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1 hour ago, lewelma said:

Sure.  Basically, any word problem that asks you to form an equation.  You can get the answer using primary school methods, but when being tested for an algebra exam, the goal is to code it into algebra so that you develop the skills required as the problems get harder.

For example

Amy bought two seedlings for her garden. One variety was 8cm tall when she bought it and grew at a rate of 2.5 cm per week. The other variety was 5 cm tall and grew at a rate of 3 cm each week. After a certain number of weeks the two seedlings were the same height. Form an equation to work out when the two seedlings were the same height.

My students would not know that you *multiply* 2.5 by the number of weeks.  So they don't know to write that term as 2.5w.  So can't write the equation 8+2.5w=5+3w.  Their approach would be guess and check, and if required to show their work, they would put it into a table if you were lucky.  Basically, they would see it as repeated addition 8+2.5=10.5; 10.5+2.5=13, and they would just go up on each side until the two answers matched.  If the word problem required 2 decimal accuracy, they would be in trouble. 

All of my students are particularly bad at recognizing division, as they see it as repeated addition to get up to an answer because that is easier to do in your head. 

 

 

To be fair, that’s a correct solution. I have a lot of trouble with kids who think there’s only one correct way to do things. They got trained out of their mathematical confidence.

How would you encourage the algebraic way without making kids think this is wrong? 

 

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In NZ, the national assessment is called "applying algebraic procedures in solving problems" so if you solve the problem without using algebraic procedures you can't pass the test.  The goal is not to get a correct answer, the goal is to learn how to apply algebraic procedures in solving problems. 

As to how to teach this, I'll try to write more about that later.

Edited by lewelma

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2 hours ago, square_25 said:

To be fair, that’s a correct solution. I have a lot of trouble with kids who think there’s only one correct way to do things. They got trained out of their mathematical confidence.

How would you encourage the algebraic way without making kids think this is wrong? 

Not lewelma, but I thought I would throw in my two cents.

I would always validate that the guess and check (or other correct method they came up with) was correct and good thinking on their part.  But then I would challenge their method by tweaking the problem a little: How tall will the first sapling be after 20 weeks?  Which sapling will reach 1 meter tall first?  How long will that take?  How many more weeks will it take the other sapling to reach 1 meter tall?  etc.

Using the guess and check/chart method, each of those problems would have to be approached more of less independently.  I would try to lead the student to the idea that the algebraic equation is useful because it provides easy answers to all those questions and countless more.  NOT that the guess and check method is bad, but simply that it has its strengths and the algebraic equation has its strengths and it is important to start to learn when each method makes the most sense.  You could compare this to finding the area of a shape via dividing into a grid versus multiplying the side lengths.  There certainly are some convoluted shapes that might be better approached with the grid method, but for rectangles whose areas can easily be found by multiplication, the grid method would be inefficient and prone to errors.

I would also talk about the goals of math.  Certainly right answers are one goal, but perhaps even more importantly, math is a method of communicating.  For example, if you were to buy a sampling from the garden supply store, which would be more convenient, them giving you a huge chart that showed how tall the sapling would be at 1 week, and 2 weeks, and 3 weeks, and ....  OR would it be more convenient for them to let you know that to find the height you just had to multiply the number of weeks by 2.5 and then add 8?  And what if you were communicating with a computer?  More and more often math is being used to tell a computer what to do.  So what if you were programming a video game and needed to tell the computer how fast to let a character travel based on how much weight they were carrying.  It would be a waste of time and space to make a chart of every possibility, but with one equation you could tell the computer exactly what you wanted it to do.

Wendy

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1 minute ago, wendyroo said:

Not lewelma, but I thought I would throw in my two cents.

I would always validate that the guess and check (or other correct method they came up with) was correct and good thinking on their part.  But then I would challenge their method by tweaking the problem a little: How tall will the first sapling be after 20 weeks?  Which sapling will reach 1 meter tall first?  How long will that take?  How many more weeks will it take the other sapling to reach 1 meter tall?  etc.

Using the guess and check/chart method, each of those problems would have to be approached more of less independently.  I would try to lead the student to the idea that the algebraic equation is useful because it provides easy answers to all those questions and countless more.  NOT that the guess and check method is bad, but simply that it has its strengths and the algebraic equation has its strengths and it is important to start to learn when each method makes the most sense.  You could compare this to finding the area of a shape via dividing into a grid versus multiplying the side lengths.  There certainly are some convoluted shapes that might be better approached with the grid method, but for rectangles whose areas can easily be found by multiplication, the grid method would be inefficient and prone to errors.

I would also talk about the goals of math.  Certainly right answers are one goal, but perhaps even more importantly, math is a method of communicating.  For example, if you were to buy a sampling from the garden supply store, which would be more convenient, them giving you a huge chart that showed how tall the sapling would be at 1 week, and 2 weeks, and 3 weeks, and ....  OR would it be more convenient for them to let you know that to find the height you just had to multiply the number of weeks by 2.5 and then add 8?  And what if you were communicating with a computer?  More and more often math is being used to tell a computer what to do.  So what if you were programming a video game and needed to tell the computer how fast to let a character travel based on how much weight they were carrying.  It would be a waste of time and space to make a chart of every possibility, but with one equation you could tell the computer exactly what you wanted it to do.

Wendy

 

Sounds good to me! It's interesting... I see quite a lot of high school students (often relatively high-performing one) and definitely some of them have entirely forgotten that guess and check is occasionally a totally valid method of solving a problem! But yes, it's not an effective method for problems like this. 

I feel like one answer to this problem is not to shove so much stuff into algebra in the first place. At least, that's what I'm currently trying with my daughter: to get her used to algebraic thinking at a developmentally appropriate level. Then we can start talking about things like, say, "If square represents the cost of one thing, what's the cost of 3 things?" 

Another failure mode of algebra that I've seen is that the kids somehow forget the the variables MEAN things. I teach precalc at AoPS and we go through properties of complex numbers, and I find that it's empowering for kids to get to check "conjectures" like "Is the magnitude of a sum the sum of the magnitudes?" Makes the properties come alive a bit, at least I think it does... 

Wendy, would you come back to guess and check once in a while to verify that their intuitions match the algebra? Or once you get to algebra, do you stay with algebraic techniques? 

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6 minutes ago, square_25 said:

Wendy, would you come back to guess and check once in a while to verify that their intuitions match the algebra? Or once you get to algebra, do you stay with algebraic techniques? 

Oh, I am all about double checking with guess and check...or just any method other than the one they first used to see if they get the same answer.  Not every problem, but certainly any they aren't completely sure of how to solve...and I make sure that my kids are consistently challenged with problems that stretch their abilities.

This is coming up a lot right now with my 9 year old because I am having him work through some old SAT tests.  Some of the math is just beyond what he truly knows how to elegantly solve, but I am still encouraging him to play with the numbers any way he knows how to see if he can at least eliminate a couple wrong answer choices.

For example, one problem that came up recently was:
Which of the following is an equation of a circle in the xy-plane with center at (0 , 4) and a radius with endpoint (4/3 , 5)?
It then listed 4 answer choices in standard (x – h)^2 + (y – k)^2 = r^2 form.

He is well versed in linear equations, but he had never encountered the equation of a circle, so did not know the significance of the r squared term.  The goal wasn't to teach any new math, but rather to see how creatively he could use the math he already knows, so the first thing I had him do was graph it as accurately as possible.  From there he could clearly see that he could find the length of the radius via Pythagoras, which could lead to the y intercepts.  At that point it was just a matter of guessing and checking his way through the answer choices until he found the one that produced the correct y intercepts.

In my experience, with my kids, math concepts go through a fairly standard learning progression. 
First, the child starts spontaneously asking about and experimenting with a concept.  My 5 year old is currently thinking and talking about multiplication a lot.  His official math curriculum hasn't gotten there yet, and he isn't ready to formally learn Multiplication with a capital M, but he still thinks it is wildly funny to "trick" me with "really hard" questions like 1 * 3 billion!!! 
Second, as their understanding of the concept develops, they frequently double check themselves with older, more comfortable methods.  When Prodigy recently asked the 5 year old how many legs 5 dogs had, he quickly counted by 5s up to 20, and then he proceeded to draw the 5 dogs and their legs and count to check himself. 
Third, their math curriculum eventually teaches the topic and their knowledge starts to be formalized.  They learn the vocabulary and algorithms that describe the concepts they have been developing in their heads all along.  Now they are firmly nudged into practicing the concept over and over until their arithmetic as well as their intuition is solid.  (Of course, by this point their challenging topics of interest are much more complex, so this drill is just wrapping up loose ends on a topic that conceptually is old hat.) 
Fourth, finally, they are masters of the topic and they feel indignant that anyone would even suggest they write that step down in a math problem or double check their answer.  It is at this point that I start looking for unique and challenging applications of the topic so that their skills and facility continue to grow.  At that point with multiplication I might introduce exponents or factorials or combinations or calculating approximate tax, because all of those seem very grown up, but sneak in extra multiplication practice.

Wendy

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45 minutes ago, wendyroo said:

Oh, I am all about double checking with guess and check...or just any method other than the one they first used to see if they get the same answer.  Not every problem, but certainly any they aren't completely sure of how to solve...and I make sure that my kids are consistently challenged with problems that stretch their abilities.

This is coming up a lot right now with my 9 year old because I am having him work through some old SAT tests.  Some of the math is just beyond what he truly knows how to elegantly solve, but I am still encouraging him to play with the numbers any way he knows how to see if he can at least eliminate a couple wrong answer choices.

For example, one problem that came up recently was:
Which of the following is an equation of a circle in the xy-plane with center at (0 , 4) and a radius with endpoint (4/3 , 5)?
It then listed 4 answer choices in standard (x – h)^2 + (y – k)^2 = r^2 form.

He is well versed in linear equations, but he had never encountered the equation of a circle, so did not know the significance of the r squared term.  The goal wasn't to teach any new math, but rather to see how creatively he could use the math he already knows, so the first thing I had him do was graph it as accurately as possible.  From there he could clearly see that he could find the length of the radius via Pythagoras, which could lead to the y intercepts.  At that point it was just a matter of guessing and checking his way through the answer choices until he found the one that produced the correct y intercepts.

In my experience, with my kids, math concepts go through a fairly standard learning progression. 
First, the child starts spontaneously asking about and experimenting with a concept.  My 5 year old is currently thinking and talking about multiplication a lot.  His official math curriculum hasn't gotten there yet, and he isn't ready to formally learn Multiplication with a capital M, but he still thinks it is wildly funny to "trick" me with "really hard" questions like 1 * 3 billion!!! 
Second, as their understanding of the concept develops, they frequently double check themselves with older, more comfortable methods.  When Prodigy recently asked the 5 year old how many legs 5 dogs had, he quickly counted by 5s up to 20, and then he proceeded to draw the 5 dogs and their legs and count to check himself. 
Third, their math curriculum eventually teaches the topic and their knowledge starts to be formalized.  They learn the vocabulary and algorithms that describe the concepts they have been developing in their heads all along.  Now they are firmly nudged into practicing the concept over and over until their arithmetic as well as their intuition is solid.  (Of course, by this point their challenging topics of interest are much more complex, so this drill is just wrapping up loose ends on a topic that conceptually is old hat.) 
Fourth, finally, they are masters of the topic and they feel indignant that anyone would even suggest they write that step down in a math problem or double check their answer.  It is at this point that I start looking for unique and challenging applications of the topic so that their skills and facility continue to grow.  At that point with multiplication I might introduce exponents or factorials or combinations or calculating approximate tax, because all of those seem very grown up, but sneak in extra multiplication practice.

Wendy

 

That's interesting! I haven't seen that progression myself, but I've also been doing something fairly non-standard, so maybe that's why. For this specific thing, I just defined multiplication without doing any skip-counting, so probably my daughter didn't have the chance to play around with it herself. She also doesn't seem to have any patience with calculational drills at all: I got her to practice adding out loud for a while, but she moans so much about simple calculations on a worksheet that I haven't been doing it at all. So even though she's not solid on her times tables, we've kind of moved on conceptually. We'll probably come back and drill a bit more at some point, though. 

Speaking of combinations, I've started doing questions like that with my daughter, although we haven't gotten to combinations, just to the multiplication principle. Like, if Bertie the bear has 4 shirts and 5 pairs of pants, how many outfits does he have? We've been coloring bears and once she got the idea I started adding questions which she would do without coloring These would get her to practice the facts. 

I haven't yet figured out at what point we'll do something closer to a standard sequence... like, I'm not sure when I'll need her to actually remember multiplication. Definitely at some point before serious algebra, anyway, maybe around fractions? I'm really not sure. I know that my daughter was super taken with binary so that was how we practiced quite a lot of addition, and it did seem to get her faster in adding! 

Oh, question for you: do you introduce the standard algorithms at the usual time? (Like, adding with carrying, borrowing, etc.) I've been avoiding them entirely for now, but I imagine I'll go back and do them at some point. 

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16 minutes ago, square_25 said:

That's interesting! I haven't seen that progression myself, but I've also been doing something fairly non-standard, so maybe that's why. For this specific thing, I just defined multiplication without doing any skip-counting, so probably my daughter didn't have the chance to play around with it herself.

Yeah, I don't tend to define much for them.  My kids seem to thrive on the discovery method, so most of what I do is label discoveries ("What you are doing is called multiplication.") and ask interesting questions that challenge them to grapple with topics more deeply.  I've never taught skip counting, but all of my kids have stumbled upon the idea naturally.

She also doesn't seem to have any patience with calculational drills at all: I got her to practice adding out loud for a while, but she moans so much about simple calculations on a worksheet that I haven't been doing it at all. So even though she's not solid on her times tables, we've kind of moved on conceptually. We'll probably come back and drill a bit more at some point, though. 

My kids don't have a choice about drill.  I think arithmetic fluency is important, so I have them steadily work on it.  I use the method that Susan Wise Bauer describes as pecked to death by ducks.  Just a little bit of work, consistently done every day.  The rest of the kids' math time is spent exploring more challenging concepts and puzzles.

Oh, question for you: do you introduce the standard algorithms at the usual time? (Like, adding with carrying, borrowing, etc.) I've been avoiding them entirely for now, but I imagine I'll go back and do them at some point. 

My kids work through Math Mammoth for arithmetic (along with many other materials for deeper, conceptual learning).  So far, all of my boys have started their kindergarten year somewhere in the middle of level 1.  I start them there not because that is the level of math they are working at (they already have a firm grasp on place value and are mentally adding and subtracting two digit numbers and exploring multiplication), but because that is the level of "math writing" they can do.  They work on it for 5-10 minutes a day, and it acts as review and reinforcement as well as introducing algorithms, vocabulary and notations they might not be familiar with.  It also provides them with practice reading and working carefully and independently.  Math Mammoth introduces two digit addition and subtraction in columns along with regrouping in addition at the end of level 1, so my boys learn it sometimes during kindergarten.  Regrouping subtraction isn't covered until 2b, so my kids learn how to formally write that algorithm toward the beginning of first grade.

Since it is mostly review for my kids, we move pretty quickly and add a lot of supplements.  My second grader just finished MM4, so learned the multi-digit multiplication and long division algorithms.  He was, of course, already doing both of those operations, but only mentally or conceptually.  He often calculated them via a hodge-podge partial product method he had figured out or through logical reasoning.  Now that he knows the algorithms, he seems to use a nice balance of those and mental math and his old partial product method depending what best suits the problem.

 

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4 minutes ago, wendyroo said:

 

 

Hmmm, this might not be quoting properly. Not sure why not if it's not... 

Anyway, sounds different from what we're doing, although certainly sounds like a reasonable approach! I always tend to want to reinvent the wheel ;-). We'll see how it goes. 

Do you think doing calculational drills that are specifically calculation and nothing else is important? For example, if she's solving the system

x+y = 15, xy = 56, (well, in triangles and squares instead of x and y),

then she's practicing quite a lot of multiplication along the way, since currently we solve those by trial and error (and number sense, I suppose). But last time I gave her things like 7*8 to practice, she took longer to do it than she does a question like the one above (which generally takes quite a few multiplications for her before she gets it.) 

Actually, I had a bit of trouble when we did a lot of Beast Academy questions of form "35 + 56 = ", for some reason, because she started misusing an equals sign and I found that troubling. So that probably also dissuades me currently. I might be overly paranoid about this issue, though: I see misunderstanding of equals signs so often in older kids that I'm sensitive to it. 

Thanks for the detailed responses! I like hearing what other people are doing for their math with mathy kids. 

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14 minutes ago, square_25 said:

Do you think doing calculational drills that are specifically calculation and nothing else is important? For example, if she's solving the system

x+y = 15, xy = 56, (well, in triangles and squares instead of x and y),

then she's practicing quite a lot of multiplication along the way, since currently we solve those by trial and error (and number sense, I suppose). But last time I gave her things like 7*8 to practice, she took longer to do it than she does a question like the one above (which generally takes quite a few multiplications for her before she gets it.) 

Actually, I had a bit of trouble when we did a lot of Beast Academy questions of form "35 + 56 = ", for some reason, because she started misusing an equals sign and I found that troubling. So that probably also dissuades me currently. I might be overly paranoid about this issue, though: I see misunderstanding of equals signs so often in older kids that I'm sensitive to it. 

Thanks for the detailed responses! I like hearing what other people are doing for their math with mathy kids. 

I have four kiddos, so around here I need to foster independence. 

Something like "x+y = 15, xy = 56" is what we would work on together during problem solving time.  It might come from Beast Academy or Singapore Challenging Word Problems.  During problem solving time, we do a lot of parallel solving, so we both work on the same problem independently.  Then, when we both reach an answer (or at least make a go of it), we compare notes.  Did we get the same answer?  What methods did we each use?  If we got different answers, what support/defense can we provide for our calculation?  Can we come up with a third solving method to act as a "tie breaker"?

I try very hard not to be seen as all-knowing during problem solving time...and with Beast Academy puzzles that is often pretty easy.  I model good problem solving techniques (like correct use of the equal sign), but I also try to model resilience, perseverance, not being frustrated by wrong answers, a willingness to try one solving method, realize it is not panning out, and start down another path.

I would never leave that kind of high-level, challenging math to be done independently.  I think deeper, conceptual understanding it built through modeling, interaction, Socratic Dialogue, etc.  However, I do think there is value in my kids then spending some time working on lower level math independently.  It is not pure, tedious drill and kill, Math Mammoth is actually a very conceptual curriculum, but I do have the kids working independently through a much lower level than they are conceptually.  When they sit and tackle it all by themselves, it isn't meant as a huge arithmetic challenge, but rather as a challenge in close reading, organized thinking, diligence, focus, and self-control.  The fact that it often really cements their mathematical foundations is just icing on the cake.

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Oh, she loves systems of equations. It’s all integer solutions, of course, so it’s largely guess and check: well, intelligent guess and check, since it’s best guessing things that work for one of them! We do them so much because other than binary (which she looooves), they are her favorite thing.

I suppose maybe I do seem too all-knowing in math! I’ve spent a bit too much time teaching it in my life. I’m much more uncertain in pretty much all other subjects.

Resilience is a big goal for us as well. I think it helps that right now I’m just working one on one with her, since we can discuss a lot of stuff. (It’s just that systems wouldn’t be an issue since she’s seen so many.)

I worry about doing conceptually lower level math with her just because she complains so much. I call her a mathy kid because she’s so good at it, but she doesn’t love it unless it’s a new cool concept (like negatives or binary.) So I tend to let her steer more than I might with a kid who seemed enthused about numbers and puzzles in general. I was that kind of kid, personally. She’d rather read.

But I’ll certainly need to teach her the facts and algorithms and other basics at some point.... I should think about when. It’s a good thing to plan a bit more.

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