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Teaching properties of numbers.

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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. 

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Miquon Math and MEP are the two that jump to my mind first after reading your post, have you looked at those? 

You might also look at the dragonbox math apps. There is an algebra one meant for young kids.

 

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MEP
Right Start
Gattegno
Math U See (but in a different order and slower than the rest)
ST (Computer based)
And I'll just throw in a smidge of CSMP in here, because it teaches from a different angle and those mini-computers are a lot of fun.

Dragonbox is one of our only apps.  It definitely teaches basic algebra, but it's one of those you either want to let them dive into without knowing anything, or wait and have them dive in after understanding basic operations.  I let ds play with the series after he completed Hands on Equations.

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Most elementary math programs are going to teach the commutative and associative properties of addition and multiplication and teach them with the correct terminology. But, they are going to use numbers vs variables. So, kids learn that 2*3=3*2 . 

Edited by 8FillTheHeart
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1 hour ago, sweet2ndchance said:

Miquon Math and MEP are the two that jump to my mind first after reading your post, have you looked at those? 

You might also look at the dragonbox math apps. There is an algebra one meant for young kids.

 

I've looked at dragonbox, but I'm not impressed, I have to say. It doesn't seem to explain why you can move things around, exactly. I've taught way too many higher level kids (high school and college) that didn't quite seem to be aware of what an equals sign means to be comfortable teaching "manipulations" of equations. 

Of the programs I've looked at, Miquon has seemed closest to what I like, but my daughter is very abstract minded and doesn't need the manipulatives, so I don't feel tempted to buy it. Good to have a confirmation that it's the right kind of thing, though! What's MEP like?  

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1 hour ago, mshanson3121 said:

Not at that age really, no, because that's not developmentally appropriate. Math in the early years is best taught in concrete ways and theories that children can understand. Abstract reasoning is better left until they're older. What you're proposing is an incredibly abstract concept that they do not need yet. As the daughter of a genius whose special interest was mathematics and physics, I urge you to try to be mindful of not overloading your child in your eagerness. What seems easy and simple to you, is not necessarily so to your 6 year old. Also, there is a significant amount of evidence that formal mathematics should be delayed until a child is older. That instead, focusing on real life, concrete math is a much better way to produce children proficient in math, with far better understanding and reasoning ability. Just something to think about before trying to find curriculum that introduce algebra to a 6 year old 😉

I actually don't think those are particularly abstract. We don't use algebra. But yes, my daughter can explain why the product of two numbers is the same if you swap the order, although of course her explanation will use specific numbers. It's the same argument whatever the pair of numbers, though (just draw the array!) so I think she accepts it's true in general, although she won't think of it in terms of a, b and c. I was just giving that format for the adults on this board :-). 

I'm not sure what you mean by formal mathematics. I think I tend to err on the side of concreteness if anything, since I try to only work with whole numbers that are easy to picture, I don't like the number line (I don't think it really helps reasoning at all and seems removed from how numbers are visualized by children), and I bring all of our work back to pictures and definitions. However, that doesn't stop from working on rudiments of algebra. For example, how would you explain why the product of three numbers is the same, no matter whether you do the first pair first or the last pair? (That is, why (a*b)*c = a*(b*c).) Is there anything inaccessible about asking what you can fill in the square if we have that 

square + 2 = square*square? 

(Imagine that it's actually a square, not just me writing "square." I leave an empty space to fill in the shape.) 

Basically, my question is whether any curriculum is very focused on working with all the operations using very concrete definitions to develop "algebraic" reasoning. One issue I've seen in almost all curriculums I've looked at is that kids start using an equals sign to mean "now compute this" and not "those two things are the same," which can lead to tons of trouble later on. I started using Beast Academy as filler for when I didn't feel like making up a lesson, and my daughter started misusing an equals sign in a month, which made me worried as she'd never done it before. So then I stopped using them except for puzzles.

 

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1 hour ago, 8FillTheHeart said:

Most elementary math programs are going to teach the commutative and associative properties of addition and multiplication and teach them with the correct terminology. But, they are going to use numbers vs variables. So, kids learn that 2*3=3*2 . 

As a question, do you think kids remember WHY those are true or do they get lost in the shuffle of learning to performed operations? I'd say that out of the adults I ask, only a third can tell you why a*b = b*a, and most think that the associative property (why (a*b)*c = a*(b*c)) follows from the commutative property, which it does not. Which makes sense, since their math programs were much more focused on computation than on the whys. 

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I have only been homeschooling our dd for five years, so I don't know how it compares to other programs, but MEP is what we use. MEP will use letters in place of numbers, though I can't remember quite when that happens, perhaps Year 2? We didn't use all of Year 1, so I'm not sure. At any rate, you can look over all the materials here:

http://www.cimt.org.uk/projects/mepres/primary/index.htm

You really need to look carefully at the lesson plans, as that is where all the teaching is laid out. It is fairly teacher intensive. I'm not sure if it is what you are looking for, but we are finding it to be challenging in a good way.

 

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21 minutes ago, square_25 said:

As a question, do you think kids remember WHY those are true or do they get lost in the shuffle of learning to performed operations? I'd say that out of the adults I ask, only a third can tell you why a*b = b*a, and most think that the associative property (why (a*b)*c = a*(b*c)) follows from the commutative property, which it does not. Which makes sense, since their math programs were much more focused on computation than on the whys. 

My kids do. I have taught all of them the same way and 7 of them are either adults or in alg up.  Honestly, any math program is only as good as the person teaching it and the student learning from it. I use Horizons math for elementary school which gets knocked as being focused on computation vs the whys (which I disagree with, btw). But, my kids have never struggled with theory or math. My oldest is a chemE and one of my ds's is at Berkeley for grad school in theoretical physics. 

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).

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Just now, mshanson3121 said:

You may not think they're abstract. You're an adult with full comprehension. Your child however, is not 😉  However, that aside, there is a very big difference between teaching commutative property with numbers, and teaching the abstract reasoning of a*b = b*a using variables. One is concrete (with numbers), one is not. In your post you specifically called it algebra and used it with variables, hence my reply.

By "formal" math, I mean a curriculum. Curriculum is not necessary and in fact, is often detrimental (long term) in the younger years. If you actually study the historical teaching of mathematics, it was withheld until students were much older (teens). When we move into the 1800s, even early 1900s, while formal instruction started younger (which had little to do with math and more to do with the institute),  for the first few years, math was oral, and was based on manipulatives and real life problem solving scenarios. Not drilling rote facts. There was minimal written work. Rows of math problems on paper, to a large degree is abstract. Manipulatives in hand and disucssing real life situations, is far more concrete and easily understood in a child's life.

Studies have shown that withholding formal math (curriculum based math like workbooks, Beast Academy etc...), and instead focusing on real life math, has better outcomes. If you wonder why people lack algebraic thinking, it's because they lack a concrete foundation. They have spent their years being drilled on equations and stupid word problems, instead of using math, manipulating math, seeing it in their life etc...

What studies do you mean? I don't think this has been particularly well studied, mostly because people are very hesitant to do this as a large-scale experiment. Cite, please? 

I agree that lacking a concrete foundation is precisely the problem, so we probably agree on a lot! But what exactly makes a question like 3+4 or 3*4 not concrete? If you think of it as just putting items together (which I do), they are very concrete, and my daughter thinks they are, too. 

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12 minutes ago, 8FillTheHeart said:

My kids do. I have taught all of them the same way and 7 of them are either adults or in alg up.  Honestly, any math program is only as good as the person teaching it and the student learning from it. I use Horizons math for elementary school which gets knocked as being focused on computation vs the whys (which I disagree with, btw). But, my kids have never struggled with theory or math. My oldest is a chemE and one of my ds's is at Berkeley for grad school in theoretical physics. 

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).

I teach for AoPS, so I did get some Beast Academy books to try out. I really like the puzzles, but I'm not sure they are a good fit otherwise. I like more review than they do. And I'm not sure that the hard problems actually communicate important concepts (other than the idea that it's OK to get stuck on a problem, which is definitely an important idea, but I feel like one that my daughter was already aware of.) Nonetheless, I do plan to continue using their puzzles. 

You have a good point that the program matters a lot less than the student and the teacher's comfort with the concepts. I'm pretty sure my daughter would do more or less fine with most programs, since she's not having a hard time with anything, and she's the kid of two mathematicians to begin with. 

What did you like about Horizons? 

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48 minutes ago, knitgrl said:

I have only been homeschooling our dd for five years, so I don't know how it compares to other programs, but MEP is what we use. MEP will use letters in place of numbers, though I can't remember quite when that happens, perhaps Year 2? We didn't use all of Year 1, so I'm not sure. At any rate, you can look over all the materials here:

http://www.cimt.org.uk/projects/mepres/primary/index.htm

You really need to look carefully at the lesson plans, as that is where all the teaching is laid out. It is fairly teacher intensive. I'm not sure if it is what you are looking for, but we are finding it to be challenging in a good way.

 

Interesting, thank you! Those are nifty. I appreciate the link. 

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1 minute ago, mshanson3121 said:

 

Perhaps one of the best studies on the subject, which dates back to the 1920s, and has since been well reviewed, is the Benezet experiment, which you can easily find online. There have been other through the years as well. Not only that, the topic of early academics (and why it is not recommended) as a whole has been extensively studied.

It's not the question itself, it's how it's delivered. 3+4 on paper is far more abstract than, at dinner time saying, I have 3 plates on the table, can you add four more please? How many plates do we have now? One form encourages procedural understanding, the other conceptual.

I'm not sure I agree. If you talk about the 3+4 on the paper or you draw a picture for it, it becomes quite concrete. 

I've seen the Benezet experiment. It's suggestive but very small, and it's not clear to me how it translates into real life. For example, I'm sure it heavily depends on the mathematical literacy of the person who's discussing the math with the kids. 

Also, I found that worksheets were incredibly helpful to connect concepts. The picture

o o o o o               o o o 

can be described in different ways (like, 3+5 = 8 and 8 - 5 = 3 and 8 - 3 = 5) and I found that this kind of exercise was very hard for us to build into day to day life. Similarly, thinking about why a*b is equal to b*a required a specific picture which just didn't come up all the time in our life. And I've definitely seen fruits of those visual exercises when we do mental math nowadays. Like, if she gets 3 dollars a week and we talk about how long it takes her to save up for a $60 toy, she thinks of "How many 3s do I need to make 60?" and then after we test out enough numbers, she realizes that it's the same thing as "What is the number that multiplied by 3 gives 60?" and the fact that she knows order doesn't matter with multiplication really helps her out with the reasoning here, since it's easier to test numbers by taking three of them rather than taking that many threes. Am I making sense here? 

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14 minutes ago, square_25 said:

I teach for AoPS, so I did get some Beast Academy books to try out. I really like the puzzles, but I'm not sure they are a good fit otherwise. I like more review than they do. And I'm not sure that the hard problems actually communicate important concepts (other than the idea that it's OK to get stuck on a problem, which is definitely an important idea, but I feel like one that my daughter was already aware of.) Nonetheless, I do plan to continue using their puzzles. 

You have a good point that the program matters a lot less than the student and the teacher's comfort with the concepts. I'm pretty sure my daughter would do more or less fine with most programs, since she's not having a hard time with anything, and she's the kid of two mathematicians to begin with. 

What did you like about Horizons? 

Horizons suits me as a teacher and is spiral. I dislike mastery programs. But, many people don't like spiral and claim it just jumps around. I can see the big picture that Horizons is teaching. For example, they teach regrouping in a simple straightforward approach (for example, changing 3 ten rods and 6 one rods to 2 ten rods and 16 one rods) long before they teach borrowing in subtraction so that kids do completely understand what they are doing. My kids are fine with just the images in the text bc they hate manipulatives.

I am not a BA fan (or at least the 2A book I purchased). Neither do I like SM. I think they overcomplicate very simple concepts and word things in a way that often obscures the simple question they are asking. I also prefer solving algebraically vs bar diagrams, so I like HOE better than SM's approach.

Edited by 8FillTheHeart
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1 minute ago, 8FillTheHeart said:

Horizons suits me as a teacher and is spiral. I dislike mastery programs. But, many people don't like spiral and claims it just jumps around. I can see the big picture that Horizons is teaching. They teach regrouping in a simple straightforward approach (for example, changing 3 ten rods and 6 one rods to 2 ten rods and 16 one rods) long before they teach borrowing in subtraction so that kids do completely understand what they are doing. My kids are fine with just the images in the text bc they hate manipulatives.

I am not a BA fan (or at least the 2A book I purchased). Neither do I like SM. I think they overcomplicate very simple concepts and word things in a way that often obscures the simple question they are asking. I also prefer solving algebraically vs bar diagrams, so I like HOE better than SM's approach.

Ah, yeah, that's how I teach subtraction. I'm honestly not sure I'm going to teach "borrowing" or "carrying" until my daughter has to take a timed test... I don't find them conceptually all that useful. 

I prefer images to manipulatives, at least with my older girl (my younger one is 2.5, so we'll see how that goes.) 

What did you think BA overcomplicated, if you don't mind me asking? We have books 2A, 2B and 2C right now. My daughter loves the comics, but I'm not really using it as math right now. 

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Maybe overly complicates isn't the correct word choice. My Dd had already learned Roman numerals through 1000 so when she saw them in 2A she wanted to go through and answer everything in Roman numerals vs the basic base 10 concepts they were trying to teach. Part of the problem was that I bought the book when she was in 2nd grade but already advanced in math. 

But still, I wasn't impressed but the teaching in the book. I have been teaching elementary math non-stop now since 1994 and I know what I like and what I don't. And, I know the big picture of where we want to end up. I could not see my spending more $$ on the texts when Dd finished it in a few days.  I hear that the upper elementary texts are different, but I have decided to just stick with Horizons and HOE bc I know how to teach them easily and my kids have all been mathematically solid.

I'm on the fence about the AoPS pre-alg text. That is one of only a couple upper level books that I don't already own. One ds jumped into AoPS at the intermediate text and never looked back. His equally gifted sister detested the alg text and begged to go back to Foerster's, so we did. My 11th and 7th graders are not AoPS level math students, but my current 3rd grader is. She will be on par with her older brother and ready for math beyond elementary level around age 10.  The difference is he loved math and everything about it. She, otoh, is all about music and violin. Not sure it will be worth the additional time it will take or not. I need to wait and see how she is functioning in a yr. 

I have seen how my kids have developed. They all tend to gravitate toward their natural interests,so that is my cue. My Dd who is now a college sophomore could have used AoPS texts and have graduated at a similarly advanced level in math (multivariable, diifEQ, linear 1&2) but had zero desire to do so. She wanted to focus on languages and literature and is still that way today. So, dropping them wasn't an issue.

i sort of let my kids be the guiding force. If they want more, I give more. I have a minimum level of expectation, but I am not going to push them to do beyond what I think is necessary for who they are. Whether or not they want to spend more time deriving and in theory and working for hrs on math daily is completely individual. I am perfectly content with my kids who aren't inclined to spend hrs immersed in theory to stay in texts focused on application. 

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I'm not an experienced mathematician or instructor yet (oldest is six). But my son primarily used Miquon for grades 1/2 plus a smattering of other stuff. Just the other day he started writing his problems from BA4 algebraically with variables in an attempt to say he didn't have to actually solve them. He manipulates numbers on either side of the equals sign, I showed that initially with a balance scale and written balance scale problems. Is that part of what you mean? 

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23 minutes ago, Sarah0000 said:

I'm not an experienced mathematician or instructor yet (oldest is six). But my son primarily used Miquon for grades 1/2 plus a smattering of other stuff. Just the other day he started writing his problems from BA4 algebraically with variables in an attempt to say he didn't have to actually solve them. He manipulates numbers on either side of the equals sign, I showed that initially with a balance scale and written balance scale problems. Is that part of what you mean? 

Hmm, perhaps :-). What do you mean about manipulating numbers on either sides of the equals? 

We haven't done proper variables yet, just shapes that get filled! What were the BA4 problems like? 

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OP, I think that MEP would work very well for what you are looking for.  Even Year 1 material has lots of "meat" in it for a child who is ready to think about relationships between numbers.  Please don't judge the program based on just the worksheets--lots of  teaching and additional problems are in the Teacher's Notes. 

Here's a link...http://www.cimt.org.uk/projects/mepres/primary/index.htm

Miquon is excellent as well.

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2 hours ago, 8FillTheHeart said:

Maybe overly complicates isn't the correct word choice. My Dd had already learned Roman numerals through 1000 so when she saw them in 2A she wanted to go through and answer everything in Roman numerals vs the basic base 10 concepts they were trying to teach. Part of the problem was that I bought the book when she was in 2nd grade but already advanced in math. 

But still, I wasn't impressed but the teaching in the book. I have been teaching elementary math non-stop now since 1994 and I know what I like and what I don't. And, I know the big picture of where we want to end up. I could not see my spending more $$ on the texts when Dd finished it in a few days.  I hear that the upper elementary texts are different, but I have decided to just stick with Horizons and HOE bc I know how to teach them easily and my kids have all been mathematically solid.

I'm on the fence about the AoPS pre-alg text. That is one of only a couple upper level books that I don't already own. One ds jumped into AoPS at the intermediate text and never looked back. His equally gifted sister detested the alg text and begged to go back to Foerster's, so we did. My 11th and 7th graders are not AoPS level math students, but my current 3rd grader is. She will be on par with her older brother and ready for math beyond elementary level around age 10.  The difference is he loved math and everything about it. She, otoh, is all about music and violin. Not sure it will be worth the additional time it will take or not. I need to wait and see how she is functioning in a yr. 

I have seen how my kids have developed. They all tend to gravitate toward their natural interests,so that is my cue. My Dd who is now a college sophomore could have used AoPS texts and have graduated at a similarly advanced level in math (multivariable, diifEQ, linear 1&2) but had zero desire to do so. She wanted to focus on languages and literature and is still that way today. So, dropping them wasn't an issue.

i sort of let my kids be the guiding force. If they want more, I give more. I have a minimum level of expectation, but I am not going to push them to do beyond what I think is necessary for who they are. Whether or not they want to spend more time deriving and in theory and working for hrs on math daily is completely individual. I am perfectly content with my kids who aren't inclined to spend hrs immersed in theory to stay in texts focused on application. 

Makes sense! I'm not sure to what extent I'll follow my daughter's lead, because she's only 6, and because my sense of what a minimum reasonable amount of math and science is probably pretty skewed by who I am (I think I would expect my kids to be fluent in calculus and statistics: calculus because it's cool and not very hard, statistics because it's useful day to day.) At the moment, math comes very easy to her, but she's more into books: she constantly has her nose in a book, but she doesn't do math for fun. 

On the other hand, the things she really liked in math were really conceptual (so far, she's really liked negative numbers, prime numbers, and binary), so I'm curious whether she'll get more excited about math later on, as you start running into cooler and cooler stuff. Certainly one thing I definitely agree with AoPS about is that when you learn math, you should go wide: I plan to tinker with all sorts of whole number stuff before getting into fractions, because bits of combinatorics and number theory are actually surprisingly developmentally appropriate for a small kid. 

That's funny about the Roman numerals! I thought the pirate numbers were cute, but that's certainly not how I taught base 10 (which we did a while ago): I just told her how the decimal system works, then we used to. I suppose I didn't motivate it much, but I'm not sure the motivation would have made sense to her before using it anyway. 

What did your son like about the AoPS text, and what did your daughter dislike? 

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The BA4 problems weren't algebra problems at all; they were solving for missing exponents. However my son wrote the answer as an algebra problem instead assigning the missing exponent as n. I'm just saying that it seems to me he understands things algebraically if he can do that so how algebraic concepts are covered in the programs we've used must be decent. Miquon introduces it fairly early, MM has some related problems and does include balance scales with shapes and numbers in Multiplication 2, BA introduces traditional but simple algebra in year 4 I think it was, maybe year 3. 

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3 minutes ago, Zoo Keeper said:

OP, I think that MEP would work very well for what you are looking for.  Even Year 1 material has lots of "meat" in it for a child who is ready to think about relationships between numbers.  Please don't judge the program based on just the worksheets--lots of  teaching and additional problems are in the Teacher's Notes. 

Here's a link...http://www.cimt.org.uk/projects/mepres/primary/index.htm

Miquon is excellent as well.

Thank you! I'm getting quite a lot of MEP recommendations, and looking at the worksheets, they are actually quite cute. My only quibble from what I've seen is that I just don't like the number line all that much :P. But that's probably something that's completely unavoidable, so I can skip around that. 

Does Miquon really require the Cuisenaire rods, or can one do without? I just haven't found with my daughter that she needs hands on stuff: she's very visual, so we get by perfectly well by drawing things on paper. 

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I'd say the CSMP minicomputer lessons (CSMP is fun to teach anyway) and Hands on Equations after a year of that.

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Just now, square_25 said:

Does Miquon really require the Cuisenaire rods, or can one do without? I just haven't found with my daughter that she needs hands on stuff: she's very visual, so we get by perfectly well by drawing things on paper. 

 

Many of the worksheets are designed to be used with the rods, but if she's an advanced student, you can probably do without.

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Just now, Sarah0000 said:

The BA4 problems weren't algebra problems at all; they were solving for missing exponents. However my son wrote the answer as an algebra problem instead assigning the missing exponent as n. I'm just saying that it seems to me he understands things algebraically if he can do that so how algebraic concepts are covered in the programs we've used must be decent. Miquon introduces it fairly early, MM has some related problems and does include balance scales with shapes and numbers in Multiplication 2, BA introduces traditional but simple algebra in year 4 I think it was, maybe year 3. 

Interesting! I haven't even thought about when we're going to move from shapes to letters. In terms of algebraic reasoning, I think what I want is a solid sense of the properties. Like, eventually one winds up needing things like a*(b+c) = a*b + a*c and I suppose when I think about "algebraic reasoning," what I want is for a kid to understand that a, b, and c are actually numbers, and that algebraic manipulations stem from visualizable facts about numbers. For example, as I said, my daughter knows why a*b = b*a, so I feel like when we eventually move to variables, that fact will not feel difficult. My goal is to have her get a feel for these facts so they don't feel like random memorizations. 

I think I also expect fluency in terms of moving from variables to numbers: I often have students who can't seem to figure out that they can CHECK identities by plugging in some numbers. I think variables feel very mysterious to them, as opposed to as "containers" for actual numbers. So that's another thing I'd want to work on. 

Oh, another thing I see all the time, even from very motivated AoPS students, is not being too clear on what an equals sign is. Like, I'd pose questions like "We have this equation, is it still true if we do the same thing on both sides?" and people are not jumping up and down yelling YES. That makes me feel like something's going wrong with their algebraic reasoning: it's pretty clearly true whatever the "same thing" is, as long as it's defined! This is why I started worrying when my daughter began to use an equals sign to mean "I performed a calculation" and not "These two are the same." 

Sorry about the essay! I have these fairly sophisticated goals that are apparently fairly hard to describe. Miquon always does look like a good program, so thanks for confirming that. 

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29 minutes ago, square_25 said:

Thank you! I'm getting quite a lot of MEP recommendations, and looking at the worksheets, they are actually quite cute. My only quibble from what I've seen is that I just don't like the number line all that much :P. But that's probably something that's completely unavoidable, so I can skip around that. 

Does Miquon really require the Cuisenaire rods, or can one do without? I just haven't found with my daughter that she needs hands on stuff: she's very visual, so we get by perfectly well by drawing things on paper. 


Is there a specific reason you really want to avoid c-rods?  They are actually quite a superb math tool, even if you think your daughter doesn't need them right now.  My son is reluctant when it comes to manipulatives, but even he appreciates c-rods when the concepts become more abstract and to be able to visualize relationships in an instant.  It's not just a hands on tool.  They reinforce in multiple visual ways, too.
You may be able to get by without, but I think you'd be doing you both a disservice if you decided not to have any manipulatives available if she does find them useful.  I'm going to link this video to show what the creator of the rods intended them for in a 1st grade classroom.  Again, yes, you can get by, but I would think a mathematician would see viable tools for what they are.

 

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28 minutes ago, square_25 said:

Interesting! I haven't even thought about when we're going to move from shapes to letters. In terms of algebraic reasoning, I think what I want is a solid sense of the properties. Like, eventually one winds up needing things like a*(b+c) = a*b + a*c and I suppose when I think about "algebraic reasoning," what I want is for a kid to understand that a, b, and c are actually numbers, and that algebraic manipulations stem from visualizable facts about numbers. For example, as I said, my daughter knows why a*b = b*a, so I feel like when we eventually move to variables, that fact will not feel difficult. My goal is to have her get a feel for these facts so they don't feel like random memorizations. 

I think I also expect fluency in terms of moving from variables to numbers: I often have students who can't seem to figure out that they can CHECK identities by plugging in some numbers. I think variables feel very mysterious to them, as opposed to as "containers" for actual numbers. So that's another thing I'd want to work on. 

Oh, another thing I see all the time, even from very motivated AoPS students, is not being too clear on what an equals sign is. Like, I'd pose questions like "We have this equation, is it still true if we do the same thing on both sides?" and people are not jumping up and down yelling YES. That makes me feel like something's going wrong with their algebraic reasoning: it's pretty clearly true whatever the "same thing" is, as long as it's defined! This is why I started worrying when my daughter began to use an equals sign to mean "I performed a calculation" and not "These two are the same." 

Sorry about the essay! I have these fairly sophisticated goals that are apparently fairly hard to describe. Miquon always does look like a good program, so thanks for confirming that. 

I don't think they sound like sophisticated goals. They sound like basic mathematical reasoning skills to me. 

My kids learn how to work with simple variables and performing basic operations on both sides of the equation in 2nd grade. Simple things like n+3=10, subtract 3 from both sides, n=7, check 7+3=10.

 

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49 minutes ago, square_25 said:

What did your son like about the AoPS text, and what did your daughter dislike? 

Ds lives in his head doing thought experiments and theorizing. (His goals in grad school are all about theory. He was describing the research team he is joining next semester....honestly I don't understand much.) But spending time (often hrs) jogging or shooting hoops thinking about how to approach solving a problem is his idea of fun. 

Dd, otoh, loves analyzing literature and following allusion trails. She spent hrs reading books like Les Mis (in French.) 

For both of them, the passion of the other is akin to torture. Dd just wanted to be taught directly the what's and how's and then explained the why's. She completely understands the whys. She just doesn't want to spend the time trying to discover them herself.

Math is a tool to her. Math is a puzzle to put together for ds. (Which is actually a funny allusion for me to use bc dd builds puzzles constantly for fun as a destressor in college. She is getting 8 new ones, including 2000 piece ones, for Christmas!)

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2 hours ago, HomeAgain said:


Is there a specific reason you really want to avoid c-rods?  They are actually quite a superb math tool, even if you think your daughter doesn't need them right now.  My son is reluctant when it comes to manipulatives, but even he appreciates c-rods when the concepts become more abstract and to be able to visualize relationships in an instant.  It's not just a hands on tool.  They reinforce in multiple visual ways, too.
You may be able to get by without, but I think you'd be doing you both a disservice if you decided not to have any manipulatives available if she does find them useful.  I'm going to link this video to show what the creator of the rods intended them for in a 1st grade classroom.  Again, yes, you can get by, but I would think a mathematician would see viable tools for what they are.

 

They seem like a totally reasonable tool, but I'm not sure how they are better than just drawing pictures for the visual child. She's never liked manipulatives of any sort that much. We started out counting using pennies and such, and she very quickly moved onto pictures and fingers, and uses those still. She's quite abstract minded: if I draw a box for her and put a 10 in the middle of it, she is happy enough to treat it as 10 things. 

It's not that I'm against them, and it's possible that I'll wind up getting them for my younger daughter, depending whether she has trouble with simply visual reasoning. The nice thing about visual reasoning, though, is she can do it anywhere and it's very independent: she doesn't need specific tools. So, I guess, I don't see any reason to add a new tool when the tools we have work well already. 

Am I missing something? What are they going to enable us to do that a picture cannot? 

Edited by square_25

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2 hours ago, 8FillTheHeart said:

Ds lives in his head doing thought experiments and theorizing. (His goals in grad school are all about theory. He was describing the research team he is joining next semester....honestly I don't understand much.) But spending time (often hrs) jogging or shooting hoops thinking about how to approach solving a problem is his idea of fun. 

Dd, otoh, loves analyzing literature and following allusion trails. She spent hrs reading books like Les Mis (in French.) 

For both of them, the passion of the other is akin to torture. Dd just wanted to be taught directly the what's and how's and then explained the why's. She completely understands the whys. She just doesn't want to spend the time trying to discover them herself.

Math is a tool to her. Math is a puzzle to put together for ds. (Which is actually a funny allusion for me to use bc dd builds puzzles constantly for fun as a destressor in college. She is getting 8 new ones, including 2000 piece ones, for Christmas!)

Interesting! Did your daughter ever find any aspects of math intriguing or fun? Like, were there any topics that caught her fancy or no? 

I used to love jigsaw puzzles, by the way! I sympathize with her there :-). But I do think of math as a puzzle as well! 

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2 hours ago, 8FillTheHeart said:

I don't think they sound like sophisticated goals. They sound like basic mathematical reasoning skills to me. 

My kids learn how to work with simple variables and performing basic operations on both sides of the equation in 2nd grade. Simple things like n+3=10, subtract 3 from both sides, n=7, check 7+3=10.

 

Yes... they certainly seem like reasonable goals to me! But I think they must be deemphasized in an average school or something, because I don't tend to see that kind of proficiency in most kids I teach. Students on average have rather troubled relationships with equals signs, with variables, with functions, with graphs... really all kinds of abstract concepts that I don't think of as very tricky, but which were apparently taught by rote in a way that made the students able to manipulate things without understanding what they actually are. 

As an example, I taught lots of calculus classes in college where people would be able to differentiate x^2 and get 2x, but wouldn't be able to tell what exactly the derivative at a given value of x tells us about the original function. Which is... not a very useful skill. I mean, it helped them pass test, but I'd rather people knew what f'(x) meant and how to estimate it, instead of remembering the algorithms (preferably, they'd remember both. But if I had to pick one thing for them to remember...) 

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6 minutes ago, square_25 said:

They seem like a totally reasonable tool, but I'm not sure how they are better than just drawing pictures for the visual child. She's never liked manipulatives of any sort that much. We started out counting using pennies and such, and she very quickly moved onto pictures and fingers, and uses those still. She's quite abstract minded: if I draw a box for her and put a 10 in the middle of it, she is happy enough to treat it as 10 things. 

It's not that I'm against them, and it's possible that I'll wind up getting them for my younger daughter, depending whether she has trouble with simply visual reasoning. The nice thing about visual reasoning, though, is she can do it anywhere and it's very independent: she doesn't need specific tools. So, I guess, I don't see any reason to add a new tool when the tools we have work well already. 

Am I missing something? What are they going to enable us to do that a picture cannot? 

Consistency.
When you have a consistent image representing a concept (be it numerical value or area or whatnot) your brain can use that image all the way from simple to more complex concepts.  Yes, you can draw pictures, but you miss the level of quick recall that a consistent image would give.  And as you point out, your child is highly visual.  You also miss out on the discovery/manipulation of numbers when you limit yourself to only what you think of before you see it.  The rods let you quickly show and find many different relationships.

There is a really good reason why Montessori and the like rely on consistent application and why it is a very developmentally appropriate tool for elementary work and beyond. 

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42 minutes ago, square_25 said:

Interesting! Did your daughter ever find any aspects of math intriguing or fun? Like, were there any topics that caught her fancy or no? 

I used to love jigsaw puzzles, by the way! I sympathize with her there :-). But I do think of math as a puzzle as well! 

She actually really enjoyed calculus. (She really dislikes most science, though.) She says she misses math and has contemplated starting back in taking math, but I am not encouraging her. She is not going to pursue anything with math and she really needs to follow the rabbit trails toward something she will want to pursue long term!!

This dd has really struggled with what she wants to do when she grows up.  😉  She started off as an international business major with a focus on France and double in finance. and minors in French and Russian.  Then she decided to switch to accounting with minors in French and Russian.  She worked an 18 hr/wk accounting internship this past semester and decided there was no way she could be an accountant b/c she did not want to do what they did every day.  So, now she is Russian and French major with information systems and computational linguistics (or something like that) minors.

Like I said, she is fully capable; she just doesn't march to the same drummer as her brothers.

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55 minutes ago, square_25 said:

They seem like a totally reasonable tool, but I'm not sure how they are better than just drawing pictures for the visual child. She's never liked manipulatives of any sort that much. We started out counting using pennies and such, and she very quickly moved onto pictures and fingers, and uses those still. She's quite abstract minded: if I draw a box for her and put a 10 in the middle of it, she is happy enough to treat it as 10 things. 

It's not that I'm against them, and it's possible that I'll wind up getting them for my younger daughter, depending whether she has trouble with simply visual reasoning. The nice thing about visual reasoning, though, is she can do it anywhere and it's very independent: she doesn't need specific tools. So, I guess, I don't see any reason to add a new tool when the tools we have work well already. 

Am I missing something? What are they going to enable us to do that a picture cannot? 

 

It takes longer to redraw pictures you are thinking about than it does to move manipulatives.

CSMP uses a couple of pictorial "languages" to teach and it sounds right up your dd's alley. Google "CSMP Materials."

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56 minutes ago, HomeAgain said:

Consistency.
When you have a consistent image representing a concept (be it numerical value or area or whatnot) your brain can use that image all the way from simple to more complex concepts.  Yes, you can draw pictures, but you miss the level of quick recall that a consistent image would give.  And as you point out, your child is highly visual.  You also miss out on the discovery/manipulation of numbers when you limit yourself to only what you think of before you see it.  The rods let you quickly show and find many different relationships.

There is a really good reason why Montessori and the like rely on consistent application and why it is a very developmentally appropriate tool for elementary work and beyond. 

 

Can you give me an example of something that'd be easier to see with rods than without? As I said, she has never been that excited about physical manipulatives. We've mostly relied on dots to represent numbers, and occasionally boxes with a number written on them (which I imagine doesn't work for all kids, but worked well for her.) 

 

12 minutes ago, Rosie_0801 said:

 

It takes longer to redraw pictures you are thinking about than it does to move manipulatives.

CSMP uses a couple of pictorial "languages" to teach and it sounds right up your dd's alley. Google "CSMP Materials."

 

Thanks for the recommendations! I will :-). 

 

16 minutes ago, 8FillTheHeart said:

She actually really enjoyed calculus. (She really dislikes most science, though.) She says she misses math and has contemplated starting back in taking math, but I am not encouraging her. She is not going to pursue anything with math and she really needs to follow the rabbit trails toward something she will want to pursue long term!!

This dd has really struggled with what she wants to do when she grows up.  😉  She started off as an international business major with a focus on France and double in finance. and minors in French and Russian.  Then she decided to switch to accounting with minors in French and Russian.  She worked an 18 hr/wk accounting internship this past semester and decided there was no way she could be an accountant b/c she did not want to do what they did every day.  So, now she is Russian and French major with information systems and computational linguistics (or something like that) minors.

Like I said, she is fully capable; she just doesn't march to the same drummer as her brothers.

 

I love math, but I wouldn't want to be an accountant! I wouldn't discourage her from taking math... I think having some quantitative stuff on your transcript is actually rather handy for a number of fields. But again, I'm biased ;-). 

Does she have any idea what appeals to her as a job at all? 

 

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1 hour ago, square_25 said:

 

Can you give me an example of something that'd be easier to see with rods than without? As I said, she has never been that excited about physical manipulatives. We've mostly relied on dots to represent numbers, and occasionally boxes with a number written on them (which I imagine doesn't work for all kids, but worked well for her.) 

 

 


Multiplication of large numbers and/or decimals, for one, if you aren't inclined to see the examples in the video I linked.  When you link a visual of a tenth of a tenth, absolutely it's easier to understand the multiplication.  And once you understand that, visualizing multiplication and division of other fractions.
My oldest used Math U See, which uses a similar rod and base ten system.  Having the consistent image helped him visualize how to find square roots (we have special ones for those, but the same colors) and not have to rely on a calculator.  He could quickly multiply large numbers by physically putting one palm on top of the other as a mental note while he saw the blocks in his head.  He actually went on to be quite good at math.

I'm just saying you really should keep an open mind.  To disregard tools completely when your child is 6 may not work best for long term.  Having them available doesn't hurt either of you, but refusing to even consider them can come back to haunt you.  You may be good at math, but being good at teaching math can be a different ball game.

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Just now, HomeAgain said:


Multiplication of large numbers and/or decimals, for one, if you aren't inclined to see the examples in the video I linked.  When you link a visual of a tenth of a tenth, absolutely it's easier to understand the multiplication.  And once you understand that, visualizing multiplication and division of other fractions.
My oldest used Math U See, which uses a similar rod and base ten system.  Having the consistent image helped him visualize how to find square roots (we have special ones for those, but the same colors) and not have to rely on a calculator.  He could quickly multiply large numbers by physically putting one palm on top of the other as a mental note while he saw the blocks in his head.  He actually went on to be quite good at math.

I'm just saying you really should keep an open mind.  To disregard tools completely when your child is 6 may not work best for long term.  Having them available doesn't hurt either of you, but refusing to even consider them can come back to haunt you.  You may be good at math, but being good at teaching math can be a different ball game.

Well, I've been teaching math for many years, in fact... I've taught for AoPS for 4 or 5 years now, I taught in graduate school, I taught during my post doc. I'm not at all new to teaching, although I've largely taught older students. 

I think the example I saw in the video was, we have 36 objects, one of us has 9, the other has twice that much, how many does the third person have? My daughter can do questions like that very quickly in her head. She's very proficient with mental math. 

I'm absolutely keeping an open mind, but we've doing math lessons with her for 2 years now, she's quite advanced, and before I introduced something new to her that she doesn't seem to require, I'd have to see some concrete benefit. I agree that it'll be helpful for her to see pictures of a tenth of a tenth, but we've already talked about things like half of a half, and it's hard for me to imagine that she won't figure out a tenth of a tenth. 

I'm wondering, are you suggesting that I introduce rods even thought she isn't struggling using visuals? Do you think a visual method is a priori less effective? I generally look for new tools if I feel like we're stuck or not progressing. So I'm absolutely not ruling these out, but I'm just not seeing a need right now. 

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1 hour ago, square_25 said:

I love math, but I wouldn't want to be an accountant! I wouldn't discourage her from taking math... I think having some quantitative stuff on your transcript is actually rather handy for a number of fields. But again, I'm biased ;-). 

Does she have any idea what appeals to her as a job at all? 

Right now she thinks she want to pursue special collections librarian. She is fluent in French and her goal is to be fluent in Russian by graduation. I'm not worried about her finding a path she wants to take. She just needs to spend time exploring them. I just know her heart is in languages, literature, and culture.

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By the end of the Miquon series the worksheets don't depict or reference the rods as much. And you don't have to use them either way. My son stopped using manipulatives regularly at around 4.5 but I still occasionally use the rods to introduce a new concept such as long division. It's quicker to use the rods then to draw pictures for every division. 

He could not tell you the definition of the laws or verbalize the properties in such a concise way, but he does understand the concepts and uses them. It's evident when he reduces fractions or finds multiples or any number of other things. I think what you're trying to say is you want to teach math conceptually and not focus on rote algorithms or memorization. If that's the case you'll see a lot of support for that position on this forum. 

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3 hours ago, 8FillTheHeart said:

Right now she thinks she want to pursue special collections librarian. She is fluent in French and her goal is to be fluent in Russian by graduation. I'm not worried about her finding a path she wants to take. She just needs to spend time exploring them. I just know her heart is in languages, literature, and culture.

I hope she finds what she likes soon! I'm glad she's getting a chance to explore. 

23 minutes ago, Sarah0000 said:

By the end of the Miquon series the worksheets don't depict or reference the rods as much. And you don't have to use them either way. My son stopped using manipulatives regularly at around 4.5 but I still occasionally use the rods to introduce a new concept such as long division. It's quicker to use the rods then to draw pictures for every division. 

He could not tell you the definition of the laws or verbalize the properties in such a concise way, but he does understand the concepts and uses them. It's evident when he reduces fractions or finds multiples or any number of other things. I think what you're trying to say is you want to teach math conceptually and not focus on rote algorithms or memorization. If that's the case you'll see a lot of support for that position on this forum. 

I definitely want to teach conceptually, and I've liked what I've seen on this forum in that regard. I might be a bit more ambitious, though: I want her to fully internalize the reasons behind the relationships. Like, I want her to know why multiplication is commutative and associative, and why the distributive property holds, and why two negative numbers add to a negative number, etc. I also want rudiments of algebra, phrased in a developmentally appropriate way: that means that we don't only calculate values of expressions, but we also find solutions to a given set up. 

I've seen older students really struggle when they don't know the reasons for these things. For example, if you've internalized that multiplication is commutative, but not that there's a reason, you'll have real trouble with matrices, where multiplication isn't commutative.  If you've internalized that you you can "move" things from one side to another, but not why, you'll have trouble with new operations, like taking the conjugate of both sides with complex numbers. (The last one I've seen a lot. Student manipulate equations without the sense that they are doing the same thing to both sides, exactly: they think of it more as following rules and less as "well, we perform the same operation on both sides, and on one of the sides, some things cancel.") 

You know, I'm not sure I'm planning to do long division or stacked multiplication for a good long while :-). I kind of like her getting to play with multiplication and division without ever having to memorize an algorithm. She's come up with all sorts of fun tricks without much prompting (and some tricks with prompting, like switching the order of multiplication.) 

I'm going to assume you used Miquon? How long did you use it for, and did you like it? 

Edited by square_25

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5 hours ago, square_25 said:

Yes... they certainly seem like reasonable goals to me! But I think they must be deemphasized in an average school or something, because I don't tend to see that kind of proficiency in most kids I teach. Students on average have rather troubled relationships with equals signs, with variables, with functions, with graphs... really all kinds of abstract concepts that I don't think of as very tricky, but which were apparently taught by rote in a way that made the students able to manipulate things without understanding what they actually are.  

As an example, I taught lots of calculus classes in college where people would be able to differentiate x^2 and get 2x, but wouldn't be able to tell what exactly the derivative at a given value of x tells us about the original function. Which is... not a very useful skill. I mean, it helped them pass test, but I'd rather people knew what f'(x) meant and how to estimate it, instead of remembering the algorithms (preferably, they'd remember both. But if I had to pick one thing for them to remember...) 

 

From my experience when the kids were in NYC elementary schools, and speaking to their friends (and occasionally helping those friends with homework) it's not so much these these things are de-emphasized as that they're badly taught.

For example, I don't remember being taught the terms "commutative property" or "associative property" until 8th grade, as preparation for the Regents. Nowadays the standard is for kids to "know this" in the second or third grade... except that they don't seem to test that the kids actually know this, so what happens is kids end up with this idea that "when you do commutative property, you write that x+y=y+x". It's like some sort of boring and pointless trick with no contest whatsover instead of the understanding that "associative property is the name we give to the fact that it doesn't matter what order you add numbers in, you always get the same result." Which you'd think wouldn't harm them so much so long as they know that fact as well... except that it adds to a general sense that mathematics is frustratingly vague and pointless.

A while back I was going over Algebra with the younger kiddo, and a friend was there, and as I always do I reminded my kid that the equals sign means that THIS SIDE is the same as THAT SIDE, that is, that they're EQUAL. And the friend goes "OH! Is that why sometimes they have the answer on the other side instead of the right side?" Yes, sweetie, glad I could clear that up for you.

The truth is that a lot of elementary teachers in math are awful at arithmetic. We're talking about several generations of schoolteachers who got subpar instruction in the subject, learned algorithms by rote and never fully understood it, and then went on to pass their vague understanding and slight fear of math to their students. If you try to fix things you get enormous resistance from parents who went through the same system and don't understand the point of taking the slow route rather than having kids simply memorize algorithms as rapidly as possible, and most of the teachers either don't understand it (and so implement it badly), or are vague and half-hearted about it because they'd rather do what they're used to. There are exceptions, and there are others who finally learned the whys of arithmetic in adulthood and would love to fix things for their charges... but this is an ongoing issue and has been for at least 80 years, probably longer. The standards are written badly too, but that's another issue.

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15 minutes ago, Tanaqui said:

 

From my experience when the kids were in NYC elementary schools, and speaking to their friends (and occasionally helping those friends with homework) it's not so much these these things are de-emphasized as that they're badly taught.

For example, I don't remember being taught the terms "commutative property" or "associative property" until 8th grade, as preparation for the Regents. Nowadays the standard is for kids to "know this" in the second or third grade... except that they don't seem to test that the kids actually know this, so what happens is kids end up with this idea that "when you do commutative property, you write that x+y=y+x". It's like some sort of boring and pointless trick with no contest whatsover instead of the understanding that "associative property is the name we give to the fact that it doesn't matter what order you add numbers in, you always get the same result." Which you'd think wouldn't harm them so much so long as they know that fact as well... except that it adds to a general sense that mathematics is frustratingly vague and pointless.

A while back I was going over Algebra with the younger kiddo, and a friend was there, and as I always do I reminded my kid that the equals sign means that THIS SIDE is the same as THAT SIDE, that is, that they're EQUAL. And the friend goes "OH! Is that why sometimes they have the answer on the other side instead of the right side?" Yes, sweetie, glad I could clear that up for you.

The truth is that a lot of elementary teachers in math are awful at arithmetic. We're talking about several generations of schoolteachers who got subpar instruction in the subject, learned algorithms by rote and never fully understood it, and then went on to pass their vague understanding and slight fear of math to their students. If you try to fix things you get enormous resistance from parents who went through the same system and don't understand the point of taking the slow route rather than having kids simply memorize algorithms as rapidly as possible, and most of the teachers either don't understand it (and so implement it badly), or are vague and half-hearted about it because they'd rather do what they're used to. There are exceptions, and there are others who finally learned the whys of arithmetic in adulthood and would love to fix things for their charges... but this is an ongoing issue and has been for at least 80 years, probably longer. The standards are written badly too, but that's another issue.

Yes to all of these. I've seen the equals signs confusion more time than I can count. And, again, I mean with middle school and high school aged kids who are fairly advanced in math. I don't find I can clear it up by saying it once, either. I should figure out a way to talk about it in an early class somehow, I guess. Like many other surprising things, I should probably not assume that people know it. 

This reminds me that when I tried to talking to my daughter about the commutative property of addition, she just kind of didn't know what I meant! It seemed too obvious. Maybe this one's better if you also compare it to subtraction, which doesn't have the commutative property... but of course, you can't do that if you never write down things like 2 - 4 (which we did even before we did negatives, since I wanted it to be clear to her that the second number was taken away from the first number. We'd write a question mark for this one, to show that we didn't know how to do this yet. Then she begged to be told what the question marks were, which is how we wound up doing negatives.) 

But anyway, the commutative property of multiplication is much neater, since unlike addition, it's really not obvious! I really enjoyed working on that one with her. 

Yeah, the sense that "mathematics is a series of random hoops and black boxes" is exactly the feeling I've encountered from kids drilled in either algorithms or pointless properties. 

Do you mean the Common Core standards are written badly? I've taken a look and I'm not impressed, but I'd be curious to hear your take on it. 

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He used the first part of Miquon for preschool, then dabbled in the last year of Miquon in Kindergarten. He played with multiplication/division for about two years before I introduced long multiplication/division at the end of kindergarten. He also used parts of Singapore 1-3, LoF Elementary, MM, and Mep. Now that he's in first grade he's more independent and so he pretty much sticks with Beast Academy and I do enrichment stuff with him separately.

I've not seen the problems you are describing. It sounds like a teaching issue and not necessarily a curriculum issue. If you know what to teach and how you can do that with whatever curricula you otherwise like. 

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I suggest using a conceptual math program like Singapore PM, RightStart, Miquon, MEP, or similar. I can’t soeak to the 2nd grade level, but for 3rd grade on Beast Academy builds from the conceptual basis in a way that requires and reinforces the need to understand the concepts behind the math. None of these programs rely on or can be done with only memorizing algorithms, and they all present multiple methods for problem solving (which itself reinforces using the properties and concepts of math over plug-and-chug output). 

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12 hours ago, square_25 said:

Yes to all of these. I've seen the equals signs confusion more time than I can count. And, again, I mean with middle school and high school aged kids who are fairly advanced in math. I don't find I can clear it up by saying it once, either. I should figure out a way to talk about it in an early class somehow, I guess. Like many other surprising things, I should probably not assume that people know it. 

This reminds me that when I tried to talking to my daughter about the commutative property of addition, she just kind of didn't know what I meant! It seemed too obvious. Maybe this one's better if you also compare it to subtraction, which doesn't have the commutative property... but of course, you can't do that if you never write down things like 2 - 4 (which we did even before we did negatives, since I wanted it to be clear to her that the second number was taken away from the first number. We'd write a question mark for this one, to show that we didn't know how to do this yet. Then she begged to be told what the question marks were, which is how we wound up doing negatives.) 

But anyway, the commutative property of multiplication is much neater, since unlike addition, it's really not obvious! I really enjoyed working on that one with her. 

Yeah, the sense that "mathematics is a series of random hoops and black boxes" is exactly the feeling I've encountered from kids drilled in either algorithms or pointless properties. 

Do you mean the Common Core standards are written badly? I've taken a look and I'm not impressed, but I'd be curious to hear your take on it. 

I know this was not addressed to me, but here’s my $.02. The standards attempted to take learning/teaching strategies and methods from nations who did well on the PISA and present them to American schools. I appreciate the intent, and think they were going in the right direction, but then there was a disconnect when writing the CCSS.  It seems a lot like Nightmare Before Christmas - someone from American math land saw the beauty of Asian math land and wanted to bring it home. But American educators, textbook companies, testing services, policy makers, etc just didn’t grasp the nuance of “Asian math” (or notice the cultural reinforcement of the methods, the simplified beauty in delivery, the developmental age appropriateness), brought in their own familiar perspectives and approaches, and created something that terrifies many when they were least expecting it.

 

eTA When two of my kids transitioned into CCSS math at public school in 4th and 9th they were not only way ahead of the game, they understood what was going on better than the teachers. Both spent lots of team teaching their peers. This was because they had a solid foundation in conceptual math (that wasn’t directed by CCSS) and actually *knew* why the commutative property of multiplication works, and why you can us partial quotients to do Long division in your head, etc.

Edited by Targhee
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Do you mean the Common Core standards are written badly? I've taken a look and I'm not impressed, but I'd be curious to hear your take on it. 

 

They could be worse, but there's a general tendency to push things down to lower grades that aren't necessarily appropriate for it, and to emphasize breadth over depth.

I think they're far better than many state standards were before CC was introduced! And I certainly think the "omg common core!!!" nonsense is... well, nonsense. Common core is just a list of things to be learned at each grade, not a list of how to teach them. But they could be a lot better.

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4 hours ago, Sarah0000 said:

He used the first part of Miquon for preschool, then dabbled in the last year of Miquon in Kindergarten. He played with multiplication/division for about two years before I introduced long multiplication/division at the end of kindergarten. He also used parts of Singapore 1-3, LoF Elementary, MM, and Mep. Now that he's in first grade he's more independent and so he pretty much sticks with Beast Academy and I do enrichment stuff with him separately.

I've not seen the problems you are describing. It sounds like a teaching issue and not necessarily a curriculum issue. If you know what to teach and how you can do that with whatever curricula you otherwise like. 

You mean the equals sign issue, or the not knowing why things are true issue? The equals sign issue is rather subtle at this age, but it's definitely pervasive by middle school and high school. I can give you an example of a problem that might stump a kid who has it, if you like. 

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2 hours ago, Targhee said:

I suggest using a conceptual math program like Singapore PM, RightStart, Miquon, MEP, or similar. I can’t soeak to the 2nd grade level, but for 3rd grade on Beast Academy builds from the conceptual basis in a way that requires and reinforces the need to understand the concepts behind the math. None of these programs rely on or can be done with only memorizing algorithms, and they all present multiple methods for problem solving (which itself reinforces using the properties and concepts of math over plug-and-chug output). 

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. 

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2 hours ago, Targhee said:

I know this was not addressed to me, but here’s my $.02. The standards attempted to take learning/teaching strategies and methods from nations who did well on the PISA and present them to American schools. I appreciate the intent, and think they were going in the right direction, but then there was a disconnect when writing the CCSS.  It seems a lot like Nightmare Before Christmas - someone from American math land saw the beauty of Asian math land and wanted to bring it home. But American educators, textbook companies, testing services, policy makers, etc just didn’t grasp the nuance of “Asian math” (or notice the cultural reinforcement of the methods, the simplified beauty in delivery, the developmental age appropriateness), brought in their own familiar perspectives and approaches, and created something that terrifies many when they were least expecting it.

 

eTA When two of my kids transitioned into CCSS math at public school in 4th and 9th they were not only way ahead of the game, they understood what was going on better than the teachers. Both spent lots of team teaching their peers. This was because they had a solid foundation in conceptual math (that wasn’t directed by CCSS) and actually *knew* why the commutative property of multiplication works, and why you can us partial quotients to do Long division in your head, etc.

 

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. 

 

40 minutes ago, Tanaqui said:

 

They could be worse, but there's a general tendency to push things down to lower grades that aren't necessarily appropriate for it, and to emphasize breadth over depth.

I think they're far better than many state standards were before CC was introduced! And I certainly think the "omg common core!!!" nonsense is... well, nonsense. Common core is just a list of things to be learned at each grade, not a list of how to teach them. But they could be a lot better.

 

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.  

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