize number properties.  And then when you explicitly describe number properties later, they simply feel a sense of "rightness" in what you are explaining.  When we are formalizing certain concepts now in AOPS, I will even occasionally pull out the rods and say something like, "All these words just mean this:" and then demonstrate.  

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Some kids process patterns and see them mathematically, but not all kids do. One of my kids "discovered" multiplication simply by observation way before he had the written language of multiplication taught to him. He saw patterns everywhere: Lego blocks, window panes, rows of cookies on a cookie sheet.  He "taught" me his discovery by telling me that he knew how many things there were if he knew the numbers of rows and the number of things in the row. Our first conversation like this took place while baking cookies. He told me that if we put 5 rows of 4 cookies that meant we would have 20 cookies, but if we put 4 rows of 3 cookies we'd only have 12. I started asking him how he knew that and he told me how when he looked at things he could tell just how many there were by knowing their numbers  in the different directions. He had never been taught the concept of multiplication, but he saw it, knew it, and understood it. 

But, how he learned and processed information is not equivalently transferable to other kids. Just bc he saw those patterns mathematically does not mean I can teach other kids to process visual images the same way. I can demonstrate to them what he is seeing and they can learn it, but that is definitely not the same thing as having them think the same way he does. He thinks in big pictures and finds the pieces and understands how they all interconnect. But, equally, I have kids who have to be taught all the pieces and taught how they go together in order for them to see the big picture.

Pt being, we shouldn't assume that what we witness in our kids is representative of specific teaching techniques or necessary for all students. That same kid would have hated using manipulatives bc once he saw things written down in numbers, his brain automatically processed what they represented and were doing.  He could "see" them in pictures without having to see them. Then there is my granddaughter who can't process any written math concepts without seeing them in manipulatives over and over and over in order to cement the concept being taught (and then it is still questionable if she actually understands.) Neither are representative of how they are being taught.

There isn't a single right approach other than what works for the individual child (and the teacher). 

 

Edited by 8FillTheHeart
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7 hours ago, Sarah0000 said:

You don't have to explain everything first. One of the benefits of concrete manipulatives is the opportunity to discover math concepts on their own. They are not necessary for that, but add additional opportunities. There have been many, many times my kid has said he figured out something in a way I never taught him, or utilized a concept I never taught him. Half the time I feel like I'm just playing catch up trying to teach him the words to use to discuss these things. I'm not sure how unusual my kid is in that regard, his younger brother seems to be following the same way. 

I don’t think providing definitions deprives a kid of playing with the ideas. Certainly the way she does multiplication is her own thing: like, she splits things up into groups however she likes. 

For older kids, I find that sturdy definitions provide grounding. They are something to go back to when stuck. And I’ve seen kids founder from not knowing whether something was a concept or a formula. So I try to teach my daughter that everything is defined, and then we show that our definition is equivalent to other stuff. Of course, if you work enough with equivalent stuff, it becomes intuitive that they are the same. But the definition is important. Again, I know many more kids who can calculate a derivative than to tell me what it means. And I’ve seen similar failure modes even when the teachers are mathematically competent.

So I guess I’m a big believer in definitions as something to be given and be allowed to play with. I think this way of thinking is helpful for later mathematical learning.

Edited by square_25

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My background is in teaching upper level biology, not math, but I've taught my 7th grader through Singapore 6, assorted preA/Algebra (mostly AoPS, but we've used others, too) and we're currently doing AoPS Geometry, my 4th grader is currently in Singapore 5, and I've been a weekly homework helper at an afterschool program for underpriviledged kids for 5 years and often help with math.  I've got some not-entirely-coherent thoughts about some of what has been discussed.  

First, over-reliance on manipulatives or drawing can be a problem.  I've seen kids who, despite understanding basic addition/subtraction in 1st grade, become unable to do it in subsequent grades because they've been forced to draw everything until they've become dependent on it.  That being said, manipulatives can be great for helping kids see equivalencies.  We just use cube blocks, and for younger I would put out 10 blocks and then move them one at a time to show that 9+1 was 10...move one block...and 8+2 is 10... etc.  It helped when she got to subtraction.  One of Singapore's mental math techniques is that, instead of traditional 'borrowing', you take 1 of the tens, subtract from that, and add the difference back to the ones.  It sounded hard to teach, but with blocks it was easy for her to see.  

As for teaching multiple methods to the same type of problem, Singapore did it and it didn't seem to confuse my kids.  But, the kids that I tutor are often convinced that you're doing an entirely different sort of problem using magic techniques when the setup is different.  One difference may be that singapore teaches different techniques, tells when you they are 'best fits' to the problem, but then lets you use whatever you want, while the homework help kids are told to solve problems following a particular example, and the next day are given a different style of example problem. They don't understand the concepts behind what they're doing.  They also don't have a good grasp of the arithmetic, so, for example, so much energy is going into skip counting to get the multiplication right that they don't have the energy to learn different ways to do 2 digit x2 digit multiplication.  When we learned multiplication, older quickly grasped what we were doing and thought that the usual algorithm was the most efficient way to get to the answer.  As I taught it to younger, explaining that, say, in 63 x 12, first we'd multiply 3x2, then 60x 2, to get 126, etc, she fairly quickly asked if we could do 120 + 6 + 30 + 600.  She knows the traditional method, but still prefers this because she can do more in her head.  

I've also found that I can't really predict what will seem complicated to my kids.  One struggles to factor because, while they can do multiplication fast enough to do their work, they actually do a lot of calculation - like, 6x9 is 60-6.  They have great understanding of how math works, but we're finding that the 'work around that i didn't know they were using' is causing some minor problems now.  Fortunately, this kid is thrilled to get to play some computer speed games, so it's an easy fix.  Other kid was truly flying through math (negatives, perfect squares, simple mental algebra) until we got to long division - we were both startled by the stall.  After a few days, I thought to word it differently (when you divide 3 into 285, instead of subtracting 27 from 28, we subtracted 270 since the 9 is in the tens place and 90 x 3 is 270).  It worked for both kids and they have great understanding, although they confuse anybody that they try to help because nobody knows what they're doing.  

I can't do this with the kids that I tutor, but at home I'm finding that, if I have a good book to get us started and I have enough knowledge to know if what they do is mathmatically legitimate or be able to offer an alternate way to deal with things that are confusing, they do fine.  The struggles that you see in your students are years in the making, and seem unlikely to be things that your child would have no matter what book you use because you'd catch it when it's a small mistake early on.  The kids that I 'homework help'  just keep moving on despite misunderstandings and I can only imagine what a mess it will be for some of them as they get older.  

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20 hours ago, square_25 said:

 

It's interesting, I found a lot of the "OMG Common Core!" stuff so annoying that I reflexively assumed I'd like the standards. I basically like the idea of having standards! But then I looked at what they were and... meh. I'm not sold. They don't go in the order I think is sensible, and I definitely have no interest in following their sequence. As you say, they don't seem all that developmentally appropriate.  

 

Common Core math is meant for students who are on or below grade level.  In NY, we have an accel path required by state law...8th Alg at the least.  Of course the upper few percent can compress and accelerate more, just as private school students do, and finish at least three semesters of Calc, if not Linear Alg.  Unfortunately due to zip coding and lack of funding, few public schooled student have that opportunity.  Common Core, per creator Jason Zimba, is designed to make the student ready for Calc after 12th grade.  Big conflict right there. Here is one opinion: https://www.lohud.com/story/opinion/contributors/2014/07/26/common-core-math-sidetracks-chance-take-calculus-high-school-supplanting-th-grade-math/13186153/ and another https://www.jamesgmartin.center/2017/03/common-core-damages-students-college-readiness/.

Our experience in public school in NY was that the standards were fine, but several grade levels behind where the students actually were...so class was mostly review for the 25%  with educated parents (ie had at least Regents Advanced Diploma from high school all the way up to PhD), and parents were buying internet or Community College courses so the students would be ready for a STEM major.  The CC was so slow I couldn't even do the cub scout math requirements with the students who weren't being tutored in elementary unless I taught the math. The Regents Physics teacher also had to devote two weeks to teach the required math, since the Common Core had not progressed far enough. The K-8 teachers are competent, but challenged by having about 6 grade levels difference between all the students in the included classroom. Its simply not possibly to have tiered discussion in the K-4 years due to the range of developmentally appropriate attention spans present. 

 

Edited by HeighHo

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On 12/21/2018 at 8:44 AM, 8FillTheHeart said:

 

The most theory based program you are going to find for elementary school is going to be Beast Academy and its AoPS complement for upper level math.  SM/MiF are your go-to Asian maths. HOE is an excellent supplement for 3rd grade level math (through about 5th or 6th grade level). HOE Verbal Book is all simple alg based word problems (iow, solved using algebraic methodology with manipulative s on a ball and beam vs bar diagrams of Asian math.)

Then the most basic differences are going to be program set up: mastery, spiral, incremental. Examples of your basic programs that fall into those categories are Math Mammoth (mastery), Horizons (spiral), and Saxon (incremental).

 

someone needs to quote and frame this post, or pin it, or something.  What a succinct explanation of the basic math options!

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Regarding C-rods: they seem like a fantastic tool! I was thinking of teaching a math class at some point locally (we're currently in NYC), and if I do, I may very well get a few sets so that the kids have something to work with when I ask questions. So I'm not in any way dismissing them as a teaching prop!

However, I am very mathy and I have a mathy daughter who's never shown a ton of interest in manipulatives. She's also advanced: she knows all the four operations very well (I think she more or less knows the multiplication table up to 10*10, and uses lots of and lots of tricks for figuring products out), and we've also done a number of enrichment things. For instance, we learned about negative numbers already, and she adds, subtracts, and multiplies them fluently (I wasn't planning to do them that young, but we talked about the fact that we don't know what 2 -4 is yet, and after mentioning this for a few months, she really wanted to know what exactly this mysterious 2 - 4 number was, so I explained, and she got it.) The recent thing she was really into was the binary system of numbers, and we wound up counting in binary (in writing) past 300, which is actually not as easy at it sounds, because at that point you're up to 9 digits! We discovered lots of neat binary patterns as well: like, multiplying by 2 adds a 0 to the end of the number (just like in base ten, but with a two instead of a ten!) 

I asked her this morning to show me why 7*3 is 3*7, and she drew me an array of 7 by 3 dots, and showed me the seven 3s, as well as the the three 7s. The thing we've been most recently working with is associativity of multiplication: like, explaining why (2*3)*4 = 2*(3*4). We also do a lot of "systems" of equations: for example, fill in the triangle and square if you have the system of equations: 

triangle = square + 2, square =  2*triangle (imagine the shapes are actually shapes, not words.) 

She doesn't manipulate the equations yet, but she's quite good at using mental tricks to find solutions. I also like this approach for teaching properties, because you do an equation like 

2*7 + triangle * 7 = 21*7, 

and then you can think about what should go in the triangle and how counting by 7s helps you out here. 

Anyway, I guess what I'm saying is that I'm not seeing a need for a new tool given where we are, and given that she didn't like using manipulatives much when we did a few years ago: we moved to pencil and paper (and sometimes fingers) because she just wasn't that interested. She's very abstract-minded and I think she "sees" the patterns in her head. I do as well, so I understand that kind of thinking very well. If I got her colorful rods, she's probably get distracted building things with them :-). 

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2 hours ago, ClemsonDana said:

My background is in teaching upper level biology, not math, but I've taught my 7th grader through Singapore 6, assorted preA/Algebra (mostly AoPS, but we've used others, too) and we're currently doing AoPS Geometry, my 4th grader is currently in Singapore 5, and I've been a weekly homework helper at an afterschool program for underpriviledged kids for 5 years and often help with math.  I've got some not-entirely-coherent thoughts about some of what has been discussed.  

First, over-reliance on manipulatives or drawing can be a problem.  I've seen kids who, despite understanding basic addition/subtraction in 1st grade, become unable to do it in subsequent grades because they've been forced to draw everything until they've become dependent on it.  That being said, manipulatives can be great for helping kids see equivalencies.  We just use cube blocks, and for younger I would put out 10 blocks and then move them one at a time to show that 9+1 was 10...move one block...and 8+2 is 10... etc.  It helped when she got to subtraction.  One of Singapore's mental math techniques is that, instead of traditional 'borrowing', you take 1 of the tens, subtract from that, and add the difference back to the ones.  It sounded hard to teach, but with blocks it was easy for her to see.  

As for teaching multiple methods to the same type of problem, Singapore did it and it didn't seem to confuse my kids.  But, the kids that I tutor are often convinced that you're doing an entirely different sort of problem using magic techniques when the setup is different.  One difference may be that singapore teaches different techniques, tells when you they are 'best fits' to the problem, but then lets you use whatever you want, while the homework help kids are told to solve problems following a particular example, and the next day are given a different style of example problem. They don't understand the concepts behind what they're doing.  They also don't have a good grasp of the arithmetic, so, for example, so much energy is going into skip counting to get the multiplication right that they don't have the energy to learn different ways to do 2 digit x2 digit multiplication.  When we learned multiplication, older quickly grasped what we were doing and thought that the usual algorithm was the most efficient way to get to the answer.  As I taught it to younger, explaining that, say, in 63 x 12, first we'd multiply 3x2, then 60x 2, to get 126, etc, she fairly quickly asked if we could do 120 + 6 + 30 + 600.  She knows the traditional method, but still prefers this because she can do more in her head.  

I've also found that I can't really predict what will seem complicated to my kids.  One struggles to factor because, while they can do multiplication fast enough to do their work, they actually do a lot of calculation - like, 6x9 is 60-6.  They have great understanding of how math works, but we're finding that the 'work around that i didn't know they were using' is causing some minor problems now.  Fortunately, this kid is thrilled to get to play some computer speed games, so it's an easy fix.  Other kid was truly flying through math (negatives, perfect squares, simple mental algebra) until we got to long division - we were both startled by the stall.  After a few days, I thought to word it differently (when you divide 3 into 285, instead of subtracting 27 from 28, we subtracted 270 since the 9 is in the tens place and 90 x 3 is 270).  It worked for both kids and they have great understanding, although they confuse anybody that they try to help because nobody knows what they're doing.  

I can't do this with the kids that I tutor, but at home I'm finding that, if I have a good book to get us started and I have enough knowledge to know if what they do is mathmatically legitimate or be able to offer an alternate way to deal with things that are confusing, they do fine.  The struggles that you see in your students are years in the making, and seem unlikely to be things that your child would have no matter what book you use because you'd catch it when it's a small mistake early on.  The kids that I 'homework help'  just keep moving on despite misunderstandings and I can only imagine what a mess it will be for some of them as they get older.  

 

Yes... your experience with your homework help kids sounds very familiar to me! I see kids move forward without understanding concepts all the time, and it definitely wreaks havoc with their mathematical competence and confidence. I am sure you're right that given that I'm very comfortable with all the concepts, whatever curriculum we used would be fine. However, I do like feeling like I'm not reinforcing bad things! I'm absolutely sure that somehow the Beast Academy books got my daughter to misuse an equals sign, because she had never done it before, and isn't doing it anymore now that we talked about it. Now, to be fair, we were jumping around in BA since she already knows the material, so it's possible that it would have been better if we took it slower. On the other hand, I was not comfortable with how many of their problems were of the "compute this" form. I do think that lots of questions where you compute things and an equals signals "you should now write down the answer for the computation" give the wrong idea.

I really loved the BA puzzles, so I'm not dismissing the curriculum as a whole. I might very well buy the next set, too, just for the puzzles. It's hard to get everything right in a curriculum! But personally, when I was designing her work, I was spending way more time making her write down her own equations and fill in equations and otherwise use an equals sign in a flexible way, and I haven't found that to be the case in general. However, having looked around, I think both MEP and Miquon seem like they might be a good fit for a conceptual, visual kid. I've heard of Miquon but not of MEP, so this has been a useful conversation for me :-).  

By the way, I also found that I couldn't predict precisely where we'd get stuck. Like, I had no idea multiple methods for subtraction would be a block, but they really were. And again, she did learn them eventually... but they weren't straightforward the way I thought they'd be. We also had more trouble with division than I thought we would. In general, I've found that the "reverse" operations (the ones that don't commute, come to think of it!) give us more trouble than the "forward" operations like addition and multiplication. To be fair, I haven't done the standard algorithm for either multiplication or division yet... I feel like those become easier when a kid is older, so I'm not in any hurry. So we might very well get stuck on one of those! 

 

 

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2 hours ago, HeighHo said:

 

Common Core math is meant for students who are on or below grade level.  In NY, we have an accel path required by state law...8th Alg at the least.  Of course the upper few percent can compress and accelerate more, just as private school students do, and finish at least three semesters of Calc, if not Linear Alg.  Unfortunately due to zip coding and lack of funding, few public schooled student have that opportunity.  Common Core, per creator Jason Zimba, is designed to make the student ready for Calc after 12th grade.  Big conflict right there. Here is one opinion: https://www.lohud.com/story/opinion/contributors/2014/07/26/common-core-math-sidetracks-chance-take-calculus-high-school-supplanting-th-grade-math/13186153/ and another https://www.jamesgmartin.center/2017/03/common-core-damages-students-college-readiness/.

Our experience in public school in NY was that the standards were fine, but several grade levels behind where the students actually were...so class was mostly review for the 25%  with educated parents (ie had at least Regents Advanced Diploma from high school all the way up to PhD), and parents were buying internet or Community College courses so the students would be ready for a STEM major.  The CC was so slow I couldn't even do the cub scout math requirements with the students who weren't being tutored in elementary unless I taught the math. The Regents Physics teacher also had to devote two weeks to teach the required math, since the Common Core had not progressed far enough. The K-8 teachers are competent, but challenged by having about 6 grade levels difference between all the students in the included classroom. Its simply not possibly to have tiered discussion in the K-4 years due to the range of developmentally appropriate attention spans present. 

 

 

Hmmmm, interesting. We're in NYC right now, actually, so that's relevant to us (although we haven't tried school here at all. We were in Austin, Texas for kindergarten.) From reports, things are if anything even more behind in Austin... my daughter starting coming home saying "math is too hard" last year, and eventually we figured that was because she found drawing pictures for every single addition question (which she either memorized or counted on her hands) incredibly onerous. Eventually, we figured out that she wouldn't get in trouble if she just drew a picture for her hands under "How did you figure this out?" Never mind that she only used her hands for the harder problems, anyway, and had a plethora of mental tricks by then. I asked for some acceleration for her, and they had her add bigger numbers, which... was just as boring but also harder, so then of course she did that once and didn't want to do it again. 

Not taking algebra by grade 8 seems like a serious problem. And stuffing in all the hard topics into grade 10 and 11 seems like a recipe for disaster. Is the reason for this that some kids aren't ready for algebra before grade 9? I can believe that, since there's (as you say) a really wide range of developmentally appropriate abilities. Is there anything stopping schools from accelerating or at least having some accelerated courses? 

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1 hour ago, square_25 said:

If I got her colorful rods, she's probably get distracted building things with them :-). 

You might be surprised by what she can come up with on her own by just building with them. It may help her see the patterns in geometry and the beauty in calculus even if she doesn't quite understand yet that is what she is doing. But those memories of just playing with the rods may very well stick with her when she does run into those concepts later in her math instruction and it just intuitively makes sense because she saw the real application of it playing with the rods. You could do that with just about any blocks really, I'm not saying you need to run out and get some c-rods right now or she will be missing something, just saying that playing and building with them isn't necessarily not a learning activity.

1 hour ago, square_25 said:

Is there anything stopping schools from accelerating or at least having some accelerated courses? 

 

Mostly money from what I've seen. They need to spend so much money cleaning the mess they created in elementary school by not teaching things clearly and teachers passing on their math-phobia that those who made it through and still somehow ended up ahead get the short in of the stick because there isn't enough money to devote to accelerated instruction for the few students who could benefit when the majority of students are barely on par or need remedial help.

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2 hours ago, sweet2ndchance said:

You might be surprised by what she can come up with on her own by just building with them. It may help her see the patterns in geometry and the beauty in calculus even if she doesn't quite understand yet that is what she is doing. But those memories of just playing with the rods may very well stick with her when she does run into those concepts later in her math instruction and it just intuitively makes sense because she saw the real application of it playing with the rods. You could do that with just about any blocks really, I'm not saying you need to run out and get some c-rods right now or she will be missing something, just saying that playing and building with them isn't necessarily not a learning activity.

 

Hmmmm... well, if I do teach a class, I'm sure I'll wind up with a set around the house :-). So we'll see. She's generally not been super interested in playing with patterns herself, though: she has an easy time with math, but doesn't explore on her own yet. But you may be right! 

I'm not sure I see how rods are applicable to calculus, though! Calculus is so continuous... and manipulatives of that form are so very discrete :-). (I love visual aids for math of all levels, they are my favorite things, but for calculus they don't tend to be whole number oriented. How would you use them?) 

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I think you may be thinking a little too concretely lol. Like said, blocks of any kind could really be used rather than cuisenaire but the advantage of cuisenaire in this situation is that they do represent numbers in an way. If she makes the pattern of a spiral with them, the pattern of numbers that the spiral is composed of is more clear than if it were made with blocks with no sort of numeric representation. It may spark an interest in her to understand how those numbers are related and why, which can be explained by complex higher math concepts (logarithmic spirals, Fibonacci sequences...) or she might build something that leads to a discussion of golden ratios....

My point wasn't that she needs manipulatives to learn calculus at a young age. I was more suggesting that her playing with the blocks rather than solving specific problems with them isn't necessarily devoid of learning opportunities. There is math in building a sound structure. There is math in making patterns and pictures. There is math in nature and in stories and many other places one wouldn't expect to find it. Math is more than just solving equations and in my experience the more a child is allow to just play with math manipulatives of all kinds with no expectations and of their own free will, the more likely they are to intuitively understand math concepts later on because even when they didn't realize it, there was a pattern and math to how they played with those manipulatives.

Of course some kids just aren't into playing and building and experimenting and that's fine. But when you suggested that she doesn't like manipulatives but then also suggested that she would "play with" rather than "use" the manipulatives, I just wanted to throw out there for your consideration that play can be just as stimulating for a young mind as work.

Here is a quote from an article I came across that kind of sums up my thinking:

"Droujkova – and many other academics – suggest that the best way to teach children mathematical ideas (whether simple or complex) is through play. Play allows children to integrate understanding, connect experiences, explore possibilities and solve problems. It also lets them engage their natural curiosity and take control of their own learning. As a social activity, play provides a context in which children can share knowledge and figure out patterns together.

Even with all its social and educational benefits, it seems a stretch to declare that simply playing games is enough to teach children “advanced” mathematics. But Droujkova’s aim isn’t to have five year-olds solving complex equations: it’s to provide a broad informal foundation upon which an understanding of more difficult ideas can be built. First come the fundamentals – skills like logical thinking and pattern making – then the identification and dissection of patterns, and then, finally the grasping of more abstract ideas."  (emphasis added by me)

My kids all had lots of time to play with and experiment with math manipulatives of  all kinds, including cuisenaire rods. They were not reserved only for school lessons. The younger children of course had more time to play with these tools than the older children because we sometimes didn't have them around the house when the older children were little. Not all my younger children ended up mathy because of it but they all definitely had an easier time grasping concepts and I attribute that to all the time they spent "playing" with math manipulatives.

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Why C-rods and not Legos? Or other blocks? They are all counting and building materials. LEGOs are very explicitly numerical and she likes them.

This is too close to magical thinking for me. Kids learn by play. But they also benefit from having their ideas organized by adults.

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Yeah it could be legos, or attribute blocks or any other block and I did specifically say that in my original post about this topic. And no, a child is not going magically understand how to solve an integral or derive an equation by playing with any thing, toys or manipulative. (I assume that is what you meant by "magical thinking".) But that informal play does have value and while some of it can and should be augmented by an adult helping them see the connections, sometimes just having that informal experience is what makes the light bulb come on later on.

That's all I was trying to say. You are certainly welcome to disagree with me.

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6 hours ago, square_25 said:

 

Not taking algebra by grade 8 seems like a serious problem. And stuffing in all the hard topics into grade 10 and 11 seems like a recipe for disaster. Is the reason for this that some kids aren't ready for algebra before grade 9? I can believe that, since there's (as you say) a really wide range of developmentally appropriate abilities. Is there anything stopping schools from accelerating or at least having some accelerated courses? 

 

Its all politics.

State law requires districts to offer accel beginning in grade 8. It does not require districts to offer enough seats to all qualified students. Nor dos it specify which subjects should have accel classes.  The law requires that compelled students have a full schedule of appropriate classes....cash strapped districts will offer the child care,nutrition, and art electives to college prep high schoolers rather than AP or Honors, or they will defy the law and put students in multiple study halls for 12th. 

Your best bet is do not be in a school where your child does not fit the demographic academically...the union limitations on 'preps' mean that there is no tiering for the above average student. In general, plan on teaching math at home unless you get into a district with enough resources to do more than the minimum.

 

 

Edited by HeighHo

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19 minutes ago, mshanson3121 said:

 

Not nearly as much as you might think. In fact, studies show that this tends to stunt them, more than benefit them. Peter Gray would be a beneficial read for you. I also would caution you to remember that your child is 6 years old. If she loves math that's great, but you're going to need to be cautious of burn out down the road.

What studies? I haven’t seen anything this definite. I’ll take a look at Peter Gray. Any specific recommendations?

I think she spends less time doing math than an average kid in school. It’s just tailored to what she’s able to do and I follow her lead on what she likes. For what it’s worth, she loves binary and the (basic version of) systems of equations we do best. She dislikes pure computation and is a lot less motivated by puzzles than I was.

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I'm going to echo @mshanson3121 in terms of your dd being 6. My 6 yr old 1st graders spend about 90 mins total on anything school related that is being directed by me. That means reading, phonics, handwriting, and math. It does not take much time to cover what is appropriate for them at that level.  

We are very interest-led and child-driven homeschoolers. I am also a huge advocate for imaginative play developing higher order critical thinking skills. I am not sure about the study being mentioned and I am not referring to teaching, but there is definitely research showing that for young kids dramatic and imaginative play develops higher order thinking skills to a different degree than academic activities. My little kids spend most of their day playing with minimal amts of time focused on academics.  Time spent on school work when they are little has not at all equated with their long-term progression, unless less is more. 🙂

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I haven’t seen anything like conclusive research on this. The things I see are always comparisons with totally different countries like Denmark or the Benezet experiment. There certainly aren’t good controlled studies. Which basically means that people take what they want from the research.

We do maybe 2 hours of academics in the morning, plus piano practice in the evening. We’re in no way overloading our daughter. She’s been very content this year as compared to kindergarten: she likes all the down time.

As for what’s appropriate for a kid at age 6.... it varies by kid. She likes what we do in math right now, and that’s what’s important to me. I certainly won’t introduce ideas she isn’t ready for.

Anyway, thanks for your input, everyone :-). I got some good ideas. 

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38 minutes ago, mshanson3121 said:

Nm

Please link me a concrete study? I can’t look through 50 links, many of which are opinion pieces. 

To be concrete, I haven’t seen a controlled study, or even observational studies on broadly similar groups. I’m not closed minded, I’d be interested.

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On 12/20/2018 at 10:48 PM, square_25 said:

Hey all! 

I've just started homeschooling my 6 year old daughter this year. I'm a mathematician by training, so I've been making up math lessons myself, and I haven't seen any curriculums that approach it how I do. 

Are there any curriculums where the main point is to teach properties of numbers and do "algebraic" exercises at the correct developmental level? For example, does anyone spent a lot of time focusing on why a*b = b*a for any a and b? Do they explain why a*(b*c) = (a*b)*c? 

I've looked around, but not a ton, so I'm curious if there's something I've missed. 

I'm going to get back to this.  This was the original question you asked, square_25.

Yes, yes there are.  And they are all developmentally appropriate, using specific concrete math tools and methods meant for young children.

However, your question turned, and to answer you honestly, I do not think there is any viable curriculum that does what you are asking it to do: focus on explanations and information given explicitly with a minimal of play and developmentally appropriate methods.

Best wishes to you in your quest for such an item.  Please feel free to research the various methods and curricula listed in this thread if you are ever looking for something different.  We will absolutely help if you are interested, but it has long been a thought on this board that it is a folly to argue and present points to someone who is not receptive or interested enough to do a google search of their own.

May your holidays be joyful and full of peace.
 

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53 minutes ago, HomeAgain said:

I'm going to get back to this.  This was the original question you asked, square_25.

Yes, yes there are.  And they are all developmentally appropriate, using specific concrete math tools and methods meant for young children.

However, your question turned, and to answer you honestly, I do not think there is any viable curriculum that does what you are asking it to do: focus on explanations and information given explicitly with a minimal of play and developmentally appropriate methods.

Best wishes to you in your quest for such an item.  Please feel free to research the various methods and curricula listed in this thread if you are ever looking for something different.  We will absolutely help if you are interested, but it has long been a thought on this board that it is a folly to argue and present points to someone who is not receptive or interested enough to do a google search of their own.

May your holidays be joyful and full of peace.
 

Well, interestingly, I haven’t had people give examples to me of how the curriculums do address these.

Question 1: what is developmentally inappropriate about the work I’ve described doing? Can you please be specific? 

Question 2: how do you think a developmentally appropriate curriculum explains why a*(b*c) is (a*b)*c? I agree, it’ll be with numbers, not with variables. But what’s the explanation?

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3 minutes ago, square_25 said:

 

Question 2: how do you think a developmentally appropriate curriculum explains why a*(b*c) is (a*b)*c? I agree, it’ll be with numbers, not with variables. But what’s the explanation?

I'm pretty sure that, somewhere along the line, Singapore starts teaching kids to do this through factoring - 18 x 5 = 9 x 2 x 5 = 9 x10  

I can't guarantee it, since I've seen other materials interspersed wth Singapore, but that curriculum is VERY into working with 10s.  In addition it teaches students to break numbers apart to make ten - 14 + 9 = 13 + 1 + 9 = 13 + 10 which introduces some of the same ideas. Students also break down numbers - 257 = 200 + 50 + 7 which can be 200 +57 of 250 +7

But, counter to your preference for a definition first, I think that students do it with actual numbers very early, and it's much later when they go back and give the commutative and associative properties names. My kids are pretty good at extrapolating a concept to bigger numbers, so I'm sure that we did the 14 + 9 example with the cube blocks so that they could see that there were clearly the same number of blocks there, but that mentally it's easier to add 10 to something than to need to carry.  

Again, though, this sort of thing, which with Singapore + blocks seems to have been grasped quickly by my kids, is completely confusing to my homework help kids.  If my own kid can't remember what 6 x 4 is, they quickly realize that you can do 6 x 2 x 2.  My homework help kids don't understand what I'm doing when I try to get them to do it.  It's as if their common core curriculum manages to take the ideas used in Singapore math, like number bonds, and teach them in a way that makes them completely useless for later application.  

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We definitely do it with numbers first!! I don’t define things with variables :-). I suppose in some sense I don’t give a formal definition, if a definition requires variables. I suppose what I meant by a definition is that I like having a dominant way to think of a concept that I come back to. For me, I come back to repeated addition when we multiply :-).

My daughter certainly started *using* the associative property of multiplication before I showed it to her. But I actually got a little stumped thinking about to explain the precise pattern to her! And you need the associative property for purely practical reasons, since otherwise you have to write your multiplications with parentheses (technically, 3*4*5 isn’t even defined until you know it works out the same whether you do (3*4)*5 or 3*(4*5).)

I don’t plan to name the properties for her :-). I don’t think it matters if she knows those names for a good long while. But I guess even though she figured out the pattern herself, I do like presenting a helpful visual for it? At least I found it helpful with why 4*7 is 7*4 and the like. I found the picture helped her stay connected to what the various things we’re playing with MEAN and not just operating by rote. Like, we had a point at which she used that fluently but forgot why, so we went back to the picture to make sure she wasn’t using a confusing black box when reasoning.

 

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On 12/22/2018 at 3:43 PM, Targhee said:

 I am not blaming teachers - most elem teachers don’t have to take conceptual math courses...

This fact really bothers me. Why are the college math requirements for teachers so minuscule? Recently, a young friend changed his major to education solely because the math requirements were so much lower than his dream field. The math requirements for his education degree are shocking: 3 hours of general studies math (below college algebra).  His previous major required much more math (calculus and statistics) even though his previous major was a decidedly non-mathy field.

My mom recently retired. She had worked at several different universities, and one thing she did was teach those who were teachers (not college students) how to teach math. She was constantly amazed at how little math comprehension they had. Decades ago, she also started a national program to encourage girls to pursue the fields of math and science. She has confided in me that this program (and others like it) actually caused an unintended consequence that has damaged our education system. This may be an over-simplification, but she basically said that in earlier decades, the smart women would become teachers, and the not-so-smart women would become secretaries. Now, the smart women generally don't become teachers. They become engineers, scientists, doctors, etc. So the intelligence level of our teachers has decreased over the years. You are not really supposed to talk about this kind of thing, but it is true. 

It is a shame when young students understand math better than their teachers. We should not be encouraging people who are deficient in math to become teachers by making the college math requirements so low.

Anyway, this is really just a vent, and I don't want to derail the thread.

 

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1 hour ago, square_25 said:

Well, interestingly, I haven’t had people give examples to me of how the curriculums do address these.

Question 1: what is developmentally inappropriate about the work I’ve described doing? Can you please be specific? 

Question 2: how do you think a developmentally appropriate curriculum explains why a*(b*c) is (a*b)*c? I agree, it’ll be with numbers, not with variables. But what’s the explanation?

You have been provided lists.  I take it you do not need someone to do all your research for you, but can look at samples and research on your own time to determine how they teach, especially since some of those resources are free and available with a click.  Others have a wealth of information if you type in the name and operation you want to see in a google search. 

I cannot possibly answer your questions any more succinctly than I have.  You are asking for mounds of research and information to be presented to you on a silver platter so that you can dismiss it all.  A good teacher is a good learner.  Feel free to take the time and enhance your education with an open mind and a focus on the brain development at various stages, as well as how to teach math with a strong base so that a child understands future operations better.  And do not assume the rest of us do not know what we are talking about - nor that abilities are unique.  There is an entire subforum here dedicated to accelerated learners and how to meet their needs.  It may even behoove you to poke around there a bit and look at what people are using.

 

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Now that my vent is over about the minuscule college math requirements for teachers , I thought I would actually answer the question. : )

I just think that you are over-thinking this. I know this because I am also mathy and tend to over-think this stuff, too. You stated that you just started homeschooling your daughter, age 6. I used to agonize over these questions, too, but I have discovered that which product we use doesn't really matter as much as we think it does, especially if the teacher and student are proficient in a subject. As others have stated, it is the teacher and the student that matter and not as much the material. If you are teaching it, and she understands it, that's great. If she truly doesn't like/need manipulatives and you would rather draw something, that's fine, too. It sounds like you are doing great, so just keep doing what you are doing if it works. You can also change to a program if you are tired of making it up yourself. Just look at samples of products that have been mentioned and pick what is closest to your style. You can always tweak it.

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33 minutes ago, HomeAgain said:

You have been provided lists.  I take it you do not need someone to do all your research for you, but can look at samples and research on your own time to determine how they teach, especially since some of those resources are free and available with a click.  Others have a wealth of information if you type in the name and operation you want to see in a google search. 

I cannot possibly answer your questions any more succinctly than I have.  You are asking for mounds of research and information to be presented to you on a silver platter so that you can dismiss it all.  A good teacher is a good learner.  Feel free to take the time and enhance your education with an open mind and a focus on the brain development at various stages, as well as how to teach math with a strong base so that a child understands future operations better.  And do not assume the rest of us do not know what we are talking about - nor that abilities are unique.  There is an entire subforum here dedicated to accelerated learners and how to meet their needs.  It may even behoove you to poke around there a bit and look at what people are using.

 

My personal sense of what’s developmentally appropriate is that it needs to be super concrete. That is also what I found poking around. I actually don’t find the number line very developmentally appropriate for that reason, since it feels like an abstraction :-).

The thing we’ve disagreed the most on so far is manipulatives, as far as I can tell. Would you say that one has to use manipulatives to have a developmentally appropriate program?

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1 minute ago, square_25 said:

My personal sense of what’s developmentally appropriate is that it needs to be super concrete. That is also what I found poking around. I actually don’t find the number line very developmentally appropriate for that reason, since it feels like an abstraction :-).

The thing we’ve disagreed the most on so far is manipulatives, as far as I can tell. Would you say that one has to use manipulatives to have a developmentally appropriate program?

No. But for MOST kids beginning with a concrete object, then moving on to pictorial, and then on to symbolic is appropriate provides the best foundation. My oldest did mental calculations at 4 before ever receiving math instruction (except the concept that addition is the total, subtraction is taking away, and multiplication is repeated addition).  And she did it on her own, not from a prompt (eg I didn’t ask her how many of such and such, she cane to me in a “did you know ___?” Fashion).  She was unable to explain her thinking or how she came a solutions until she was 11 or 12, despite accurately doing her math without any written work. This is not the norm, even if common among “mathy” people. She did it, but I would never require that of my other kids even if they *could* do it.

What is developmentally appropriate is not just this progression either of concrete first and abstract last.  Much of what is inappropriate about CCSS for math has to do with requiring young kids to write an explanation for how they solved the problem and why they chose the strategy.  It seems twice as inappropriate when your child is accelerated in a math and are very math intuitive.

Also, as 8 said above, just because a child *can* do something doesn’t mean it is appropriate for their growing mind.  That’s another topic in itself.  

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4 minutes ago, Targhee said:

No. But for MOST kids beginning with a concrete object, then moving on to pictorial, and then on to symbolic is appropriate provides the best foundation. My oldest did mental calculations at 4 before ever receiving math instruction (except the concept that addition is the total, subtraction is taking away, and multiplication is repeated addition).  And she did it on her own, not from a prompt (eg I didn’t ask her how many of such and such, she cane to me in a “did you know ___?” Fashion).  She was unable to explain her thinking or how she came a solutions until she was 11 or 12, despite accurately doing her math without any written work. This is not the norm, even if common among “mathy” people. She did it, but I would never require that of my other kids even if they *could* do it.

What is developmentally appropriate is not just this progression either of concrete first and abstract last.  Much of what is inappropriate about CCSS for math has to do with requiring young kids to write an explanation for how they solved the problem and why they chose the strategy.  It seems twice as inappropriate when your child is accelerated in a math and are very math intuitive.

Also, as 8 said above, just because a child *can* do something doesn’t mean it is appropriate for their growing mind.  That’s another topic in itself.  

We did actually start with concrete objects at 4, and she used them in preschool and kindergarten. She’s just always been super lukewarm about them, hence my lack of enthusiasm. But we had a whole bag of “counting pennies” for half a year. We used LEGOs as well.

Yes, I really dislike the explanation writing! What would your daughter say if asked to explain? Mine’s very verbal and can tell you (“Well, 3*8 is just 3*9 with one 3 taken away, and 3*9 is 27, so it’s 24.”) I don’t think she could write it down, though. My second is also super verbal and will probably be able to explain, when she’s big enough.

Is there something I’ve described that seems inappropriate? We aren’t using variables, we’re only doing very concrete things so far.

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45 minutes ago, Skippy said:

Now that my vent is over about the minuscule college math requirements for teachers , I thought I would actually answer the question. : )

I just think that you are over-thinking this. I know this because I am also mathy and tend to over-think this stuff, too. You stated that you just started homeschooling your daughter, age 6. I used to agonize over these questions, too, but I have discovered that which product we use doesn't really matter as much as we think it does, especially if the teacher and student are proficient in a subject. As others have stated, it is the teacher and the student that matter and not as much the material. If you are teaching it, and she understands it, that's great. If she truly doesn't like/need manipulatives and you would rather draw something, that's fine, too. It sounds like you are doing great, so just keep doing what you are doing if it works. You can also change to a program if you are tired of making it up yourself. Just look at samples of products that have been mentioned and pick what is closest to your style. You can always tweak it.

I definitely overthink! We’ve actually been doing afterschool type math for a while, so this is thankfully a continuation for us. But I’ve recently started looking around more: I guess because she’s being officially homeschooled now, and also because I ran into the BA issue and was curious if other curriculums are more mindful of it. 

Thanks for the encouragement! How old are your kids, and what have you used, if you don’t mind me asking?

I agree with you about the math teachers vent. It’s shocking that the weaker math students who are math phobic go on to introduce kids to math. I don’t need math teachers to know higher math, but they should see the beauty in numbers and be conceptually fluent!

I peeked at the Liping Ma book, by the way (it’s expensive so I haven’t bought it yet) and the explanations by the Chinese teachers are from what I’ve seen much more consonant with what I think of as a reasonable math education. Whereas the American teacher responses are just appalling 😕 

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38 minutes ago, square_25 said:

I actually don’t find the number line very developmentally appropriate for that reason, since it feels like an abstraction :-).

 

I actually can't think of a curriculum that has been mentioned to you here that uses a number line without first having the child build a number line using objects (blocks, bricks, c-rods, counters, you can choose whichever suits you). It's fine to move on to abstract concepts once the child has the concrete foundation of using objects. Building their own number lines with objects is something typically done in preschool and kindergarten. By first grade, it is usually assumed that the child already has this foundational knowledge or that the teacher can scaffold it in on their own if it is missing.

Your original question was for a curriculum that explains associative and communitive properties in a developmentally appropriate way to a 6yo. Most curriculum writers do that with manipulatives. Most children learn better by seeing and doing rather than a long winded algebraic explanation. You do that by playing with the manipulatives. You show that no matter what order you add the numbers together, the answer is the same. You show them with manipulatives that a * (b * c) yields the same answer as (a * b) * c. If your particular child would learn better from a long winded explanation, then by all means find your favorite algebra book and lecture away. Your child would not be a typical 6yo if she learns that way and no elementary math curriculum would be suitable for her.

Are you looking for curriculum options or validation that your child is atypical and your teaching paradigm is superior to what is available?

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9 minutes ago, square_25 said:

Thanks for the encouragement! How old are your kids, and what have you used, if you don’t mind me asking?

Two are college graduates, and one is in third grade. I have used and liked Miquon. You may find it relevant that I did not use the rods with the first two children, and if I remember correctly, they completed the whole program. I bought the rods for the third child, and she liked them. So from my standpoint, you can work it either way successfully. 

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