The Money Mustache Community
Other => Off Topic => Topic started by: Sugaree on November 14, 2018, 08:40:19 AM
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This was brought up on another topic and to keep from derailing that one I'll start this here:
I'm a fan of the Common Core worksheets that my (Kindergarten) son has been bringing home. Especially the math. I've seen the arguments against how addition and subtraction are taught on FB and it makes perfect sense to me. That is how I've always done math in my head (adding to get to the next ten, adding the tens, and then adding whatever is left). Having sort of grown up in a military town (the base closed when I was in HS) I saw first hand how it could be good thing that kids be on the same general glidepath if they are moving around a lot. Someone on the other thread pointed out that it wasn't necessarily the standards, but rather how they are being taught. I can understand that. My personal theory is that the implementation of the new curriculum was flawed in a lot of school districts, either purposefully or not.
For those of you who have kids/teach kids/know kids, how has Common Core worked out for your schools? Is it really the debil?
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So, I don't teach in schools anymore, but I work intimately with Common Core (and other state standards)- and have had many meetings with the writers of the standards. I think there are some flaws to it, especially in the lowest levels, but what Common Core is not is a curriculum. There is almost nothing in the Common Core (math, I know nothing about ELA) that tells you HOW to teach the standards. So if you don't like what is going on in the classroom, it is unlikely to have anything to do with Common Core. If you do like what is going on in the classroom, it also is unlikely to have much to do with Common Core.
The one thing Common Core does that is different is stress conceptual understanding over rote memorization. That upsets some people because they can't help their kids because they honestly don't know the why. They just remember the how. But while it might be weird to learn some of these "why's" in elementary school, it helps with the "how" later on. For example- kids often have trouble with Algebra and multiplying polynomials because they see it as a different task than multiplication of whole numbers. But if multi-digit whole number multiplication is thought of as partial products, they learn the distributive property needed to multiply polynomials. So their understanding in high school matches what they learned early on. They don't have to re-learn.
The common core standards are publicly available. What is in them might surprise you if you think of it as a curriculum.
http://www.corestandards.org/Math/
If your state "does not do common core" you might also be surprised how similar the standards are. Some states are truly unique standards. But many have just stuck a new cover on common core and called it their own.
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So, I don't teach in schools anymore, but I work intimately with Common Core (and other state standards)- and have had many meetings with the writers of the standards. I think there are some flaws to it, especially in the lowest levels, but what Common Core is not is a curriculum. There is almost nothing in the Common Core (math, I know nothing about ELA) that tells you HOW to teach the standards. So if you don't like what is going on in the classroom, it is unlikely to have anything to do with Common Core. If you do like what is going on in the classroom, it also is unlikely to have much to do with Common Core.
The one thing Common Core does that is different is stress conceptual understanding over rote memorization. That upsets some people because they can't help their kids because they honestly don't know the why. They just remember the how. But while it might be weird to learn some of these "why's" in elementary school, it helps with the "how" later on. For example- kids often have trouble with Algebra and multiplying polynomials because they see it as a different task than multiplication of whole numbers. But if multi-digit whole number multiplication is thought of as partial products, they learn the distributive property needed to multiply polynomials. So their understanding in high school matches what they learned early on. They don't have to re-learn.
The common core standards are publicly available. What is in them might surprise you if you think of it as a curriculum.
http://www.corestandards.org/Math/
If your state "does not do common core" you might also be surprised how similar the standards are. Some states are truly unique standards. But many have just stuck a new cover on common core and called it their own.
I think this is the part that I like about what my son is doing. We work on it a little at home too. Like, right now he's very interested in the concept of x in math. I tried to explain x using a box. We counted out a certain number of balls and then he turned around and I put part of them in the box labeled x. He figured out pretty quickly that x was equal to whatever portion of the balls were missing from what we started with. But he keeps asking me things like "what is 100 + x?" so I don't think he's quite grasped the idea that x can be anything and is likely to be different in every equation. He's also been asking a lot about multiplication and I'm trying to explain that multiplication is just groups of things. Like three groups of three crayons is 9 crayons rather than just memorizing that 3 x 3 = 9.
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We're doing something similar here in Ontario: trying to teach kids how to understand numbers instead of just plugging them into a formula.
Otherwise, you end up with dumb teachers who mark a "diagram of 3x5" wrong because "It means 3 groups of 5, NOT 5 groups of 3!".
Heavy sigh.
The result is that my sons do arithmetic the way I do: 97 + 14? It takes 3 to reach 100, with 11 left over, so it's 111.
Say what? Don't you have to write it out and do all the carrying?
The teachers are frustrated, but this is mostly because the vast majority of humanity was taught to be scared of numbers at a young age and consequently can't do math very well. (or, for that matter, and more relevantly, personal finances).
Toque.
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I prefer learning why over just memorizing how, but there's a place for rote memorization and basic math is one of those places. It really seems some of the math stuff is trying to learn to run before you can walk. Take this first grade requirement:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
That's great, but if every time a kid needs to solve a basic subtraction problem they're reduced to deriving the answer instead of just knowing it, they're not going to have a good time. And it isn't until second grade that you get:
CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
So great, you've essentially taught the first grader the basics of algebra: you can take a subtraction problem of 10 - 8 = x, add eight to both sides of the equation to get 10 = x + 8, then solve for x by counting up from 8 until you get to 10. But you only need to know basic sums by the time you're done with second grade. And the requirements only dictate that kids need strategies to do subtraction.
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I prefer learning why over just memorizing how, but there's a place for rote memorization and basic math is one of those places. It really seems some of the math stuff is trying to learn to run before you can walk. Take this first grade requirement:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
That's great, but if every time a kid needs to solve a basic subtraction problem they're reduced to deriving the answer instead of just knowing it, they're not going to have a good time. And it isn't until second grade that you get:
CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
So great, you've essentially taught the first grader the basics of algebra: you can take a subtraction problem of 10 - 8 = x, add eight to both sides of the equation to get 10 = x + 8, then solve for x by counting up from 8 until you get to 10. But you only need to know basic sums by the time you're done with second grade. And the requirements only dictate that kids need strategies to do subtraction.
You're taking a "for example" and making it the concrete only way to do something. It doesn't say that. It says students should understand the relationship between addition and subtraction- and that's an awesome thing, because like you said it is the basis of Algebra. It doesn't mean that is the only way students should ever approach subtraction. You've pulled from the "Operations and Algebraic thinking" strand. Of course these will be algebra based.
Even when I was in school 30 years ago we learned "fact families"- but we just memorized them. 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1. No one taught me to understand why these families worked together, or the relationship between addition and subtraction and how it could help me solve a difficult subtraction problem that I didn't have memorized.
Look at numbers and operations strand
CCSS.MATH.CONTENT.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Relationship between addition and subtraction is ONE strategy given. You could also subtract using models or using place value.
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As someone who holds degrees in math fields but only has nieces and nephews, common core math is much, much better than the rote memorization method that most people have been taught. It teaches how to think about numbers and their relationships and how to solve problems.
For people who know the formulaic methods, it can seem really silly or tedious, but only because you haven't really learned math =P
What is boils down to is teaching kids real math and how to conceptualize the relationship between numbers earlier rather than teaching them just arithmetic/calculation. It will help them later on when they have to start doing things like algebra, trig and calculus where conceptualization what is happening is critical to understanding.
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I like the common core math stuff. It's way better than being taught to just memorize stuff, which is mostly what I remember until I figured out how to do something more similar to the common core approach.
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What is boils down to is teaching kids real math and how to conceptualize the relationship between numbers earlier rather than teaching them just arithmetic/calculation. It will help them later on when they have to start doing things like algebra, trig and calculus where conceptualization what is happening is critical to understanding.
It's interesting that you say that because I've heard a lot of (non-educator) people say that teaching arithmetic this way makes it "impossible" for a kid to learn the higher maths.
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Common Core math hit our school when my older kid hit second grade. I had some difficulty that year because:
1. It was new to the teacher, who was new to 2nd grade AND common core was new
2. It was obviously new to my kid
3. I didn't learn it that way.
It probably took me 2 weeks of reading his worksheets before I finally understood what they were doing.
But man, after that? I could do SO MUCH MORE math in my head than ever before! And I'm an engineer. It seems like the Common core way is more like how my Asian friends learned math.
Unfortunately, I forgot it after that but now I have a second chance with kid #2.
SO many of my FB friends are all butt-hurt about it. "Our old ways or better!" "But what about everyone else who doesn't learn that way??" Well, dummy, they learn the "old way" in 3rd grade anyway - this gives them a few different techniques. Still kid #1 in 7th can do more math in his head than me.
I'm also getting more info about ELA, and I really like those changes too. Reading for comprehension, etc. They alternate between fiction and non-fiction also, so they are learning quite a bit. Everything seems to aimed at more of a deep understanding.
When I was a kid, I had straight A's. I tested like a boss! I could memorize a ton of shit! But...maybe my comprehension was not all that great. Luckily, I got past that before college. Nobody wants an engineer designing a nuclear reactors or a transistor who doesn't understand the physics or chemistry.
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Common Core had two big issues which have unfortunately tainted what was actually a really good idea.
First, the implementation times were way too short. Common Core is a whole new way of thinking about education, yet classroom teachers were expected to revise their instruction and get ready for high-stakes testing in just a couple years...and remember these are all people who were taught with earlier methods, and may not be comfortable with conceptual math learning themselves (people who go into elementary education often have gifts that are different from math fluency). No wonder a lot of them were unhappy about it, which trickled down to parents and classrooms. I work in a field adjacent to K-12 education, and when we revamped our program it was a three-year process plus another couple years practicing for everyone to get solid on it.
Second, educational leaders didn't do a good job communicating what Common Core actually was. So any change or problem at the classroom level got blamed on it, fueled by the high-pressure implementation issues. I remember there were complaints about specific books kids were reading that were blamed on Common Core. Common Core gives some examples of texts you can use and what you can do with them, but it's up to teachers and districts to decide what they're actually going to include.
Frankly, I really like what they were trying to do. I am not a stupid person, but I couldn't get past pre-calculus because memorization only got me so far and my teachers hadn't done a good job of setting me up to be a conceptual thinker in math--and that closed off a lot of potential majors for me. I wish I had had that kind of instruction. The literacy standards were meant to encourage the kind of reading and writing skills, in multiple subjects, that my academic friends tell me are sadly lacking in their college students.
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SO many of my FB friends are all butt-hurt about it. "Our old ways or better!"
I remember sitting at the table yelling at my Mom "no- you're doing it wrong" learning addition and multiplication.
So apparently they change the method every generation or so. And whatever the new way is, is never received well by those who didn't do it that way.
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Second, educational leaders didn't do a good job communicating what Common Core actually was. So any change or problem at the classroom level got blamed on it, fueled by the high-pressure implementation issues. I remember there were complaints about specific books kids were reading that were blamed on Common Core. Common Core gives some examples of texts you can use and what you can do with them, but it's up to teachers and districts to decide what they're actually going to include.
Frankly, I really like what they were trying to do. I am not a stupid person, but I couldn't get past pre-calculus because memorization only got me so far and my teachers hadn't done a good job of setting me up to be a conceptual thinker in math--and that closed off a lot of potential majors for me. I wish I had had that kind of instruction. The literacy standards were meant to encourage the kind of reading and writing skills, in multiple subjects, that my academic friends tell me are sadly lacking in their college students.
Common core was an excellent scapegoat for all kinds of things!
That's why it makes me laugh when states who have "left common core" just rebrand the standards with barely any changes. But they can at least say they don't use common core.
I've heard some people say that the ELA standards doesn't leave enough room for fiction at the higher grades. But I don't know much about them.
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SO many of my FB friends are all butt-hurt about it. "Our old ways or better!"
I remember sitting at the table yelling at my Mom "no- you're doing it wrong" learning addition and multiplication.
So apparently they change the method every generation or so. And whatever the new way is, is never received well by those who didn't do it that way.
Yeah, I'm not sure this is better. There may be a number of people whose brains can work this way, and some who can't.
When I try to explain the short cuts methods, people look at me in complete bafflement. Like "99x99". You don't have to multiply all those 9s. You just do 100x100 and subtract the edges off the square. Done. Whuh....?
Trying to teach everyone the techniques that work for math nerds might not come out right. I haven't seen enough evidence either way.
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I prefer learning why over just memorizing how, but there's a place for rote memorization and basic math is one of those places. It really seems some of the math stuff is trying to learn to run before you can walk. Take this first grade requirement:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
That's great, but if every time a kid needs to solve a basic subtraction problem they're reduced to deriving the answer instead of just knowing it, they're not going to have a good time. And it isn't until second grade that you get:
CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
So great, you've essentially taught the first grader the basics of algebra: you can take a subtraction problem of 10 - 8 = x, add eight to both sides of the equation to get 10 = x + 8, then solve for x by counting up from 8 until you get to 10. But you only need to know basic sums by the time you're done with second grade. And the requirements only dictate that kids need strategies to do subtraction.
You're taking a "for example" and making it the concrete only way to do something. It doesn't say that. It says students should understand the relationship between addition and subtraction- and that's an awesome thing, because like you said it is the basis of Algebra. It doesn't mean that is the only way students should ever approach subtraction. You've pulled from the "Operations and Algebraic thinking" strand. Of course these will be algebra based.
Even when I was in school 30 years ago we learned "fact families"- but we just memorized them. 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1. No one taught me to understand why these families worked together, or the relationship between addition and subtraction and how it could help me solve a difficult subtraction problem that I didn't have memorized.
No, I'm saying subtraction of single digit numbers from an arbitrary number without thinking anything through is something I believe everybody should be able to do. I don't believe anybody should have to apply any thought to solve a problem like 10 - 8. You should simply know. CC is saying learn strategies to derive answers to subtraction problems like this. And at no point is subtraction by memory a requirement in CC like single digit addition is. So, sure, some may still get to memorize these things, but it isn't prescribed.
Also, please note that I never said kids shouldn't learn the strategies to derive answers to subtraction problems. Or that learning the basics of algebra early isn't a good thing. I just think the very basic small number stuff should be committed to memory, that it can and should be done early, and it's an important building block to being able to do more difficult math efficiently.
If pressed, I'll readily admit that if I had to choose one of deriving an answer or just knowing the answer, deriving will always win. But both is an option here, both is superior, and both aren't required.
Look at numbers and operations strand
CCSS.MATH.CONTENT.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Relationship between addition and subtraction is ONE strategy given. You could also subtract using models or using place value.
I agree that you should be able to explain the reasoning if asked, but this is still single digit subtraction, with a zero tossed on the end to spice things up a little. If they've been taught places, I assert nobody should need a strategy to solve any problem that can be defined under this requirement. Yes, you should be able to provide one, but actually doing so is overkill. You should be able to look at the problem and just have an answer.
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As someone who holds degrees in math fields but only has nieces and nephews, common core math is much, much better than the rote memorization method that most people have been taught. It teaches how to think about numbers and their relationships and how to solve problems.
For people who know the formulaic methods, it can seem really silly or tedious, but only because you haven't really learned math =P
What is boils down to is teaching kids real math and how to conceptualize the relationship between numbers earlier rather than teaching them just arithmetic/calculation. It will help them later on when they have to start doing things like algebra, trig and calculus where conceptualization what is happening is critical to understanding.
+1
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I prefer learning why over just memorizing how, but there's a place for rote memorization and basic math is one of those places. It really seems some of the math stuff is trying to learn to run before you can walk. Take this first grade requirement:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
That's great, but if every time a kid needs to solve a basic subtraction problem they're reduced to deriving the answer instead of just knowing it, they're not going to have a good time. And it isn't until second grade that you get:
CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
So great, you've essentially taught the first grader the basics of algebra: you can take a subtraction problem of 10 - 8 = x, add eight to both sides of the equation to get 10 = x + 8, then solve for x by counting up from 8 until you get to 10. But you only need to know basic sums by the time you're done with second grade. And the requirements only dictate that kids need strategies to do subtraction.
You're taking a "for example" and making it the concrete only way to do something. It doesn't say that. It says students should understand the relationship between addition and subtraction- and that's an awesome thing, because like you said it is the basis of Algebra. It doesn't mean that is the only way students should ever approach subtraction. You've pulled from the "Operations and Algebraic thinking" strand. Of course these will be algebra based.
Even when I was in school 30 years ago we learned "fact families"- but we just memorized them. 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1. No one taught me to understand why these families worked together, or the relationship between addition and subtraction and how it could help me solve a difficult subtraction problem that I didn't have memorized.
No, I'm saying subtraction of single digit numbers from an arbitrary number without thinking anything through is something I believe everybody should be able to do. I don't believe anybody should have to apply any thought to solve a problem like 10 - 8. You should simply know. CC is saying learn strategies to derive answers to subtraction problems like this. And at no point is subtraction by memory a requirement in CC like single digit addition is. So, sure, some may still get to memorize these things, but it isn't prescribed.
Also, please note that I never said kids shouldn't learn the strategies to derive answers to subtraction problems. Or that learning the basics of algebra early isn't a good thing. I just think the very basic small number stuff should be committed to memory, that it can and should be done early, and it's an important building block to being able to do more difficult math efficiently.
If pressed, I'll readily admit that if I had to choose one of deriving an answer or just knowing the answer, deriving will always win. But both is an option here, both is superior, and both aren't required.
Look at numbers and operations strand
CCSS.MATH.CONTENT.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Relationship between addition and subtraction is ONE strategy given. You could also subtract using models or using place value.
I agree that you should be able to explain the reasoning if asked, but this is still single digit subtraction, with a zero tossed on the end to spice things up a little. If they've been taught places, I assert nobody should need a strategy to solve any problem that can be defined under this requirement. Yes, you should be able to provide one, but actually doing so is overkill. You should be able to look at the problem and just have an answer.
But you are talking about 6 year olds.
To develop a skill that "anybody should be able to do" requires scaffolding. They don't just automatically know.
If we just teach them to memorize 10-8 = 2; but they don't know why, will they be able to apply it when we ask them what 33829 - 29738 is? How many things should they just simply know?
Do you know how many people have no clue "why" borrowing works? That limited level of thinking prevents them from doing higher level mathematics.
What good is it to simply "know 10-8" if you don't understand what is going on, so then you have no idea what to do when you get to 6x^2 - 5x + 7 - 18x^2 = 3x - 4 ? Or heck even $5.00 - $3.18. And that's EXACTLY the same as 500 - 318. But I know adults can't do it, because I see it regularly at grocery stores!
Yes, some 3 year olds can subtract, but if you were actually in a first grade classroom, I think you'd be shocked at the lack of understanding many kids exhibit. 5-2 is difficult, even when you try to show them counting on their fingers. Because they don't understand what those numbers mean.
Numbers are abstract thinking.
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I prefer learning why over just memorizing how, but there's a place for rote memorization and basic math is one of those places. It really seems some of the math stuff is trying to learn to run before you can walk. Take this first grade requirement:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.
That's great, but if every time a kid needs to solve a basic subtraction problem they're reduced to deriving the answer instead of just knowing it, they're not going to have a good time. And it isn't until second grade that you get:
CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
So great, you've essentially taught the first grader the basics of algebra: you can take a subtraction problem of 10 - 8 = x, add eight to both sides of the equation to get 10 = x + 8, then solve for x by counting up from 8 until you get to 10. But you only need to know basic sums by the time you're done with second grade. And the requirements only dictate that kids need strategies to do subtraction.
You're taking a "for example" and making it the concrete only way to do something. It doesn't say that. It says students should understand the relationship between addition and subtraction- and that's an awesome thing, because like you said it is the basis of Algebra. It doesn't mean that is the only way students should ever approach subtraction. You've pulled from the "Operations and Algebraic thinking" strand. Of course these will be algebra based.
Even when I was in school 30 years ago we learned "fact families"- but we just memorized them. 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1. No one taught me to understand why these families worked together, or the relationship between addition and subtraction and how it could help me solve a difficult subtraction problem that I didn't have memorized.
No, I'm saying subtraction of single digit numbers from an arbitrary number without thinking anything through is something I believe everybody should be able to do. I don't believe anybody should have to apply any thought to solve a problem like 10 - 8. You should simply know. CC is saying learn strategies to derive answers to subtraction problems like this. And at no point is subtraction by memory a requirement in CC like single digit addition is. So, sure, some may still get to memorize these things, but it isn't prescribed.
Also, please note that I never said kids shouldn't learn the strategies to derive answers to subtraction problems. Or that learning the basics of algebra early isn't a good thing. I just think the very basic small number stuff should be committed to memory, that it can and should be done early, and it's an important building block to being able to do more difficult math efficiently.
If pressed, I'll readily admit that if I had to choose one of deriving an answer or just knowing the answer, deriving will always win. But both is an option here, both is superior, and both aren't required.
Look at numbers and operations strand
CCSS.MATH.CONTENT.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Relationship between addition and subtraction is ONE strategy given. You could also subtract using models or using place value.
I agree that you should be able to explain the reasoning if asked, but this is still single digit subtraction, with a zero tossed on the end to spice things up a little. If they've been taught places, I assert nobody should need a strategy to solve any problem that can be defined under this requirement. Yes, you should be able to provide one, but actually doing so is overkill. You should be able to look at the problem and just have an answer.
But you are talking about 6 year olds.
To develop a skill that "anybody should be able to do" requires scaffolding. They don't just automatically know.
If we just teach them to memorize 10-8 = 2; but they don't know why, will they be able to apply it when we ask them what 33829 - 29738 is? How many things should they just simply know?
Do you know how many people have no clue "why" borrowing works? That limited level of thinking prevents them from doing higher level mathematics.
What good is it to simply "know 10-8" if you don't understand what is going on, so then you have no idea what to do when you get to 6x^2 - 5x + 7 - 18x^2 = 3x - 4 ? Or heck even $5.00 - $3.18. And that's EXACTLY the same as 500 - 318. But I know adults can't do it, because I see it regularly at grocery stores!
Yes, some 3 year olds can subtract, but if you were actually in a first grade classroom, I think you'd be shocked at the lack of understanding many kids exhibit. 5-2 is difficult, even when you try to show them counting on their fingers. Because they don't understand what those numbers mean.
Numbers are abstract thinking.
Agreed. Teaching memorization is fine when it works. But sometimes everyone makes mistakes. You should always have a strategy to prove or double check anything that you've memorized, and there's no reason that kids can't be taught this.
My son is close to five now and is learning to add and subtract right now. Sometimes I ask him what seven plus two is and he tells me eight. If all that you've learned is memorization, you can't verify your answer and check if your memory is wrong. It's just wrong. When I tell him to try again, he'll hold up two fingers and say "Start at seven, eight, oh, it's nine!"
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But you are talking about 6 year olds.
To develop a skill that "anybody should be able to do" requires scaffolding. They don't just automatically know.
If we just teach them to memorize 10-8 = 2; but they don't know why, will they be able to apply it when we ask them what 33829 - 29738 is? How many things should they just simply know?
No doubt they need some explanation. You'd think I just said memorize some shapes and tell me the missing shape. Counting on fingers or having a pile of objects is going to be required as an introduction to what subtraction means. Yes, most kids need things to be more concrete to get the concept. But all of that is largely getting past the question of what is a number.
So to answer how many things somebody should just know, in the context of subtraction, it's again my assertion that it's just subtraction of a single digit number from another number. So, I guess for 1st graders, stick with single digits in both places and only subtract from larger or equal numbers, giving you 55 things to remember. Get places down by second grade and give them the other 45 cases. So 101 things to simply know. 100 single digit subtraction cases and one rule with place digits. You need to know 101 things in order to subtract arbitrary numbers without thinking about it.
And of course they're not going to get 33829 - 29738 if they only know 10 - 2. But after we're done with places and single digit subtraction, this and every other subtraction problem is easy. Mechanical even. Because that's how subtraction works.
Should you know how numbers are composed of digits, each of which represents some power of your base? Yes, of course, and in much simpler terms when all you know is base 10. But you're going nowhere if you have to derive the appropriate result for each operation rather than knowing it offhand.
Do you know how many people have no clue "why" borrowing works? That limited level of thinking prevents them from doing higher level mathematics.
What good is it to simply "know 10-8" if you don't understand what is going on, so then you have no idea what to do when you get to 6x^2 - 5x + 7 - 18x^2 = 3x - 4 ? Or heck even $5.00 - $3.18. And that's EXACTLY the same as 500 - 318. But I know adults can't do it, because I see it regularly at grocery stores!
Tons of people, but it isn't like the why hasn't been taught since what, the 50s? Tom Lehrer wrote his tune about new math around then, so it's been at least that long. So it's not like the innumerate adults under 60 that you're observing weren't given the tools by being taught only rote subtraction. Nor am I suggesting that only rote is appropriate. But we appear to be dropping the rote side from being required, which if not picked up elsewhere, will limit how quickly one can do basic operations.
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I agree that understanding more conceptually how numbers work is a fantastic goal. However, in my experience, the age at which CC requires these conceptual methods is far too young, and kids minds aren't ready for it at that age. In the early grade school years, kids (or at least my kids) do better with math facts/memorization. Once they have achieved mastery of those, the conceptual methods eventually come instinctively. If they don't, they can be taught later. Before CC (5 years ago), my 1st grader excelled at math. He was probably a solid year ahead. In his 2nd grade year, the conceptual methods were introduced, and we watched his math skills actually *regress*. Simple addition or subtraction problems became an exercise in frustration, because instead of simply writing down 15-8=7, he had to spend a lot of time trying to figure out which of the various conceptual methods he needed to apply. In effect, rather than promoting a better understanding of numbers, the CC math had merely supplied him with an array of more complex and confusing formulas.
This fact was a significant (though by far not the only) reason we elected to homeschool our kids after that year. We went back to a "math facts first" approach, and saw immediate and dramatic improvement. Fast forward a couple of years, and we found that he had, on his own, developed his own mental shortcuts very similar to those taught by CC. They also happen to resemble the same tricks that I developed at a similar age. Now, I realize this is a single (very personal and biased) datapoint. Well, it's several datapoints now, because we have several school-aged kids, and have seen the same sequence happen with each--once they have gained mastery with the simple math facts, the intuitive/conceptual approach comes on its own.
I wonder if, when developing the CC math standards, the thought process was "Kids who are good with math use these conceptual methods, so we should teach all kids these conceptual methods so they will be good at math." If this is the case, it sounds like a case of confusing correlation with causation. I agree that there's a very strong relationship between the conceptual methods and a strong grasp of math in general, but feel that it is a mistake to try to teach those methods at the age currently required. Certainly (IME) kids' brains aren't sufficiently developed at that age. But even without that, I'm not convinced that teaching the conceptual methods actually promotes conceptual understanding. Or, put another way, perhaps the conceptual methods that PWAGAM (people who are good at math) use are a result of an intuitive understanding of math, and not the cause.
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I agree that understanding more conceptually how numbers work is a fantastic goal. However, in my experience, the age at which CC requires these conceptual methods is far too young, and kids minds aren't ready for it at that age. In the early grade school years, kids (or at least my kids) do better with math facts/memorization. Once they have achieved mastery of those, the conceptual methods eventually come instinctively. If they don't, they can be taught later. Before CC (5 years ago), my 1st grader excelled at math. He was probably a solid year ahead. In his 2nd grade year, the conceptual methods were introduced, and we watched his math skills actually *regress*. Simple addition or subtraction problems became an exercise in frustration, because instead of simply writing down 15-8=7, he had to spend a lot of time trying to figure out which of the various conceptual methods he needed to apply. In effect, rather than promoting a better understanding of numbers, the CC math had merely supplied him with an array of more complex and confusing formulas.
This fact was a significant (though by far not the only) reason we elected to homeschool our kids after that year. We went back to a "math facts first" approach, and saw immediate and dramatic improvement. Fast forward a couple of years, and we found that he had, on his own, developed his own mental shortcuts very similar to those taught by CC. They also happen to resemble the same tricks that I developed at a similar age. Now, I realize this is a single (very personal and biased) datapoint. Well, it's several datapoints now, because we have several school-aged kids, and have seen the same sequence happen with each--once they have gained mastery with the simple math facts, the intuitive/conceptual approach comes on its own.
I wonder if, when developing the CC math standards, the thought process was "Kids who are good with math use these conceptual methods, so we should teach all kids these conceptual methods so they will be good at math." If this is the case, it sounds like a case of confusing correlation with causation. I agree that there's a very strong relationship between the conceptual methods and a strong grasp of math in general, but feel that it is a mistake to try to teach those methods at the age currently required. Certainly (IME) kids' brains aren't sufficiently developed at that age. But even without that, I'm not convinced that teaching the conceptual methods actually promotes conceptual understanding. Or, put another way, perhaps the conceptual methods that PWAGAM (people who are good at math) use are a result of an intuitive understanding of math, and not the cause.
Funny how variable things are. My daughter has been doing CC since 1st grade and it's worked really well for her. Now she's in 6th grade and they put her in with the 7th graders because her math skills are so strong. Guess it just goes to show, some kids do better with the "facts first, concepts later" like your kid, while others do better with the "concepts first" approach like my kid.
What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
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What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
There is. It's called homeschooling! :P
I guess this exposes another issue entirely, specifically the fact that kinda have different learning styles, and the traditional public school model is very much a one-size fits all (or most. Or maybe a plurality. Or some).
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What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
It is really hard to teach 30 different kids in one classroom 30 different ways (multiply by 7 classes a day...) But this is one of the reasons students get taught different approaches- to find the one that works for them. Unfortunately, it sometimes confuses kids because they have to go through the ones that don't work for them.
I've never met a teacher that didn't differentiate instruction though. And I've observed many hundreds of classrooms in my line of work. Classrooms aren't "my way or the highway"
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What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
It is really hard to teach 30 different kids in one classroom 30 different ways (multiply by 7 classes a day...) But this is one of the reasons students get taught different approaches- to find the one that works for them. Unfortunately, it sometimes confuses kids because they have to go through the ones that don't work for them.
I've never met a teacher that didn't differentiate instruction though. And I've observed many hundreds of classrooms in my line of work. Classrooms aren't "my way or the highway"
That's been my observation too - the teachers Kira's had have all been great. And they do a far better job teaching her than I ever could. I've actually been really happy with her education in the Denver public school system.
One teacher said the hard part is not having 30 kids, that 30 kids are manageable if they are all at a similar level of ability. The hard part is when you have high performing kids AND low performing kids in the same class, that spread is really tough to handle.
The flip side of course is not that great though. If you separate the high performing kids out, then you end up with kids labeled (very early) as either "smart kids" or "dumb kids" and that's not great either, IMO.
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Great discussion. I can go along with the new methods IF the schools would only send home a textbook.
When junior comes to me and asks for help with a textbook I can quickly learn what method the teacher wants to use for this assignment.
We've had far too many evenings at the dinner table as Red Panda pointed out where junior #2 is telling me "I'm doing it wrong" or with junior #1 I'm telling junior #1 that they are "doing it wrong". My 1975 method is better - but its not what the teacher wanted b/c teacher wants to build upon this lesson and my 1975 version just throws a wrench in that plan.
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Do kids not get to take their books home anymore?
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Not mine. We were at least granted access to an online copy for a school research project over the weekend. The math workbook has the barest of examples most of the time.
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Not mine. We were at least granted access to an online copy for a school research project over the weekend. The math workbook has the barest of examples most of the time.
Same here - most of Kira's book work is now all on computer, which they access in class.
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Great discussion. I can go along with the new methods IF the schools would only send home a textbook.
When junior comes to me and asks for help with a textbook I can quickly learn what method the teacher wants to use for this assignment.
We've had far too many evenings at the dinner table as Red Panda pointed out where junior #2 is telling me "I'm doing it wrong" or with junior #1 I'm telling junior #1 that they are "doing it wrong". My 1975 method is better - but its not what the teacher wanted b/c teacher wants to build upon this lesson and my 1975 version just throws a wrench in that plan.
You can ask either the teacher, the school administration, or the PTO/PTA if they'd help you organize a "math evening" where parents can go in and have the teacher explain what they're doing and what kind of help they'd want you to be able to provide to your student. It might take a while to work out the details, but we've seen this at several of the schools we've attended. In general, schools are pretty thrilled if parents want to understand what's going on in the classroom.
(On the other hand, some teachers really don't want you to provide much help, because they'd like the feedback of knowing how many kids didn't understand what they just taught. Situations vary!)
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Do kids not get to take their books home anymore?
What's a text book?
My kids are in 7th and 1st. Neither of them has ever had a text book. Sometimes they have "workbooks". Most of the lessons for kid #1, in elementary, were photocopied handouts.
Same for kid #2 I guess, but 1st grade...not a lot of "lessons".
7th grader everything is online.
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What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
It is really hard to teach 30 different kids in one classroom 30 different ways (multiply by 7 classes a day...) But this is one of the reasons students get taught different approaches- to find the one that works for them. Unfortunately, it sometimes confuses kids because they have to go through the ones that don't work for them.
I've never met a teacher that didn't differentiate instruction though. And I've observed many hundreds of classrooms in my line of work. Classrooms aren't "my way or the highway"
That's been my observation too - the teachers Kira's had have all been great. And they do a far better job teaching her than I ever could. I've actually been really happy with her education in the Denver public school system.
One teacher said the hard part is not having 30 kids, that 30 kids are manageable if they are all at a similar level of ability. The hard part is when you have high performing kids AND low performing kids in the same class, that spread is really tough to handle.
The flip side of course is not that great though. If you separate the high performing kids out, then you end up with kids labeled (very early) as either "smart kids" or "dumb kids" and that's not great either, IMO.
Our school, and most in this area, spread the kids out within a classroom intentionally. Probably for many reasons
1. It's "fair" (to the teachers and the kids)
2. It's shown to be better for the average and lower performing kids to be in a mixed group
3. They can still give individual attention in groups - which we do by having teacher's aides and reading specialists for group times. Also, starting young, they group kids by levels. So the high kindergarten readers in all 3 classes will be grouped together during reading time. In 6th grade, we had 2 teachers. They would have pull outs for math and English. The high English kids would go with Mr. C and the low English kids would go with Mr. W. The High math kids go with Mr W for math, and the low math kids go with Mr C.
So, you get to group kids by ability, but only for certain sections. You may have a high math kid who is a low English kid, so they aren't "labeled". And the rest of the day they are grouped.
It's really tough because studies ALSO so that while grouping kids together with a variety of abilities is good for the lower/ slower kids, it can be detrimental to the bright kids. My thought is that for the most part, the bright kids are gonna be fine anyway. If yours isn't, then put them somewhere else.
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What would be cool is if there'd be a way to figure that out for each kid and then tailor the teaching approach that worked best for that particular brain.
It is really hard to teach 30 different kids in one classroom 30 different ways (multiply by 7 classes a day...) But this is one of the reasons students get taught different approaches- to find the one that works for them. Unfortunately, it sometimes confuses kids because they have to go through the ones that don't work for them.
I've never met a teacher that didn't differentiate instruction though. And I've observed many hundreds of classrooms in my line of work. Classrooms aren't "my way or the highway"
That's been my observation too - the teachers Kira's had have all been great. And they do a far better job teaching her than I ever could. I've actually been really happy with her education in the Denver public school system.
One teacher said the hard part is not having 30 kids, that 30 kids are manageable if they are all at a similar level of ability. The hard part is when you have high performing kids AND low performing kids in the same class, that spread is really tough to handle.
The flip side of course is not that great though. If you separate the high performing kids out, then you end up with kids labeled (very early) as either "smart kids" or "dumb kids" and that's not great either, IMO.
Our school, and most in this area, spread the kids out within a classroom intentionally. Probably for many reasons
1. It's "fair" (to the teachers and the kids)
2. It's shown to be better for the average and lower performing kids to be in a mixed group
3. They can still give individual attention in groups - which we do by having teacher's aides and reading specialists for group times. Also, starting young, they group kids by levels. So the high kindergarten readers in all 3 classes will be grouped together during reading time. In 6th grade, we had 2 teachers. They would have pull outs for math and English. The high English kids would go with Mr. C and the low English kids would go with Mr. W. The High math kids go with Mr W for math, and the low math kids go with Mr C.
So, you get to group kids by ability, but only for certain sections. You may have a high math kid who is a low English kid, so they aren't "labeled". And the rest of the day they are grouped.
It's really tough because studies ALSO so that while grouping kids together with a variety of abilities is good for the lower/ slower kids, it can be detrimental to the bright kids. My thought is that for the most part, the bright kids are gonna be fine anyway. If yours isn't, then put them somewhere else.
How old are your kids?
Because while this sounds great (in theory), I've met very few teachers who can pull it off effectively. And as a parent to two very bright, very wiggly boys who would ... find ways to entertain themselves when they got bored ... I'm here to tell you that when this was NOT being pulled off effectively I received a number of phone calls from the school. Which basically were "Your kid was misbehaving because he was bored and unengaged."
Also, you can only do it for so long. At some point, some kids will be doing algebra and some will still be working on adding fractions, or learning multiplication. And this will bleed into science ("why is this class moving so slowly?" and social studies, and ...)
Some of our middle school teachers made this claim ("studies show..."), but the only research I could find from the past two decades showed that doing this grouping in a single Language Arts class in (Tennessee? Louisiana? I forget, someplace in the southeast) produced results for the advanced kids that were no worse then having them read quietly by themselves in the library. Not exactly a ringing endorsement. (Also, the research paper -- which I don't recall being peer-reviewed -- used an inappropriate standardized test as the metric, because it was basically someone's thesis and they presumably didn't have any $ to test the kids using something more appropriate.) Some jack-hole from the education school here convinced the middle school teachers that teaching more mixed-ability classes was more "equitable", but in practice the teachers weren't good enough or invested enough to do the work to make the whole thing effective. The two years we were involved in this were a shit-show, and our kid basically didn't learn anything in his LA class for those years.
No, I'm not bitter about my personal experience, why do you ask? :^)
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Our school, and most in this area, spread the kids out within a classroom intentionally. Probably for many reasons
1. It's "fair" (to the teachers and the kids)
2. It's shown to be better for the average and lower performing kids to be in a mixed group
3. They can still give individual attention in groups
...
It's really tough because studies ALSO so that while grouping kids together with a variety of abilities is good for the lower/ slower kids, it can be detrimental to the bright kids. My thought is that for the most part, the bright kids are gonna be fine anyway. If yours isn't, then put them somewhere else.
In my personal experience, having a broad spectrum of kids is absolutely a negative influence on the bright kids, and not just academically. I was a bright kid in school, and also really, really bored in most of my classes. Unfortunately, as a result I learned a lot of habits that, while not detrimental in high school, have had negative consequences on me ever since. I got lazy. I didn't study hard for tests. I did my calculus homework during the first half of my physics class, and my physics homework during the second half of my physics class, while I ignored the teacher. When I got to college, those habits came back to bite me, and I really struggled during my freshman year, and some of those habits (laziness) even dog me today. I can't state positively that I would have done better in college or in the workforce, but I wonder how my life would be different had I been presented with higher expectations.
As an aside, it feels like the K-12 public education setting is a bit unique, in that it's one of the few places where people spend a significant amount of time with peers who are on a different intellectual level. After high school, that spectrum of peers narrows dramatically--either you go straight into the workforce, or you go to a trade school, or you go to college. And real life tends to segregate people even more--CS grads typically end up in an office with a bunch of other CS grads, teachers spend their time with other teachers, doctors with doctors and nurses, liberal arts majors with busboys (I kid! I kid!), tradesmen with tradesmen, etc.
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One of the things I've noticed in looking over my child's homework is that the worksheets don't just ask for an answer, they ask you to calculate the answer using a specific method. Since I don't have the textbook in front of me to read about the method, I can't easily help my kid with the homework.
That said, I have a grad degree in engineering so I can sort of mention a couple of ways to do it and so my kid is like "oh yeah, I think I've heard that before".
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
Familiarity with math facts gives you flexibility for solving problems in a variety of different ways, and it's critical for taking upper level classes. An exercise I got once in a math presentation was to solve 26 * 32 in your head as many different ways as possible, without resorting to multiplying by columns. The group found 6+ ways to solve the problem (although, honestly, they didn't all make sense to me).
More importantly, you need to know basic multiplication to be able to
- add and subtract fractions, where you need to find common denominators (or group appropriately)
- do larger scale multiplication problems
- find (or approximate) square roots
- Be able to approximate answers in general
- learn to factor quadratics (and larger polynomials) in algebra
- be able to compute area in geometry (this is a particular pet peeve of mine, because I was tutoring a kid in geometry who couldn't do even basic multiplication: so every time we tried to work on a multi-step problem she'd get completely derailed by even the simplest multiplication component).
- after you've learned to factor quadratics, you need that knowledge to be able to integrate, once you get to calculus...
In general, as the students advance and have problems that require multiple steps to solve, not being able to just do the multiplication really derails a kid's train of thought. I know it seems like they should be able to pull out a calculator and just do the multiplication, but it doesn't seem to work that way -- however if it's a near-automatic skill, they just slot it in and continue on the problem. It's not a big deal to learn these facts (for most kids) -- it just takes some repetition. There are loads of on-line and physical games designed to help memorize these facts. (I'm a big fan of multiplication tic-tac-toe on a sheet of paper, with flattened glass pieces as markers, but my kids like the "asteroids" online game.)
Also, in real adult life, are you really going to pull out your phone each time you're not sure of a sales tax calculation on a receipt? I glance at it and estimate, generally, to see if it's in the ballpark.
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How old are your kids?
Because while this sounds great (in theory), I've met very few teachers who can pull it off effectively. And as a parent to two very bright, very wiggly boys who would ... find ways to entertain themselves when they got bored ... I'm here to tell you that when this was NOT being pulled off effectively I received a number of phone calls from the school. Which basically were "Your kid was misbehaving because he was bored and unengaged."
Also, you can only do it for so long. At some point, some kids will be doing algebra and some will still be working on adding fractions, or learning multiplication. And this will bleed into science ("why is this class moving so slowly?" and social studies, and ...)
Some of our middle school teachers made this claim ("studies show..."), but the only research I could find from the past two decades showed that doing this grouping in a single Language Arts class in (Tennessee? Louisiana? I forget, someplace in the southeast) produced results for the advanced kids that were no worse then having them read quietly by themselves in the library. Not exactly a ringing endorsement. (Also, the research paper -- which I don't recall being peer-reviewed -- used an inappropriate standardized test as the metric, because it was basically someone's thesis and they presumably didn't have any $ to test the kids using something more appropriate.) Some jack-hole from the education school here convinced the middle school teachers that teaching more mixed-ability classes was more "equitable", but in practice the teachers weren't good enough or invested enough to do the work to make the whole thing effective. The two years we were involved in this were a shit-show, and our kid basically didn't learn anything in his LA class for those years.
No, I'm not bitter about my personal experience, why do you ask? :^)
My kids are 6 and 12.
So, kid #1 is in 7th grade now (out of elementary). He's a GATE kid and we kept him in our school even though we could have transferred to a dedicated GATE program.
In his class, we had a reasonably high % of GATE kids (10%), plus at least another 10-20% of kids who were high performers.
So, he was bored in kindergarten (because he'd had 2 years of preschool). His first grade teacher was FANTASTIC at giving him extra work to keep him engaged. My second kid didn't get her and she's retiring. Bummer. She just about started him on multiplication at the end of first grade but held off. He moved on to common core math in second grade, which meant he was fairly well challenged to learn the "whys" because before that...when asked how to explain the answer he'd say "I just know".
The teachers were able to adjust to his abilities BECAUSE they all worked together - so like I said, the individual grouping would take the high ELA kids from all three 3rd grade classes and they would study together during ELA time. This continued through sixth. Now, that doesn't mean there weren't challenges - we did have a couple of children who were unable to remain engaged - though they weren't necessarily "bright" kids or "not bright" kids - there didn't seem to be a relationship.
In any event, he's in junior high now, and of course, junior high divides up kids by their abilities. So he's in math compaction, honors ELA/ social studies, and honors science. The thing that I really like about his school is that they aim for mastery at all levels. So if you don't get a concept, you redo it until you do. From an ELA/ reading standpoint, everyone needs a certain number of "AR" points. But the # of points depends on the reading level of the kid. My kid and 1 or 2 others are reading at college level, so they are required to read more and more advanced books than the rest. So: not bored.
Kid #2 is in first grade. His teachers, so far, are adjusting to his level. His homework is harder and different from other kids. They already split kids out in the three classes into different reading groups. (Luckily, all 3 classes are 17-18 students this year). He's reading at a 2.8 and his math is at a 2.4. Clearly, the combination of group pull-outs, online studies (a reading and a math program), and parents and teachers who had him 2nd grade books is working.
(Also, I think schools have to adjust - our school has historically been an UMC school, but that has changed in the last 10-15 years. It took a lot of work to "adjust" to a higher % of students who were poor and English learners. You need different methods and techniques when you have a wide disparity like that.
On a more personal note, I was lucky to have teachers to do this for me too. Sadly, I changed schools in high school, and was unable to get into the most advanced math class because of how they had their schedules. I was bored silly in trig and spent most of my time teaching the other students (the teacher was the football coach, he sucked). Turns out when we turned in the book at the end of the FULL YEAR, it was designed for a semester, and we never finished it.
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<snip of my own blather>
My kids are 6 and 12.
So, kid #1 is in 7th grade now (out of elementary). He's a GATE kid and we kept him in our school even though we could have transferred to a dedicated GATE program.
In his class, we had a reasonably high % of GATE kids (10%), plus at least another 10-20% of kids who were high performers.
So, he was bored in kindergarten (because he'd had 2 years of preschool). His first grade teacher was FANTASTIC at giving him extra work to keep him engaged. My second kid didn't get her and she's retiring. Bummer. She just about started him on multiplication at the end of first grade but held off. He moved on to common core math in second grade, which meant he was fairly well challenged to learn the "whys" because before that...when asked how to explain the answer he'd say "I just know".
The teachers were able to adjust to his abilities BECAUSE they all worked together - so like I said, the individual grouping would take the high ELA kids from all three 3rd grade classes and they would study together during ELA time. This continued through sixth. Now, that doesn't mean there weren't challenges - we did have a couple of children who were unable to remain engaged - though they weren't necessarily "bright" kids or "not bright" kids - there didn't seem to be a relationship.
In any event, he's in junior high now, and of course, junior high divides up kids by their abilities. So he's in math compaction, honors ELA/ social studies, and honors science. The thing that I really like about his school is that they aim for mastery at all levels. So if you don't get a concept, you redo it until you do. From an ELA/ reading standpoint, everyone needs a certain number of "AR" points. But the # of points depends on the reading level of the kid. My kid and 1 or 2 others are reading at college level, so they are required to read more and more advanced books than the rest. So: not bored.
Kid #2 is in first grade. His teachers, so far, are adjusting to his level. His homework is harder and different from other kids. They already split kids out in the three classes into different reading groups. (Luckily, all 3 classes are 17-18 students this year). He's reading at a 2.8 and his math is at a 2.4. Clearly, the combination of group pull-outs, online studies (a reading and a math program), and parents and teachers who had him 2nd grade books is working.
(Also, I think schools have to adjust - our school has historically been an UMC school, but that has changed in the last 10-15 years. It took a lot of work to "adjust" to a higher % of students who were poor and English learners. You need different methods and techniques when you have a wide disparity like that.
On a more personal note, I was lucky to have teachers to do this for me too. Sadly, I changed schools in high school, and was unable to get into the most advanced math class because of how they had their schedules. I was bored silly in trig and spent most of my time teaching the other students (the teacher was the football coach, he sucked). Turns out when we turned in the book at the end of the FULL YEAR, it was designed for a semester, and we never finished it.
I gotta tell you, that sounds fantastic -- I wish we had that in our district! My kids are both in high school now, but the road's been a bit uneven. In particular, they eliminated the advanced option for LA in middle school just in time to screw my younger kid, and didn't replace it with anything useful.
Congrats! Now get in there and support your teachers -- because I'm pretty sure what you've got is rather rare.
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
Familiarity with math facts gives you flexibility for solving problems in a variety of different ways, and it's critical for taking upper level classes. An exercise I got once in a math presentation was to solve 26 * 32 in your head as many different ways as possible, without resorting to multiplying by columns. The group found 6+ ways to solve the problem (although, honestly, they didn't all make sense to me).
Also, in real adult life, are you really going to pull out your phone each time you're not sure of a sales tax calculation on a receipt? I glance at it and estimate, generally, to see if it's in the ballpark.
I agree that knowing how numbers work and how to problem solve are important. Those are math. But actually *doing* it, and especially memorizing the answers to "math facts" never seemed worth it to me. Some things I can do in my head, and a very few things I have memorized, but generally yes I use a calculator for anything that comes up in conversations. It's quick, it's easy, it's accurate (though I don't know why I would be calculating sales tax myself...). As for "arithmetic is the gateway to advanced math" I heard that repeatedly growing up -- but my refusal to fill my memory with "math facts" didn't prevent me from getting a degree from a Faculty of Math.
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I gotta tell you, that sounds fantastic -- I wish we had that in our district! My kids are both in high school now, but the road's been a bit uneven. In particular, they eliminated the advanced option for LA in middle school just in time to screw my younger kid, and didn't replace it with anything useful.
Congrats! Now get in there and support your teachers -- because I'm pretty sure what you've got is rather rare.
I'm not really sure how successful we are, to be honest. The advanced learners do really well at our school.
The English learners...not so much. A few years ago we were 9th out of 10 for our EL scores. Right now, we aren't much better. Maybe a little bit better.
The district has implemented new methods that seem to be working at the schools that started first. We are starting to see that improve things this year.
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
Familiarity with math facts gives you flexibility for solving problems in a variety of different ways, and it's critical for taking upper level classes. An exercise I got once in a math presentation was to solve 26 * 32 in your head as many different ways as possible, without resorting to multiplying by columns. The group found 6+ ways to solve the problem (although, honestly, they didn't all make sense to me).
Also, in real adult life, are you really going to pull out your phone each time you're not sure of a sales tax calculation on a receipt? I glance at it and estimate, generally, to see if it's in the ballpark.
I agree that knowing how numbers work and how to problem solve are important. Those are math. But actually *doing* it, and especially memorizing the answers to "math facts" never seemed worth it to me. Some things I can do in my head, and a very few things I have memorized, but generally yes I use a calculator for anything that comes up in conversations. It's quick, it's easy, it's accurate (though I don't know why I would be calculating sales tax myself...). As for "arithmetic is the gateway to advanced math" I heard that repeatedly growing up -- but my refusal to fill my memory with "math facts" didn't prevent me from getting a degree from a Faculty of Math.
You know what's interesting? The math facts that you need to know you'll end up memorizing without practicing simply by working with numbers for long enough. I got my engineering degree without ever sitting down and memorizing multiplication, addition, or subtraction tables. :P
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For those of you who have kids/teach kids/know kids, how has Common Core worked out for your schools? Is it really the debil?
I'm a teacher in Maryland. I'm a HUGE fan of the common core standards. I think the reading foundational skills in particular, are great. There is room for improvement but I've worked with many different standards and there are always problems. These are great.
I'm not a fan of the PARCC tests. We will be doing away with them in Maryland at the end of this year. I think they tried too hard to be and measure too many things, and they confused "rigor" with "incomprehensibility".
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
It's 63.
You're welcome. :)
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I've never understood the obsession with "math facts". Even as a child with good memory I refused to fill that memory with such low-value information. Some very simple things stick with you over time if you use them a lot, but the speed with which you can get them by pulling out your device... tap tap... calculation done with perfect accuracy.
Understanding numbers and abstract thought, math, problem solving, I find these valuable. Remembering what is 7 times 9? Not worth it.
It's 63.
You're welcome. :)
Exactly. We don't even need calculators now, the internet will tell us the answers. Less reason to memorize stuff. :P
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Simple nine times table shortcut:
9 x 7
Subtract one from 7 gives you 6
Count from 6 back up to 9, gives you 3
There you go, the answer is indeed 63. Works from 1 x 9 up to 10 x 9.
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Simple nine times table shortcut:
9 x 7
Subtract one from 7 gives you 6
Count from 6 back up to 9, gives you 3
There you go, the answer is indeed 63. Works from 1 x 9 up to 10 x 9.
For 9s up to 10 you just put down the finger that corresponds with the multiplier. 7th finger- leaves 6 on the left side and 3 on the right. 63. Done.
No weird subtract and count from going on. LOL.
7 x 6 was the one that always got me. Through adulthood I had the hardest time remembering it. I know it now :)
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7 x 6 was the one that always got me. Through adulthood I had the hardest time remembering it. I know it now :)
Just add 7 to 7x5. Sooo . . . 5, 10, 15, 20, 25, 30, 35 and 7 makes 42. :P
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7 x 6 was the one that always got me. Through adulthood I had the hardest time remembering it. I know it now :)
Just add 7 to 7x5. Sooo . . . 5, 10, 15, 20, 25, 30, 35 and 7 makes 42. :P
Well that's what I did. Because it was too hard to just remember 7 x 6.
Fact memorization failed me, so I had to remember another fact, and then add.
This is why fact memorization on it's own isn't useful without understanding the relationships of numbers.
To KNOW that it works to just do 7 x 5 and then add 7 more you have to understand the relationship between multiplication and addition. It's why we practice skip counting when multiplying.
It is a much more flexible approach than memorization on it's own.
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7 x 6 was the one that always got me. Through adulthood I had the hardest time remembering it. I know it now :)
Just add 7 to 7x5. Sooo . . . 5, 10, 15, 20, 25, 30, 35 and 7 makes 42. :P
Or, if it's easier to remember squares, start with 6x6 and add another 6: 36+6=42. Or start with 7x7 and subtract 7 (to get 6 7s): 49-7=42.
Or start with 10 6s (60) and subtract 3 6s (18) to get 42.
Or break the 6 into 3x2, and do 7x3x2, or 21x2 = 42.
Or draw a picture of a rectangle...
I think the idea is that it's the fact memorization that gives you enough information to begin to see the relationships. Or they arrive in tandem.
Look I get that it's not exactly fun to learn (depending on how it's presented), but for most kids it's not generally a big deal, and how can you learn to do algebra (solve x^2 + 10x +21 =0, for instance) if you don't know multiplication families? I literally just got home from working on this with a high school student who isn't really strong enough in multiplication to be able to factor 21 promptly.
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I didn't have to use any fact memorization in my answer.
- 7 x 6 is seven groups of six. Which is the same as six groups of seven. Which is the same thing as five groups of seven plus a group of seven. No memorization required.
- It's easy to count by fives because we use a decimal system. No memorization required.
- It's easy to add an additional seven to the number we just got by counting in fives. You can hold up seven fingers and just count up from 35. No memorization required.
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Well that's what I did. Because it was too hard to just remember 7 x 6.
Fact memorization failed me, so I had to remember another fact, and then add.
This is why fact memorization on it's own isn't useful without understanding the relationships of numbers.
To KNOW that it works to just do 7 x 5 and then add 7 more you have to understand the relationship between multiplication and addition. It's why we practice skip counting when multiplying.
It is a much more flexible approach than memorization on it's own.
The problem my students have is that they forget how much more they are supposed to add. They might remember 6x6 is 36, and know that to get to 6x7 they need to add one more.... something. But they aren't quite clear if they need to add another 6, or another 7!
Yes, it should be obvious. But no, it isn't obvious to some kids. The other problem is they will start at 36, add on one more 6 by counting and ... miscount! They will end up at 41 by counting on.
They won't use the fact that 6+6 is 12 to realize that they are starting with a 30 and a 6, and adding another 6, which is one more 10 and a 2... so the answer had to end in a 2.
Once kids are in 4th grade, it is almost impossible to find the time to go back and correctly remediate. They just want to "get the answer", understandably, so they can finish the worksheet.
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I was literally counting on my fingers still when I was about to start student teaching, I missed so much understanding early on, because I couldn't memorize.
I have Masters in math educaton now and work full time in a math field.
So it's possible to eventually figure it out. But it's hard to make up. I wish someone had taught me earlier why things worked rather than me having to figure out coping strategies to make up for what I missed.
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SO many of my FB friends are all butt-hurt about it. "Our old ways or better!"
I remember sitting at the table yelling at my Mom "no- you're doing it wrong" learning addition and multiplication.
So apparently they change the method every generation or so. And whatever the new way is, is never received well by those who didn't do it that way.
Yeah, I'm not sure this is better. There may be a number of people whose brains can work this way, and some who can't.
When I try to explain the short cuts methods, people look at me in complete bafflement. Like "99x99". You don't have to multiply all those 9s. You just do 100x100 and subtract the edges off the square. Done. Whuh....?
Trying to teach everyone the techniques that work for math nerds might not come out right. I haven't seen enough evidence either way.
Well don't confuse me with an imaginary square FT:) I just think of the next closest higher rounded number. I agree, the short cut methods are the best thing ever.
When I was going to school they taught several different methods all at once - like you said - if one method doesn't work for you, maybe a shortcut will set your brain on fire.
It seems weird to me to insist on one method - whyever would one do that? FWIW the memorization has faded after almost 60 years but the logic behind subtracting the edges of the cube still works. Logic always solves the problem.
Although I've always loved stuff like 25x25 = 625 - it was part of some random memorization games our math teacher played with us.
It was fun and exciting to play senseless memorization with big numbers - more fun than 7x6 for sure:)
Once you understand that you can add numbers and it is the same as multiplying them, you are good. Add the same number seven times is the same as multiplying them seven times.
You can even break down the seven into 3 plus 4 - or 5 plus 2 - it still all comes out to the same number. Amazing.
Of course, if you happen to know that 7x7 is 49, you can just deduct one seven - voila, you just figured out 7x6, much faster:).
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SO many of my FB friends are all butt-hurt about it. "Our old ways or better!"
I remember sitting at the table yelling at my Mom "no- you're doing it wrong" learning addition and multiplication.
So apparently they change the method every generation or so. And whatever the new way is, is never received well by those who didn't do it that way.
Yeah, I'm not sure this is better. There may be a number of people whose brains can work this way, and some who can't.
When I try to explain the short cuts methods, people look at me in complete bafflement. Like "99x99". You don't have to multiply all those 9s. You just do 100x100 and subtract the edges off the square. Done. Whuh....?
Trying to teach everyone the techniques that work for math nerds might not come out right. I haven't seen enough evidence either way.
Hey, somebody who thinks about math like me!
Seriously, my sister considers herself "scarred for life" because our school introduced some different style of writing when she was in K, and then abandoned it soon after. Her cohort was out of step each year because they started to learn writing in an unusual experimental style, and were never retaught later. That's sort of how the Common Core roll-out seemed in our state: pushed out prematurely, and when it hit snags it is the kids that suffer the fallout in the skills they didn't master because theory didn't pan out as expected. Move on to the next grade - teach to the test for that grade, don't have time to fill the gaps from previous years, too busy testing testing testing!
The focus on testing has triggered pushback - many districts have high numbers of parents choosing to opt their kids out of the testing. My brother's district has something like 72% opt out rates - administrators are frantic, because the state is threatening to ?? if a majority don't test. Brother and his wife are both teachers, but are adamant that their kids skip the testing. I think the testing is a waste of time better spent on teaching, and the tests are badly designed, but, eh, my kids can take them, if only so the state can get enough data to improve them (or ditch them as pointless).
Our district says the scores don't affect the kids' local grades, but they do affect resources used - DS5 was assigned ELA AIS when he scored a 2 on the ELA test one year. Three is considered acceptable, 4 mastery, 2 is not quite good enough. Given his teacher's glowing grades and comments for that class, I have to assume the standardized test was the issue (or DS5's lack of concern for that test), but regardless, DS5 had ELA pullout instruction for the year. Waste of resources.
DH has just returned to teaching HS ELA. He was actually so fed up with the Common Core hassles (from our state's botched roll-out) that he left teaching for industry for 3 years or so. He was bored silly, and missed the teaching, so jumped when the opportunity arose to get back in another school. It's only with the break from it that he's realized it was the CC part that was the frustration. Not the concept of CC, just our state's implementation. Although he doesn't like the shift to emphasize more nonfiction - he'd rather have more fiction!
Which leads me to personal differences in how each of us thinks, and learns. DH and I frequently discuss the differences in our preferences. I love the idea of more emphasis on non-fiction. Much of the fiction we read in school was not to my taste, and that lack of interest in the subject matter I now realize means I wasn't focused well on the lesson at the time. To this day, I abhor short stories and novellas (i feel like I've been dropped in the middle of a story, don't know what's going on, and it ends abruptly), but time constraints meant those were the most frequently used versions of literature in school. But a chapter out of a book on, say, the origins of our quirky number systems (we can thank the Babylonians for our time system of 12 daytime and nighttime hours with 60 minutes and 60 seconds - they used base 5 and base 12) would have been right up my alley - something useful to know, not just another story. Of course, DH is much less enamoured of math - it's all fake to him. English is his preferred language skill, mine is math and logic. We each enjoy more the one we are more fluent in. I've come to realize it's a sign of how differently we each think and learn.
There isn't just one optimal way for everyone to learn X, but there is at least one optimal way for each of us. So I like the idea of teachers showing multiple ways to do X, but I recognize that many children are black and white thinkers - they just want to be taught the "right way" to do X. Different methods can become jumbled together in their minds. It also makes testing harder - can the kids use any method, or only the one taught this week? Learning "why" helps me immensely, but others don't want the why, just the process. And I recognize that certain methods just click with you - when i tutored Calc in college I found it useful to explain things in different ways, until I found the one that matched the student's thought process. Tutoring also showed me that early mis-learned topics made it incredibly difficult to be successful in college math; order of operations was a common problem. Once identified and relearned, students who believed themselves "bad at math" improved remarkably, and suddenly felt smart.
My biggest objection to the Common Core (and education in general as it is done currently) is the whole age cohort model, with every subject taught in equal shares daily. That's not how my brain works - I'd much rather pursue in depth study into one or two related topics for a while, learn lots rather than shallowly, and then start again on something different. I might be ready for math before my peers, but only develop an interest in history later. Just because I was born in xxxx year should not determine if I study American vs European history, or Bio vs Physics in school this year. We are trying to cram more and more into the school day, but so much is superficial due to lack of time, and time spent switching mental gears with classrooms.
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I get what you're saying about the age thing. I have a different perspective. I grew up near a military base so a lot of my friends had attended 4 or 5 schools by the time they graduated from high school. It was common for someone to transfer in already having taken whatever, let's say, history class that their grade was studying that year. I'm high school, this wasn't as big of a deal because you just assign them to whatever history class that they needed to be in. But in elementary school it was easily possible that a kid gets two years of American history, but no western civ. For example, if school A taught American history in third grade and western civ in fourth grade but a child transfers to school B over the summer and school B teaches American history in fourth grade and western civ in third grade. So, I can see where having specific subjects in specific grades will help the kids who move.around a lot, for whatever reason.
One thing that my kiddo's (in kindergarten) school is doing that I like, and I don't know if it's related to the CC curriculum they are using or not, is that they teach in units and everything is kind of tied to that unit. For example, right after school started they did an apple unit. They counted and added apples in math (including a chart of how many kids preferred red to yellow to green apples), they read stories about apples, they concentrated on the letters A, P, L, and E, and they did art projects about apples. At the end of the unit they had a class apple tasting party where they sampled different apple foods.