Hi, Second Grade Teacher here. I didn't go through the whole video because, frankly, the editing style was not my jam, but I got the gist. I've also had a LOT of discussions with people over the...
Exemplary
Hi, Second Grade Teacher here. I didn't go through the whole video because, frankly, the editing style was not my jam, but I got the gist. I've also had a LOT of discussions with people over the years about how "New Math" is just a more complicated way of doing everything, and it couldn't be further from the truth. What kids are learning these days is focused on a few key concepts:
Understanding Relationships Between Numbers And Shapes
All numbers are shapes. I want to repeat that: All numbers are shapes. This is a very weird concept for those not familiar with the idea, but once you see it it's quite intuitive. It begins with 5- and 10-frames in Kindergarten. These are tools to help children understand quantities and numbers, and is a very concrete and visual way to demonstrate an abstract idea. Children learn to work within these frames in order to build up fluency and speed at working with these "friendly numbers" (a concept we will return to later). How does this relate to shapes? Consider the following: What actually is a 10-frame? (It's an array). What is an array? (In this context, it is a set of rows and columns that shows a number). What else is this array? (A shape, and eventually a model for moving into another dimension with multiplication). By having children get used to working with numbers as shapes, we are preparing them even from such a young age to work in multiple dimensions and laying a foundation for more advanced mathematics in the future.
This ties into that funny looking model the video shows. It's role is like that of a 10-frame; it shows the relationships between numbers in an equation as a shape. Most often, it's initially a triangle (relationship between 3 numbers) and we even call it a "Fact Triangle". These number facts demonstrate the relationship between addition and subtraction in a concrete way (3+1=4, 1+3=4, 4-1=3, 4-3=1). One thing that is hammered home here is that "equals" does NOT mean "answer".
Understanding Base-10
Let's return to that 10-frame again. It's an incredible tool for introducing not just numbers as shapes and arrays, but also for building an awareness of the base-10 system most of our daily mathematics relies on. Students learn to work fast within this frame, adding, subtracting, and moving numbers around quickly and accurately. In Kindergarten and First Grade, they use manipulatives to model these ideas concretely; by Second Grade, we move more and more to written models and rapidly ramp up the difficulty as students are able to wrangle more abstract concepts. In Second Grade, we are setting multiple 10-frames side-by-side and having students move quantities around to rapidly build groups of 10 and be able to perform mathematical operations with speed and accuracy that would shock most people to see in such young learners. This all builds toward giving students a strong innate sense of what base-10 means and how it can be applied to numbers once we remove the supports of the 10-frames and base-10 blocks they rely on for concrete modeling.
Another poster also mentioned about "fast facts" and that can be lumped in here; students learn and practice "ways to make 10" so they can make those calculations quickly. We also have them work on many other facts to aid processing, but making 10 is by far the most important one.
Creating Strategies And Understanding Patterns
If it seems like your Second Grader is coming home with an enormous amount of wildly varying methods for solving problems, that's because we're teaching them an enormous amount of wildly varying methods for solving problems. This is where most people get confused. What we do in the classroom is work together to discover, create, and apply strategies that make sense for the individual learner. Each student tends to settle on finding a mental strategy that works for them, and will be able to use and explain this method well because they had a part in discovering and creating this strategy. Some examples include open number lines, change diagrams, and partial sums (this is the one most people are familiar with). We are also working on allowing students to find patterns in the problems for determining which strategy will work best in a given situation. I guarantee that if you are strong in math yourself, you grew up doing the same thing, rejecting the algorithmic process that used to be taught in favor of your own constructed methods. THIS IS WHAT NEW MATH IS DOING.
Edit: I forgot to summarize with an example
Let's use an example and how one child may combine everything discussed above to solve a subtraction problem.
51 - 25 = ?
Us adults learned the algorithm: borrow a ten from the 5, add it to 1 to make 11, subtract 5 from 11 to get 6, and then subtract the 2 from the 4 in the tens place. Combine it all to get 26. This is fine, but it doesn't rely on an understanding of how the numbers actually relate to one another.
A child in my class might do the following:
A strategy they like is the open number line:
<25---------------------------------51>
They recognize that these two numbers don't exist in isolation, they are part of the same continuum. The next step for them is to identify the distance between those numbers. They can quickly, mentally do this (and record their thinking) using this number line:
<25--------30--31---------41--------51>
It's a quick series of hops along the line (-10, -10, -1, -5) that gets the same answer when quickly counting along ("10, 20, 21, 26")
51 - 25 = 26
With this form of solving the problem, it isn't an algorithm. It doesn't exist in isolation of the numbers, it's recognizing that these numbers have a relationship.
In a nutshell: We are teaching children to think about numbers in a more intuitive and relational way, rather than just memorizing tables and algorithms.
I could go on, I love discussing pedagogy, but I feel this has gotten long winded enough. If anyone has questions I'll happily answer further.
thanks for taking the time! I'll offer a response in two parts: praise, and then criticism. Praise I learned "old" math: the way it was taught is that there is a universal algorithm that is...
thanks for taking the time! I'll offer a response in two parts: praise, and then criticism.
Praise
I learned "old" math: the way it was taught is that there is a universal algorithm that is applied mechanically without thought or curiosity, to arrive at equal-sign-means-answer. What this can do is several things:
kids are drilled into efficiently identifying and applying the correct algorithm, then drilled for speed.
some kids, via the many drills, mentally and independently come up with these other manipulation strategies to help them go even faster than mechanically applying the correct algorithm, or use the manipulation to aid with steps within the algorithm.
other kids don't learn any strategies, but they still get very good at the algorithm, so they appear very mathematically competent until some creativity and understanding of "why" stuff works is needed, then they stall out
So yeah, I love new math. The 10-frame, when it was my turn to teach, was an array of single ants, row of 10 ants, 10x10 ants, and other friendly number arrays of ants. We also did number lines, leaves and nodes etc but ants worked and we went further with them.
Criticism
What we do in the classroom is work together to discover, create, and apply strategies that make sense for the individual learner.
What I don't love about new math though, is when each of the strategies are taught at length, and there's a take home portion for the kids to demonstrate competency for each even though this method is clearly not resonating with them. What we see in the video is this: the kids are given an algorithm for how to apply Triple Node Triangle, applied mechanically without thought or curiosity, then drilled for speed. We have the trappings of new math, taught in the ugly old way, then that effort is multiplied by an assortment of ways. Where once we had one brutal way of grinding kids, we now have multitudes.
What you're doing is teaching thinking strategies, encouraging creativity, respecting individual differences, and then helping each student mentally manipulate shapes faster in the way that makes the most sense to them. I can't say this is how it works for classrooms where the teachers themselves admit they always hated math, that they don't understand or want or celebrate new math, and they just assign worksheets demanding algorithmic application equal sign means answers.
I think your criticism is valid. I feel (I don't have the data) that many of my fellow educators don't respect the process. As soon as I saw what New Math was trying to do I was sold. Honestly the...
I think your criticism is valid. I feel (I don't have the data) that many of my fellow educators don't respect the process. As soon as I saw what New Math was trying to do I was sold.
Honestly the most difficult part of teaching it is the brutal reality that many students don't have the inclination to question these things and look for a deeper understanding. I don't mean this as a mark against them, it is just reality. Their curiosity and motivation lies elsewhere.
I have one student right now who is able to quickly pick up processes, but struggles to explain their thinking. This is another type of challenge. Clearly they can make sense of the math but their expressive language can't keep up. That's tough.
That does sound tough but I would imagine it must feel so amazing when you see that spark ignite :) what you're teaching them is that the world is a knowable and reasonable place, that we can grab...
That does sound tough but I would imagine it must feel so amazing when you see that spark ignite :) what you're teaching them is that the world is a knowable and reasonable place, that we can grab abstractions and turn them over and cut them up and stick em together the way we do with clay, that we can start with "I don't know" and have fun kneading it until we arrive at an Aha! I think the first group is "just tired, boss", that they have a lot of things that demand their attention and time, and they're spending cycles figuring out the fastest way to get us to go away.
Your criticism of the each strategies rings true for me and my young. It's great when it works, but my child grokked it with the first few and the was bored for a week. As well as the worse/burnt...
Your criticism of the each strategies rings true for me and my young. It's great when it works, but my child grokked it with the first few and the was bored for a week. As well as the worse/burnt out teachers basically skipping half them for one reason or another. Large classroom sizes really hurt.
And sometimes, despite the vast covering of different methods, sometimes the old methods just click. Kiddo was struggling with on particular concept for a bit outside of class, mom showed 'the old way,' and it clicked instantly and their speed and accuracy trippled overnight.
I will say I think it's this solid foundation, as well as the algebratic concepts also introduced, which enabled this kind of moment.
A humorous video based on a question from r/askmath that does a good job of explaining how to do 2nd grade math using a slightly different method than was taught to most of us. It's definitely...
A humorous video based on a question from r/askmath that does a good job of explaining how to do 2nd grade math using a slightly different method than was taught to most of us. It's definitely interesting and different than my mental method of decomposing (I'd explain it but it's a whole thing) which gets me there but takes me down a circuitous route.
Off-Topic: For those not in the US school system, 2nd Grade is the 3rd year of schooling for our children, which is 0-based with Kindergarten, then has another 12 years of required schooling with grades levels that count from 1 to 12.
The first half of the video is kinda meh but the second half is where the good stuff is. The regrouping idea is very intuitive and it's unfortunate how the worksheet only has 10 as the big number...
The first half of the video is kinda meh but the second half is where the good stuff is. The regrouping idea is very intuitive and it's unfortunate how the worksheet only has 10 as the big number over and over because it doesn't really teach the principle.
The idea is to break big numbers into friendly chunks like 59 becomes 40 + 10 + 5 + 4.
It helps to pick a number like 4, 5, or 6, to always use so that you only need to memorize a few fixed subtractions from 10.
Then when you see something like 44-27 you break it into groups which have as many similar numbers on each side as is comfortable and then you can just cross out the same numbers and subtract whichever is easiest:
Sure! That's a valid strategy as long as you have enough subtractions memorized, ie. knowing that it will be useful to take 14 instead of 10. If you don't know ahead of time and don't know where...
Sure! That's a valid strategy as long as you have enough subtractions memorized, ie. knowing that it will be useful to take 14 instead of 10.
If you don't know ahead of time and don't know where to start, a sub-strategy could be to recursively splitting in half like a binary search, ie. 44 = 22 + 11 + 6 + 3 + 2
Then you could pick the number that's closest to 7 and any left over you could subtract again from the next closest number until you have only additions left
Tangential, but I know we've got a lot of teachers here. I'm on the hunt for a good (and reasonably priced) homeschool prek syllubus to supplement bored younger sibling. We've got Hooked On...
Tangential, but I know we've got a lot of teachers here.
I'm on the hunt for a good (and reasonably priced) homeschool prek syllubus to supplement bored younger sibling. We've got Hooked On Phonics which is great for the early reading, but want something a bit more well-rounded for everything else.
The authors says "borrowing" is useless for numbers less than 20 and "you just have to know what 16-9 is", but is it really true? I don't remember what 16-9 is off the top of my head, if I have to...
The authors says "borrowing" is useless for numbers less than 20 and "you just have to know what 16-9 is", but is it really true? I don't remember what 16-9 is off the top of my head, if I have to calculate it, I always calculate it as "16-9=6+(10-9)=6+1=7". Of course, over the years I got very fast at this operation, but every time I have to subtract from a number larger then 10 I still basically do it in two steps. Do most people actually just remember results for subtractions of all numbers up to 20?
Disclosure I didn't watch the video (can't do audio atm) but that is not borrowing, Borrowing is saying "ok 6-7, no we can't do that because 6-7 is negative so take a 10 away from the next column...
16-9=6+(10-9)=6+1=7
Disclosure I didn't watch the video (can't do audio atm) but that is not borrowing, Borrowing is saying "ok 6-7, no we can't do that because 6-7 is negative so take a 10 away from the next column and change that to 16-7 now we can do 16-7=9 and yes we do just need to know this"
I made a longer comment in the thread but I want to jump in here and say that no in fact you do NOT "just need to know" that 16-7 = 9! We teach an idea called "friendly numbers" to help students...
I made a longer comment in the thread but I want to jump in here and say that no in fact you do NOT "just need to know" that 16-7 = 9! We teach an idea called "friendly numbers" to help students with this idea, since at this level they are only JUST beginning to move from a concrete approach to an abstract approach.
The most basic "friendly number" is 10. How will that help us here? Students know they want to get to the friendly number. they see 16 - 7 and immediately change it to 16 - 6 - 1 so they can reach a much easier equation to solve: 10 - 1.
That's how you avoid memorization in math and teach number relationships at the same time.
EDIT: Woops I misread your post. I need more coffee. I think we're talking in the same direction so I'll leave my post up as well.
I like how I was taught for longer numbers. Take: 7256 -4985 ----- 2271 You can go left to right here if you know that when the two digits in your column will require borrowing, you borrow from...
I like how I was taught for longer numbers. Take:
7256
-4985
-----
2271
You can go left to right here if you know that when the two digits in your column will require borrowing, you borrow from the left as always. You keep a running tally of what your number is as you move left to right in digits, so we start with an answer of 0000 and each 0 gets replaced as you go left to right:
I'm not sure why you would want to do this if you can just do normal long subtraction which is basically the same but goes right to left? Or is this approach easier to do in your head without paper?
I'm not sure why you would want to do this if you can just do normal long subtraction which is basically the same but goes right to left? Or is this approach easier to do in your head without paper?
Easier in my head. I get lost in formulas and the like. Math was my strongest subject until about middle school, then I had to not only show my work but use only the method I was presented instead...
Easier in my head. I get lost in formulas and the like. Math was my strongest subject until about middle school, then I had to not only show my work but use only the method I was presented instead of being allowed to invent my own, and I got docked again and again on my homework and tests to the point that I decided I really wasn't any good at math and I gave up trying to learn it at all, just doing enough to get by and scrape by in tests. I went from advanced placement to remedial math in 3 years, because I fully gave up.
I imagine most people who are really good at math just remember them, because they've used them so much that it's just reflex. My mental arithmetic ability has drawn comments at various times over...
I imagine most people who are really good at math just remember them, because they've used them so much that it's just reflex. My mental arithmetic ability has drawn comments at various times over the years and for me those subtractions are just recall... but that's after endless repetition. I don't remember ever making a conscious effort to memorize them or specifically being taught them when I was in school, but I do remember the breakthrough of recognizing patterns for certain combinations of minuend and subtrahend (as well as products for times tables) on my own. In my school/classes (which were outside of the US system), we did a lot of mental arithmetic testing, likely much more than the average US school today.
I learned that if you subtract nine from anything the ones digit will be one higher than it is now. So it's almost as quick as subtracting from ten. Between that and memorization I'm fine, until...
I learned that if you subtract nine from anything the ones digit will be one higher than it is now. So it's almost as quick as subtracting from ten. Between that and memorization I'm fine, until my brain hiccups and I have forgotten how math works for 30 seconds and I have to do it the long way.
There are pros and cons to memorizing it vs doing it each time in your head I suppose. I also had times table speed tests as a kid and was one of those that always finished and finished early so that was a skill of mine.
Have child that passed through the second grade. Under 20 is 'fast facts' and they want those snap memorized for faster standardized test taking. It makes sense though, because it minimizes the...
Have child that passed through the second grade. Under 20 is 'fast facts' and they want those snap memorized for faster standardized test taking.
It makes sense though, because it minimizes the distance you need to decompose. As soon as you hit 20, and then later multiples of 20 once times tables are built, will allow for super fast mental math.
Honestly I don't know. I use an approach not far from yours for 2 digit subtraction, but I veer wildly off course when you start subtracting really long digits.
Honestly I don't know. I use an approach not far from yours for 2 digit subtraction, but I veer wildly off course when you start subtracting really long digits.
(I'm not an educator, nor do I have any applicable background which would help me understand the field, that said) Does anyone know off hand if the method proposed in "Knowing and Teaching...
(I'm not an educator, nor do I have any applicable background which would help me understand the field, that said)
Does anyone know off hand if the method proposed in "Knowing and Teaching Elementary Mathematics" -- or even the "borrowing" method portrayed in-video -- have been studied for their impact on math education compared to other social factors? My understanding of the Chinese education system + culture surrounding primary/secondary/post-sec education is poor, but I'm distantly aware of make-or-break exams (e.g. the zhongkao) which -- rightfully or otherwise -- are understood as setting the entire course of a child's life, at age 15, with a single pass/fail exam. That (apparently) leads to an incredible amount of pressure (and support) for children to learn everything, ASAP, as effectively as possible ... perhaps it's possible that that has more of an impact on education outcomes than different ways of subtracting numbers?
(edit) Similarly, primary and secondary education in North America is often -- again, rightfully or otherwise -- portrayed as daycare, so that parents can work their regular job to make rent. Or, in the case of lunch programs, to feed both themselves and their children, during the school year.
Hi, Second Grade Teacher here. I didn't go through the whole video because, frankly, the editing style was not my jam, but I got the gist. I've also had a LOT of discussions with people over the years about how "New Math" is just a more complicated way of doing everything, and it couldn't be further from the truth. What kids are learning these days is focused on a few key concepts:
Understanding Relationships Between Numbers And Shapes
All numbers are shapes. I want to repeat that: All numbers are shapes. This is a very weird concept for those not familiar with the idea, but once you see it it's quite intuitive. It begins with 5- and 10-frames in Kindergarten. These are tools to help children understand quantities and numbers, and is a very concrete and visual way to demonstrate an abstract idea. Children learn to work within these frames in order to build up fluency and speed at working with these "friendly numbers" (a concept we will return to later). How does this relate to shapes? Consider the following: What actually is a 10-frame? (It's an array). What is an array? (In this context, it is a set of rows and columns that shows a number). What else is this array? (A shape, and eventually a model for moving into another dimension with multiplication). By having children get used to working with numbers as shapes, we are preparing them even from such a young age to work in multiple dimensions and laying a foundation for more advanced mathematics in the future.
This ties into that funny looking model the video shows. It's role is like that of a 10-frame; it shows the relationships between numbers in an equation as a shape. Most often, it's initially a triangle (relationship between 3 numbers) and we even call it a "Fact Triangle". These number facts demonstrate the relationship between addition and subtraction in a concrete way (3+1=4, 1+3=4, 4-1=3, 4-3=1). One thing that is hammered home here is that "equals" does NOT mean "answer".
Understanding Base-10
Let's return to that 10-frame again. It's an incredible tool for introducing not just numbers as shapes and arrays, but also for building an awareness of the base-10 system most of our daily mathematics relies on. Students learn to work fast within this frame, adding, subtracting, and moving numbers around quickly and accurately. In Kindergarten and First Grade, they use manipulatives to model these ideas concretely; by Second Grade, we move more and more to written models and rapidly ramp up the difficulty as students are able to wrangle more abstract concepts. In Second Grade, we are setting multiple 10-frames side-by-side and having students move quantities around to rapidly build groups of 10 and be able to perform mathematical operations with speed and accuracy that would shock most people to see in such young learners. This all builds toward giving students a strong innate sense of what base-10 means and how it can be applied to numbers once we remove the supports of the 10-frames and base-10 blocks they rely on for concrete modeling.
Another poster also mentioned about "fast facts" and that can be lumped in here; students learn and practice "ways to make 10" so they can make those calculations quickly. We also have them work on many other facts to aid processing, but making 10 is by far the most important one.
Creating Strategies And Understanding Patterns
If it seems like your Second Grader is coming home with an enormous amount of wildly varying methods for solving problems, that's because we're teaching them an enormous amount of wildly varying methods for solving problems. This is where most people get confused. What we do in the classroom is work together to discover, create, and apply strategies that make sense for the individual learner. Each student tends to settle on finding a mental strategy that works for them, and will be able to use and explain this method well because they had a part in discovering and creating this strategy. Some examples include open number lines, change diagrams, and partial sums (this is the one most people are familiar with). We are also working on allowing students to find patterns in the problems for determining which strategy will work best in a given situation. I guarantee that if you are strong in math yourself, you grew up doing the same thing, rejecting the algorithmic process that used to be taught in favor of your own constructed methods. THIS IS WHAT NEW MATH IS DOING.
Edit: I forgot to summarize with an example
Let's use an example and how one child may combine everything discussed above to solve a subtraction problem.
51 - 25 = ?
Us adults learned the algorithm: borrow a ten from the 5, add it to 1 to make 11, subtract 5 from 11 to get 6, and then subtract the 2 from the 4 in the tens place. Combine it all to get 26. This is fine, but it doesn't rely on an understanding of how the numbers actually relate to one another.
A child in my class might do the following:
A strategy they like is the open number line:
<25---------------------------------51>
They recognize that these two numbers don't exist in isolation, they are part of the same continuum. The next step for them is to identify the distance between those numbers. They can quickly, mentally do this (and record their thinking) using this number line:
<25--------30--31---------41--------51>
It's a quick series of hops along the line (-10, -10, -1, -5) that gets the same answer when quickly counting along ("10, 20, 21, 26")
51 - 25 = 26
With this form of solving the problem, it isn't an algorithm. It doesn't exist in isolation of the numbers, it's recognizing that these numbers have a relationship.
In a nutshell: We are teaching children to think about numbers in a more intuitive and relational way, rather than just memorizing tables and algorithms.
I could go on, I love discussing pedagogy, but I feel this has gotten long winded enough. If anyone has questions I'll happily answer further.
thanks for taking the time! I'll offer a response in two parts: praise, and then criticism.
Praise
I learned "old" math: the way it was taught is that there is a universal algorithm that is applied mechanically without thought or curiosity, to arrive at equal-sign-means-answer. What this can do is several things:
So yeah, I love new math. The 10-frame, when it was my turn to teach, was an array of single ants, row of 10 ants, 10x10 ants, and other friendly number arrays of ants. We also did number lines, leaves and nodes etc but ants worked and we went further with them.
Criticism
What I don't love about new math though, is when each of the strategies are taught at length, and there's a take home portion for the kids to demonstrate competency for each even though this method is clearly not resonating with them. What we see in the video is this: the kids are given an algorithm for how to apply Triple Node Triangle, applied mechanically without thought or curiosity, then drilled for speed. We have the trappings of new math, taught in the ugly old way, then that effort is multiplied by an assortment of ways. Where once we had one brutal way of grinding kids, we now have multitudes.
What you're doing is teaching thinking strategies, encouraging creativity, respecting individual differences, and then helping each student mentally manipulate shapes faster in the way that makes the most sense to them. I can't say this is how it works for classrooms where the teachers themselves admit they always hated math, that they don't understand or want or celebrate new math, and they just assign worksheets demanding algorithmic application equal sign means answers.
I think your criticism is valid. I feel (I don't have the data) that many of my fellow educators don't respect the process. As soon as I saw what New Math was trying to do I was sold.
Honestly the most difficult part of teaching it is the brutal reality that many students don't have the inclination to question these things and look for a deeper understanding. I don't mean this as a mark against them, it is just reality. Their curiosity and motivation lies elsewhere.
I have one student right now who is able to quickly pick up processes, but struggles to explain their thinking. This is another type of challenge. Clearly they can make sense of the math but their expressive language can't keep up. That's tough.
That does sound tough but I would imagine it must feel so amazing when you see that spark ignite :) what you're teaching them is that the world is a knowable and reasonable place, that we can grab abstractions and turn them over and cut them up and stick em together the way we do with clay, that we can start with "I don't know" and have fun kneading it until we arrive at an Aha! I think the first group is "just tired, boss", that they have a lot of things that demand their attention and time, and they're spending cycles figuring out the fastest way to get us to go away.
Your criticism of the each strategies rings true for me and my young. It's great when it works, but my child grokked it with the first few and the was bored for a week. As well as the worse/burnt out teachers basically skipping half them for one reason or another. Large classroom sizes really hurt.
And sometimes, despite the vast covering of different methods, sometimes the old methods just click. Kiddo was struggling with on particular concept for a bit outside of class, mom showed 'the old way,' and it clicked instantly and their speed and accuracy trippled overnight.
I will say I think it's this solid foundation, as well as the algebratic concepts also introduced, which enabled this kind of moment.
A humorous video based on a question from r/askmath that does a good job of explaining how to do 2nd grade math using a slightly different method than was taught to most of us. It's definitely interesting and different than my mental method of decomposing (I'd explain it but it's a whole thing) which gets me there but takes me down a circuitous route.
Off-Topic: For those not in the US school system, 2nd Grade is the 3rd year of schooling for our children, which is 0-based with Kindergarten, then has another 12 years of required schooling with grades levels that count from 1 to 12.
The first half of the video is kinda meh but the second half is where the good stuff is. The regrouping idea is very intuitive and it's unfortunate how the worksheet only has 10 as the big number over and over because it doesn't really teach the principle.
The idea is to break big numbers into friendly chunks like 59 becomes 40 + 10 + 5 + 4.
It helps to pick a number like 4, 5, or 6, to always use so that you only need to memorize a few fixed subtractions from 10.
Then when you see something like 44-27 you break it into groups which have as many similar numbers on each side as is comfortable and then you can just cross out the same numbers and subtract whichever is easiest:
(30 + 7 + 3 + 4) - (20 + 7)
cross out duplicates: 7
(30 + 3 + 4) - (20)
subtract the closest numbers: 30 - 20
10 + 3 + 4
add together the remaining chunks to get 17
Yes, this is the way! It simplifies the mental math and makes even larger equations easy to break apart and calculate.
Why break 44 to 30+7+3+4 and not just 30+7+7? Since you don't do anything with the 3 and 4 except add them again at the end.
Sure! That's a valid strategy as long as you have enough subtractions memorized, ie. knowing that it will be useful to take 14 instead of 10.
If you don't know ahead of time and don't know where to start, a sub-strategy could be to recursively splitting in half like a binary search, ie. 44 = 22 + 11 + 6 + 3 + 2
Then you could pick the number that's closest to 7 and any left over you could subtract again from the next closest number until you have only additions left
22 + 11 + (6 - 6) + 3 + (2 -1)
(22-20) + 11 + 3 + 1
2 + 11 + 3 + 1
17
Tangential, but I know we've got a lot of teachers here.
I'm on the hunt for a good (and reasonably priced) homeschool prek syllubus to supplement bored younger sibling. We've got Hooked On Phonics which is great for the early reading, but want something a bit more well-rounded for everything else.
DM me, I taught prek for 15 years. I didn't like any curriculum out there and made my own based off of state standards. I can help you do the same.
The authors says "borrowing" is useless for numbers less than 20 and "you just have to know what 16-9 is", but is it really true? I don't remember what 16-9 is off the top of my head, if I have to calculate it, I always calculate it as "16-9=6+(10-9)=6+1=7". Of course, over the years I got very fast at this operation, but every time I have to subtract from a number larger then 10 I still basically do it in two steps. Do most people actually just remember results for subtractions of all numbers up to 20?
Disclosure I didn't watch the video (can't do audio atm) but that is not borrowing, Borrowing is saying "ok 6-7, no we can't do that because 6-7 is negative so take a 10 away from the next column and change that to 16-7 now we can do 16-7=9 and yes we do just need to know this"
eg
26-7= 20 + 6 - 7 = 10 + (10 + 6) - 7 = 10 + 16 - 7 = 10 + 9 = 19
obviously you normally write in columns but I'm not gonna try to format that
I made a longer comment in the thread but I want to jump in here and say that no in fact you do NOT "just need to know" that 16-7 = 9! We teach an idea called "friendly numbers" to help students with this idea, since at this level they are only JUST beginning to move from a concrete approach to an abstract approach.
The most basic "friendly number" is 10. How will that help us here? Students know they want to get to the friendly number. they see 16 - 7 and immediately change it to 16 - 6 - 1 so they can reach a much easier equation to solve: 10 - 1.
That's how you avoid memorization in math and teach number relationships at the same time.
EDIT: Woops I misread your post. I need more coffee. I think we're talking in the same direction so I'll leave my post up as well.
This is exactly how I do all mental math.
I like how I was taught for longer numbers. Take:
You can go left to right here if you know that when the two digits in your column will require borrowing, you borrow from the left as always. You keep a running tally of what your number is as you move left to right in digits, so we start with an answer of
0000and each 0 gets replaced as you go left to right:And I think there's something just lovely about that.
I'm not sure why you would want to do this if you can just do normal long subtraction which is basically the same but goes right to left? Or is this approach easier to do in your head without paper?
Easier in my head. I get lost in formulas and the like. Math was my strongest subject until about middle school, then I had to not only show my work but use only the method I was presented instead of being allowed to invent my own, and I got docked again and again on my homework and tests to the point that I decided I really wasn't any good at math and I gave up trying to learn it at all, just doing enough to get by and scrape by in tests. I went from advanced placement to remedial math in 3 years, because I fully gave up.
I imagine most people who are really good at math just remember them, because they've used them so much that it's just reflex. My mental arithmetic ability has drawn comments at various times over the years and for me those subtractions are just recall... but that's after endless repetition. I don't remember ever making a conscious effort to memorize them or specifically being taught them when I was in school, but I do remember the breakthrough of recognizing patterns for certain combinations of minuend and subtrahend (as well as products for times tables) on my own. In my school/classes (which were outside of the US system), we did a lot of mental arithmetic testing, likely much more than the average US school today.
I learned that if you subtract nine from anything the ones digit will be one higher than it is now. So it's almost as quick as subtracting from ten. Between that and memorization I'm fine, until my brain hiccups and I have forgotten how math works for 30 seconds and I have to do it the long way.
There are pros and cons to memorizing it vs doing it each time in your head I suppose. I also had times table speed tests as a kid and was one of those that always finished and finished early so that was a skill of mine.
Have child that passed through the second grade. Under 20 is 'fast facts' and they want those snap memorized for faster standardized test taking.
It makes sense though, because it minimizes the distance you need to decompose. As soon as you hit 20, and then later multiples of 20 once times tables are built, will allow for super fast mental math.
Honestly I don't know. I use an approach not far from yours for 2 digit subtraction, but I veer wildly off course when you start subtracting really long digits.
Just noting that you double posted by mistake, I think.
I've noticed this happens sometimes when my internet connection is spotty. I'm sure it's something on my end.
(I'm not an educator, nor do I have any applicable background which would help me understand the field, that said)
Does anyone know off hand if the method proposed in "Knowing and Teaching Elementary Mathematics" -- or even the "borrowing" method portrayed in-video -- have been studied for their impact on math education compared to other social factors? My understanding of the Chinese education system + culture surrounding primary/secondary/post-sec education is poor, but I'm distantly aware of make-or-break exams (e.g. the zhongkao) which -- rightfully or otherwise -- are understood as setting the entire course of a child's life, at age 15, with a single pass/fail exam. That (apparently) leads to an incredible amount of pressure (and support) for children to learn everything, ASAP, as effectively as possible ... perhaps it's possible that that has more of an impact on education outcomes than different ways of subtracting numbers?
(edit) Similarly, primary and secondary education in North America is often -- again, rightfully or otherwise -- portrayed as daycare, so that parents can work their regular job to make rent. Or, in the case of lunch programs, to feed both themselves and their children, during the school year.