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Teaching properties of numbers.

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2 hours ago, Skippy said:

She was constantly amazed at how little math comprehension they had

I completely agree.  I tutored math in college. 90% of the students who came to me for help were education majors. Most of them just wanted to pass College Algebra 101 which was the minimum gen ed. requirement for all majors. Education majors didn't have to take any math beyond the gen ed requirement unless they minored in math education. College Algebra 101 was basically a review of Algebra 1 and 2 from high school. To say that they struggled with the concepts would be an understatement. Many of them either couldn't wrap their head around the concepts or just refused to understand because they had such a distaste for math. They didn't understand why they had to learn this stuff to teach little kids how to add and subtract. Sigh.

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3 minutes ago, Skippy said:

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. 

 

Cool, that is good to know. I remember seeing some early worksheets that referenced rods and got worried. But we’re probably past that point.

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12 minutes ago, sweet2ndchance said:

 

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?

 

I think there is a lot of misunderstanding going on in this thread :-/. I show the commutative property by drawing an array and showing that if you count the rows, it’s multiplied one way, and if you count the columns, it’s multiplied in the other. Like, here’s 3*5 = 5*3:

X X X X X

X X X X X

X X X X X

One of my earliest mathematical memories is being told why it’s true and having an ah-ha! moment. I think it’s a beautiful fact :-).

I do no long winded lecturing, and I find lecturing in general a waste of time. 

As for (a*b)*c = a*(b*c), as I said, my daughter uses it intuitively, but I just wanted to know how people would demonstrate it with the visual clarity of the commutative property. For me, it always feels tied up with volume calculations, but those seemed  hard for this age. But I love visuals for pulling things together.

By the way, what in the world would a long winded algebraic explanation look like of this? I actually have never seen one. By the time you get to algebra, it’s taken for granted.

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8 minutes ago, square_25 said:

Cool, that is good to know. I remember seeing some early worksheets that referenced rods and got worried. But we’re probably past that point.

I should mention that Miquon uses the number line pretty extensively, so you may not like that aspect of it either.

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Oh, and I just don’t like number lines in early grades. At least, I don’t think it makes things more intuitive for a kid, so I don’t really see the point. I like numbers to refer to objects at this age! 

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Just now, Skippy said:

I should mention that Miquon uses the number line pretty extensively, so you may not like that aspect of it either.

 

Yes, I saw that! How did you feel about that? Did you feel that it helped communicate concepts that were otherwise tricky? 

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Personally, I like number lines. It is sort of an introduction to the coordinate plane which is just two number lines. So much of my math education was based on the coordinate plane and graphing equations.

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1 minute ago, Skippy said:

Personally, I like number lines. It is sort of an introduction to the coordinate plane which is just two number lines. So much of my math education was based on the coordinate plane and graphing equations.

 

Yes... agreed there! But I don’t plan to graph things with her for another 3 years, I’d guess, and it feels like she’d be more ready then. I love the complex plane, too, by the way, so I do like this kind of visual aid. Just dubious about it before age 8 or 9.

I also prefer number lines when a kid knows there are things between the integers, and we are only just starting to get an intuitive sense of fractions other than halves!! 

 

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I think it would be fine to wait to introduce number lines later. I don't think she would be missing anything and could pick it up easily when the time came.

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Assosciative property is easy with either rods or just plain unit blocks.  Build your axb rectangle, then stack up c of them: (axb)xc.  Then make a bxc rectangle, and stack up a of them, and you've got ax(bxc).  So just 3D array.  

I've got a degree in physics, and I always just use our curriculum as a jumping off point.  To me, it sounds like you are doing everything right.  I don't buy into "developmentally appropriate", only "move at the child's pace".  So, in my family, what is "appropriate" for my 6yo is almost the same as that of my 4yo.  It just is, he "gets it" already.  I'm not going to not do math with him because it's not "appropriate".  And I'm not going to push my 6yo into "appropriate" math, where she would get lost.  We just move at the child's pace.  

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2 minutes ago, Monica_in_Switzerland said:

Assosciative property is easy with either rods or just plain unit blocks.  Build your axb rectangle, then stack up c of them: (axb)xc.  Then make a bxc rectangle, and stack up a of them, and you've got ax(bxc).  So just 3D array.  

I've got a degree in physics, and I always just use our curriculum as a jumping off point.  To me, it sounds like you are doing everything right.  I don't buy into "developmentally appropriate", only "move at the child's pace".  So, in my family, what is "appropriate" for my 6yo is almost the same as that of my 4yo.  It just is, he "gets it" already.  I'm not going to not do math with him because it's not "appropriate".  And I'm not going to push my 6yo into "appropriate" math, where she would get lost.  We just move at the child's pace.  

 

Hah, you know, the rods would be helpful, wouldn’t they?? I was thinking volume, and that how am I going to ever communicate that... but of course, if your units are 3D blocks, then problem solved. I might need to get a set just for that.

Yeah, kids vary SO MUCH. My older girl knew her letters at 2. My younger still mixes up some capitals at 2.5, and we haven’t even tried lowercase letters. But she spoke in full, grammatical sentences at age 2, unlike my oldest who was still referring to herself as Mimi once in a while. The range of abilities for an age group is just awe-inspiring, and is part of what made me homeschool. What do you do with a 6 year old kid who’s reading at a grade 5 level, doing math at a grade 3 level, and is socially actually a bit young for her age??? That’s a real conundrum for a teacher with 15-25 kids.

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36 minutes ago, sweet2ndchance said:

Most of them just wanted to pass College Algebra 101 which was the minimum gen ed. requirement for all majors. 

It looks like now our state schools just require one course of "Contemporary Math" or "General Math" (lower than College Algebra) for education majors. This makes no sense because it is less than our state's high school graduation requirements (4 credits, including Algebra I, II, Geometry and a fourth higher level math course). 

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2 minutes ago, Skippy said:

It looks like now our state schools just require one course of "Contemporary Math" or "General Math" (lower than College Algebra) for education majors. This makes no sense because it is less than our state's high school graduation requirements (4 credits, including Algebra I, II, Geometry and a fourth higher level math course). 

 

My goodness. That’s... depressing.

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What you're describing as far as how to teach is basically what every math curriculum we've used does, with the exception of Life of Fred. My kids are the same as yours, they don't really use manipulatives after age four except when I want to show a new concept. My oldest does mental math naturally as well and finds his own shortcuts for calculations and can describe what he's doing. Maybe you would be happier looking for supplemental math stuff to broaden her math education. You can search for that on the Accelerated Learner board in particular.

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2 hours ago, square_25 said:

 

What do you do with a 6 year old kid who’s reading at a grade 5 level, doing math at a grade 3 level, and is socially actually a bit young for her age??? That’s a real conundrum for a teacher with 15-25 kids.

Its not a conundrum when enrichment or small group instruction is allowed. It is impossible if its whole class, full inclusion and your child isn't 'average', or if its not project based.  The ex-gifted teachers have no trouble with the spread of abilities in the classroom -- until they are forced to stick to remedial or grade level.

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4 minutes ago, mshanson3121 said:

Amen.  I spoon feed a list of dozens upon dozens of articles etc... most of which have links to studies, and she can't even be bothered to look at that those... lol. So the only real conclusion is that the OP is either lazy or close minded, since she's not willing to put the effort in. What you said is so true - a good teacher is a good learner. And if you're not willing to open your mind to learning other view points and methods, well then.... fill in the blanks I guess.

I looked. I can’t answer all studies at once. Give me a specific study, and I’ll tell you whether I think its methods are good. Lots of studies are flawed or irrelevant. Some I agree with. 

This isn’t the first time I’ve thought about it. I’m relatively familiar with the literature.

Edited by square_25

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Just now, mshanson3121 said:

 

Out of curioisity, are you on the spectrum?

 

Wow. This forum is much more hostile than I thought it would be. No, I’m not. You?

 

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Just now, mshanson3121 said:

 

I meant it honestly, unlike your juvenille rhetoric. Anyways, good luck to you, and your child.

 

Is the only response to you to agree that you’re right, teaching is harmful, and manipulatives are essential? I already tried to have a substantive conversation with you. I asked you for specifics of what we’re doing wrong. I’m happy to discuss.

We value free play. We did a play-based preschool. Why is doing the work my child likes offensive to you??

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Just now, mshanson3121 said:

 

Haha, and I said or insinuated any of that... where exactly? The only defensive and argumentative one has been you, when a few of us present ideas that you don't like. 😉
Anyways, it's been real, but play time is over now 🙂 

 

The only thing I’ve pushed back on is manipulatives. I looked at the Liping Ma book (which I liked), I looked at Peter Gray’s stuff (which seems flawed if interesting), I liked the idea of MEP.

Do you think I shouldn’t do any direct teaching? What is it that bothers you about what we do? I’m honestly a bit confused.

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As far as I can tell, Peter Gray is very much on the unschooling side. Am I misreading? We are not unschoolers although we do limited academics per day, just writing and math.

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21 hours ago, square_25 said:

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.

 

In some ways it doesn't matter in the least. In others, c-rods are like a standard dialect in maths instruction for small people. It may not matter if they don't know it, but there is no advantage to not knowing it.

The CSMP minicomputers I mentioned are based on c-rods. A bright kid wouldn't need familiarity to use the minicomputers, but it helps.

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1 hour ago, HeighHo said:

Its not a conundrum when enrichment or small group instruction is allowed. It is impossible if its whole class, full inclusion and your child isn't 'average', or if its not project based.  The ex-gifted teachers have no trouble with the spread of abilities in the classroom -- until they are forced to stick to remedial or grade level.

 

Yeah, I see that! Have you seen successful project-based classes with a wide range of ability?

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1 hour ago, square_25 said:

Wow. This forum is much more hostile than I thought it would be...

Just like anyplace on the Internet, things can easily get heated. In my limited experience, it is best to have thick skin and reply with a gentle answer. It is easy to forget that, for the most part, we are all basically on the same team here.

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11 minutes ago, Skippy said:

Just like anyplace on the Internet, things can easily get heated. In my limited experience, it is best to have thick skin and reply with a gentle answer. It is easy to forget that, for the most part, we are all basically on the same team here.

 

It’s my first post here. I still need to decide whether participation is a good idea... you’re right, though. It’s best not to escalate. That never goes well.

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Frankly, people are speaking past each other in this thread for whatever reason, and I encourage OP to stay and lurk the boards... I would not consider interactions in this thread to be the common way things are done here.  

OP, you asked for sources and studies.  A huge list was provided, and you said, "Nah, I don't have time for those."  What do you expect to have happen when you do that?  

Other posters are shouting out, "Developmentally inappropriate!" while ignoring signs that this particular child described by the OP is gifted in math and being taught at her own pace and in an appropriate way.  A six year old who grasps all four operations? Of course OP wants to give her more mathematical meat to chew on.  

 

The problem is, there is a group talking about generalities- what the studies show, what is typical, etc.  And we all know no study can be applied to one child.  Studies are a jumping off point, nothing more.  And I would argue most group learning studies, in the context of homeschool, are useless anyway.  Early academics don't work in schools, but that really gives us very little idea on whether or not laying on the floor with mommy and building rectangular prisms with blocks is going to ruin a child forever.

OP, stop looking for studies and just teach the child in front of you.  As someone who has had accelerated learners, I give you only a warning: if a road block occurs, find something else to do.  Just because a child has accelerated seamlessly up to that point doesn't mean you should try to beat down barriers when they come.  Do step back and look at the meta-picture, do some geometry or art or reading or factoring or some other random topic, and circle back around.  You'll have to find your path as you go, because no curriculum is going to be just right.  That's ok.  

And as you've already recognized, kids don't accelerate evenly across all domains.  It is totally normal to have a fluent reader, doing pre-algebra, who is socially not on level, or perhaps who seems unable to remember to change socks on a regular basis (ds, I'm looking at you!).  Don't neglect the weaknesses.  It is so rewarding to focus on strengths and watch the huge leaps, but keep working on small talk and socks and whatever your child's gaps are.  

Ok, I guess that was several pieces of advice.  🙂  Sorry, I get carried away.  

 

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Oooh, I didn't mean to give the impression that I didn't have time for studies! I just have a hard time discussing a lot of studies at once :-).  I know there are pretty convincing studies that direct instruction doesn't work in preschool, and I'm very much in agreement with that. We did a play-based preschool for my older daughter, we're doing one for my toddler as well: I don't have any disagreement with that.

Some of the other studies I'm more dubious about, and it's not obvious to me they are applicable to how I teach my daughter. And as you say, very few studies are about the home setting, with individualized instruction (which is the thing I LOVE about homeschooling.) And of course... I am only teaching this one kid, and I'm going to teach her the way she seems to like being taught. And honestly, she's the daughter of two mathematicians, it's not exactly surprising she's accelerated in math! 

I'm going to attach a copy of the kind of lesson she likes best in the next message, which she calls a "shapes" lesson. I don't know why this is what she loves, but she vastly prefers it to pure calculations. (Even though of  course she HAS to do calculations to figure these out.) 

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7F41062C-C93D-4EC2-8F66-F837E53B1B4E.jpeg

The way it works, I give her the shapes equations, grouped together, and she fills in the shapes (she's filled them in already above. They were empty when I gave them to her.) The rule is that in a given question, a shape only contains one particular number. She loves these questions and is very good at them. She doesn't do algebraic manipulations to solve them, obviously: they are purely about number sense and trying things. She could probably do these every day, although I do try to branch out a bit (I try to branch out into more puzzles, like Beast Academy, or into questions with visuals that demonstrate specific ideas.) 

Edited by square_25

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6 minutes ago, HomeAgain said:

MEP does a lot of those puzzles in their year 1 curriculum.

KblankMEP.jpg

KfilledMEP.jpg

KshapeMEP.jpg

Cool! Are those on their website? I glanced through the link but I didn't see them. I should look more. 

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Just now, square_25 said:

Cool! Are those on their website? I glanced through the link but I didn't see them. I should look more. 

They are.  The way each year is set up is that you have three parts:
practice pages (student worksheets)
lesson plans (detailed breakdown of every step of every lesson, including small breaks)
copymasters (classroom activity pages that are also on the studen worksheets, but bigger for the teacher to print for an overhead projector)

I didn't always print the copymasters but we did the activities anyway.  I drew the same ones out on paper or the whiteboard for him to play with.  He did year 1 before his handwriting was small enough to do well on the practice pages.

Also, there are at least a few vendors I have found that sell square centimeter graph paper books.  They run between $2.50 and $6.  The size of the graph paper is just right for small hands to fill in and keep larger equations straight, and it's also the same size as c-rods for Singapore type bar models.

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Just now, HomeAgain said:

They are.  The way each year is set up is that you have three parts:
practice pages (student worksheets)
lesson plans (detailed breakdown of every step of every lesson, including small breaks)
copymasters (classroom activity pages that are also on the studen worksheets, but bigger for the teacher to print for an overhead projector)

I didn't always print the copymasters but we did the activities anyway.  I drew the same ones out on paper or the whiteboard for him to play with.  He did year 1 before his handwriting was small enough to do well on the practice pages.

Also, there are at least a few vendors I have found that sell square centimeter graph paper books.  They run between $2.50 and $6.  The size of the graph paper is just right for small hands to fill in and keep larger equations straight, and it's also the same size as c-rods for Singapore type bar models.

Were these puzzles parts of the copymasters? I looked at the worksheets, I think, and I didn't see those (although I saw some questions my daughter might enjoy.) 

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2 minutes ago, HomeAgain said:

Also, there are at least a few vendors I have found that sell square centimeter graph paper books.  They run between $2.50 and $6.  The size of the graph paper is just right for small hands to fill in and keep larger equations straight, and it's also the same size as c-rods for Singapore type bar models.

 

That's a good idea, thanks! Might be handy for starting geometry as well... 

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Ah-ha! Found the puzzle you were doing. Cool :-). I like what I've seen of MEP so far, so I'll use it as a change of pace for sure. 

Edited by square_25

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5 minutes ago, square_25 said:

Were these puzzles parts of the copymasters? I looked at the worksheets, I think, and I didn't see those (although I saw some questions my daughter might enjoy.) 

Yes, the copymasters.  Some are only class activities.  All are in the lesson plans (along with hands on lessons.  Worksheets are only reinforcement)

MEP appears to go slow but it is fast paced.  The entire first month is spent on number bonds to 3 and being as concrete as possible while learning how to build.  Year one only manipulates numbers up to 20, I believe.  My youngest did year 1 and part of year 2 in one school year, and went through year 4 before we moved on to something different.(MEP starts to be a bit more British than I would like)  Now he's at another jumping point and in order to continue that strong base before moving to an abstract-only program, he's going to spend the rest of elementary shoring up with Gattegno.  It requires less writing but more work than his current curriculum and has open ended lessons.

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I like working only with small numbers at first. That's what we did, actually (we started math lessons at 4.5, I think. She had already been reading for a year, which made things easier.) 

Is it just me, or are they using a square root symbol for division? Or are my eyes playing a trick on me? 

What do you mean by "more British than you would like"? And what's Gattegno? 

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7 minutes ago, square_25 said:

I like working only with small numbers at first. That's what we did, actually (we started math lessons at 4.5, I think. She had already been reading for a year, which made things easier.) 

Is it just me, or are they using a square root symbol for division? Or are my eyes playing a trick on me? 

What do you mean by "more British than you would like"? And what's Gattegno? 

Not quite a square root symbol.  And they focus on metric, rather than adding imperial measurements, plus their geometry terminology is slightly different.

Gattegno...surely..?  The video I linked on the first page?  That you addressed as though you watched and understood what was going on with the rods?  How the 6yos were comfortable with verbal algebraic problems and multiplying/dividing fractions?

There are 6 books going from "the study of numbers to 20" up through algebra/geometry.  The first two books are available online for free on issuu, and I believe linked in the free resource list in the General Education subforum.  The presentation order with Gattegno is different than any other I'd seen and they will be a nice, relaxing way to finish elementary school, I think, without getting bogged down.  He'll still use his supplemental books, but these will be the main.  I want everything well set for when he gets to higher math.

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7 minutes ago, HomeAgain said:

Not quite a square root symbol.  And they focus on metric, rather than adding imperial measurements, plus their geometry terminology is slightly different.

Gattegno...surely..?  The video I linked on the first page?  That you addressed as though you watched and understood what was going on with the rods?  How the 6yos were comfortable with verbal algebraic problems and multiplying/dividing fractions?

There are 6 books going from "the study of numbers to 20" up through algebra/geometry.  The first two books are available online for free on issuu, and I believe linked in the free resource list in the General Education subforum.  The presentation order with Gattegno is different than any other I'd seen and they will be a nice, relaxing way to finish elementary school, I think, without getting bogged down.  He'll still use his supplemental books, but these will be the main.  I want everything well set for when he gets to higher math.

 

Yes, I watched the video, but I tend to watch with things at fairly low volume: sometimes I watch when the toddler is sleeping, and we have a tiny apartment. Rewatching with headphones, I see that the guy who's presenting with the rods is Gattegno. Please don't assume I ignored what you sent? 🙂 I prefer not to have conversations where people assume I'm operating in bad faith. 

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10 minutes ago, square_25 said:

Out of curiosity, why did you prefer MEP to Gattegno at first? 

I didn't.  MEP was a third shot in the dark with my youngest. 
Tried the Math U See I used with my oldest - youngest hated it.  Then I bought Mathematical Reasoning.  We both hated it.  It was the exact opposite of MUS, but it was very much the exact opposite of MUS.  Very little direct teaching or order to the workbook.
He did like Anno's Math Games.  I was not interested in investing more money in a math curriculum for a 5yo and figured we'd do the year lightly with Anno, but that when he was ready to write I'd try MEP and see if that worked.  It was free and I could print it out, which was my whole reasoning for trying it.  Plus, it was a good balance of hands on without being overly so, unlike how he felt about MUS, and it was orderly.  I was not willing to abandon what was working so we stuck with it.  We have always supplemented.  I brought in lessons from Gattegno when it seemed appropriate, brought lessons over from MUS, lessons from Montessori, things from CSMP, he's done Life Of Fred, Hands On Equations.....He usually spends about an hour a day: 30-40 minutes on a main lesson and 20-30 minutes on fun lessons or exploring topics. This past summer we did book 1 of Gattegno for fun each day. 

Since he's now getting to the end of his Right Start book, I offered him choices again, like I did when we left MEP.  I don't to go forward too quickly.  Besides being little, he's really enjoying math and all the logic behind it.  I offered him a few different programs that I thought he would like and he picked Gattegno with LoF books on the side.  We can go back and forth in the books if necessary since neither uses any workbook.

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Yes... Free and printable definitely appeals. Have you taken a look at BA at all or does it not look good to you? I’ll have take a look at the others you mentioned, I’m not familiar with them.

Are you still using MUS with your oldest? How old are they?

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6 minutes ago, square_25 said:

Yes... Free and printable definitely appeals. Have you taken a look at BA at all or does it not look good to you? I’ll have take a look at the others you mentioned, I’m not familiar with them.

Are you still using MUS with your oldest? How old are they?

He didn't like BA.  I borrowed the 3 set when we left MEP and had him go through it.  He likes comics, but he doesn't like comics in his math.  He prefers black and white with lots of interaction.  Does that make sense?  BA felt busy and cluttered to him, like it was trying to sell the idea that math was fun.  He didn't want that.  I don't blame him.  It was the same way I think we both felt about Mathematical Reasoning.  I let him pick what he likes for the most part.  He tells me what he wants in a subject, I find samples, he looks at them and chooses one.  Mostly.  This year I chose language arts, and I slid in an extra Latin book because his choice was making me go nuts.  It's a balance.  But it also means I keep an interesting collection of books and tools so that he has the ability to learn to use them and see what he likes.

My oldest is in college for engineering. He outgrew MUS. 🙂 He really enjoyed it, though, up through pre-algebra. He was a product of a time when there was not a lot out there for homeschoolers.  I started with Saxon, which made him cry, and tried MUS after that, then AOPS when he was ready.  He went on to a public school with dual enrollment at the cc for his math classes.

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18 hours ago, HomeAgain said:

He didn't like BA.  I borrowed the 3 set when we left MEP and had him go through it.  He likes comics, but he doesn't like comics in his math.  He prefers black and white with lots of interaction.  Does that make sense?  BA felt busy and cluttered to him, like it was trying to sell the idea that math was fun.  He didn't want that.  I don't blame him.  It was the same way I think we both felt about Mathematical Reasoning.  I let him pick what he likes for the most part.  He tells me what he wants in a subject, I find samples, he looks at them and chooses one.  Mostly.  This year I chose language arts, and I slid in an extra Latin book because his choice was making me go nuts.  It's a balance.  But it also means I keep an interesting collection of books and tools so that he has the ability to learn to use them and see what he likes.

My oldest is in college for engineering. He outgrew MUS. 🙂 He really enjoyed it, though, up through pre-algebra. He was a product of a time when there was not a lot out there for homeschoolers.  I started with Saxon, which made him cry, and tried MUS after that, then AOPS when he was ready.  He went on to a public school with dual enrollment at the cc for his math classes.

 

You know, that makes a ton of sense about Beast Academy! I didn't even think about that. We'd already covered pretty much everything in the 2A, 2B, and 2C guides, so when we got them, I just gave the guides to my daughter to read whenever she wanted, and didn't try to incorporate them into our math lessons. I did try to incorporate the practice books, with rather mixed results. 

I try to be as guided by my daughter as I can be as well :-). I find there's a LOT less grumbling about practicing basic arithmetic if it's in the service of something she actually likes. That's how we wound up doing so many shapes lessons: somehow, doing 10 calculations while she tries to find two numbers that add to something and multiply to something else is OK, while doing a simple worksheet where she just multiplies and adds things isn't! Even though really the latter is easier... 

Which AoPS books did you use? I'm not super familiar with the earlier ones, although I've been using the Precalculus one for my class over and over again recently.  

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2 hours ago, square_25 said:

 

You know, that makes a ton of sense about Beast Academy! I didn't even think about that. We'd already covered pretty much everything in the 2A, 2B, and 2C guides, so when we got them, I just gave the guides to my daughter to read whenever she wanted, and didn't try to incorporate them into our math lessons. I did try to incorporate the practice books, with rather mixed results. 

I try to be as guided by my daughter as I can be as well :-). I find there's a LOT less grumbling about practicing basic arithmetic if it's in the service of something she actually likes. That's how we wound up doing so many shapes lessons: somehow, doing 10 calculations while she tries to find two numbers that add to something and multiply to something else is OK, while doing a simple worksheet where she just multiplies and adds things isn't! Even though really the latter is easier... 

Which AoPS books did you use? I'm not super familiar with the earlier ones, although I've been using the Precalculus one for my class over and over again recently.  

I feel like the puzzles are better in BA3+. I never sat down and compared BA 2 with 3 but it seems like 2 is more straightforward. BA 4 is even better. I usually just leave the stack of the years books on the table and DS does whatever. 

Have you looked at Critical Thinking Companies supplemental math books? You might like the Balance Math and More series. I do not recommend the regular math books for accelerated kids though. 

You probably don't need to wait three years to introduce variables. Just draw shapes like you have been then use a variable and show how the answer can include the variable and it's still true. It wasn't a big deal for my kid.

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19 minutes ago, Sarah0000 said:

I feel like the puzzles are better in BA3+. I never sat down and compared BA 2 with 3 but it seems like 2 is more straightforward. BA 4 is even better. I usually just leave the stack of the years books on the table and DS does whatever. 

Have you looked at Critical Thinking Companies supplemental math books? You might like the Balance Math and More series. I do not recommend the regular math books for accelerated kids though. 

You probably don't need to wait three years to introduce variables. Just draw shapes like you have been then use a variable and show how the answer can include the variable and it's still true. It wasn't a big deal for my kid.

Mind giving an example? Like, the answer winds up as x=3 instead of filling in the square?

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1 hour ago, Sarah0000 said:

I feel like the puzzles are better in BA3+. I never sat down and compared BA 2 with 3 but it seems like 2 is more straightforward. BA 4 is even better. I usually just leave the stack of the years books on the table and DS does whatever. 

Have you looked at Critical Thinking Companies supplemental math books? You might like the Balance Math and More series. I do not recommend the regular math books for accelerated kids though. 

You probably don't need to wait three years to introduce variables. Just draw shapes like you have been then use a variable and show how the answer can include the variable and it's still true. It wasn't a big deal for my kid.

 

Replying to everything else now (I was on my phone before, so didn't feel like typing a lot!): it's good to hear that the puzzles get better! In general, I feel like AoPS is really excellent at providing harder questions, that's a big strength of theirs. We'll probably get the other books as well, then, because the puzzles are really fun as enrichment. (Besides, I'm pretty sure I can get them as freebies, so why not? 😛

I haven't seen the Critical Thinking Company's books, no! I'll have to take a look. 

I just asked my daughter "If x stands for a number, and x plus 4 is 7, then what's x?" and she answered without thinking "x is 3," so I see what you mean about variables! I'm going to have to think about whether we should switch to variables or not... I sometimes feel like shapes would help out even college kids conceptualize the idea that "a variable is just a stand-in for a number" (I occasionally feel like tattooing that on my forehead as a constant reminder), but on the other hand, if we started working with variables soon, I could keep reminding her about that... food for thought, thank you for the suggestion. 

Did you ever wind up having any conceptual variable issues after introducing them? 

 

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