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Help with 6th grade math!


blessedmom3
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With a bar diagram:

[   P   ] = Paul's Age
[   P   ][   P   ][   P   ][   P   ][   P   ] = Dad's Age (5 times Paul's Age)

[  P   ][ 3 ] = Paul's Age in 3 Years
[   P   ][ 3 ][   P   ][ 3 ][   P   ][ 3 ][   P   ][ 3 ] = Dad's Age in 3 Years (4 times Paul's Age)

So if this is Dad's Age in 3 years:
[   P   ][ 3 ][   P   ][ 3 ][   P   ][ 3 ][   P   ][ 3 ]
then, we can subtract 3 to find a different expression for Dad's current age:
[   P   ][ 3 ][   P   ][ 3 ][   P   ][ 3 ][   P   ]

So, we now have two ways to write Dad's current age:
[   P   ][   P   ][   P   ][   P   ][   P    ] = Dad's Age
[  P  ][ 3 ][  P  ][ 3 ][  P  ][ 3 ][  P  ] = Also Dad's Age

We set them equal, 5P = 4P + 9
Paul is currently 9, Dad is currently 45.  (Paul will be 12 and Dad will be 48.)

 

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50 minutes ago, blessedmom3 said:

She drew some bars like she learned in Singapore Math. First, one bar (unit) for Paul and 5 bars for Dad. Then she drew Paul one bar and dad 4 bars. She added + 3 years then she got stuck. Honestly, I don't know how to solve it without a traditional algebra equation but there must be a simpler way.

Is that the order she did it in: drew the new Paul/Dad bars *and then* added three years to each?  If so, that may be where she is getting into trouble.  (Also, if she's doing it on graph paper or is otherwise a real stickler for drawing the bars perfectly to scale, she could be having problems, too, because to draw it accurately on graph paper requires you to already have solved the problem (I've spent 15min attempting it in Excel and still haven't solved it yet), but to do a rough not-to-scale sketch is the work of a minute and illustrates what's going on perfectly.)

Here's what I did.  I first drew the Now Paul/Dad bars:

NOW                                        
Paul's Dad          
Paul                                  

Then I added on the 3 years:

+3 YEARS                                          
Paul's Dad           3
Paul   3                                

Then I re-partitioned the new diagram into fourths (fudging the spacing a bit):

+3 YEARS                                          
Paul's Dad                                         3
Paul   3                                


The key thing to notice is that the new units are equal to the old units plus 3:

+3 YEARS                                          
Paul's Dad   3   3   3           3
Paul   3                                

This might help show it better:

Paul's Dad now            
Paul's Dad +3   3   3   3           3
Paul + 3   3                                
Paul now                                    


So 4 units + 9 = 5 units => Therefore 1 unit = 9 years.

Paul is 9 years old, and his dad is 45 years old.

 

ETA: If she re-partitions the bars *before* adding on three to both ages, she might be missing how the new unit relates to the old unit.  Especially if she re-partitioned Paul's Dad's *current* age into fourths and appends the +3 years to the end, so that the +3 years *isn't* part of the new fourths.

 

 

Edited by forty-two
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I've never done bar diagrams. However, I do talk things out. So let me explain how I'd do it. 

We know the following: 


Dad's current age =  5* Paul's current age.

Dad's age in 3 years = 4*(Paul's age in 3 years). 

 

OK, but Dad's age in 3 years is obviously 3 more than his current age. That means that 4*(Paul's age in 3 years) is 5*Paul's current age plus 3. 

Of course, Paul's age in 3 years is 3 more than it is now, which means that if you quadruple it, you get 4*Paul's current age plus another 12. And that must be equal to 5*Paul's current age plus 3, which means that 4*Paul's current age plus 9 is 5*Paul’s current age. That means Paul is 9. 

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6 hours ago, Not_a_number said:

I've never done bar diagrams. However, I do talk things out. So let me explain how I'd do it. 

We know the following: 


Dad's current age =  5* Paul's current age.

Dad's age in 3 years = 4*(Paul's age in 3 years). 

 

OK, but Dad's age in 3 years is obviously 3 more than his current age. That means that 4*(Paul's age in 3 years) is 5*Paul's current age. 

Of course, Paul's age in 3 years is 3 more than it is now, which means that if you quadruple it, you get 4*Paul's current age plus another 12. And that must be equal to 5*Paul's current age, which means that Paul's current age must be 12. 

Except Paul’s current age is 9.

If Paul was currently 12, then Dad would be 60.  And in 3 years Paul would be 15 and Dad would be 63, which is not 4 * 15. 

Using the bar diagram helps show that 4*(Paul's age in 3 years) minus 3 is 5*Paul's current age. 

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1 hour ago, wendyroo said:

Except Paul’s current age is 9.

If Paul was currently 12, then Dad would be 60.  And in 3 years Paul would be 15 and Dad would be 63, which is not 4 * 15. 

Using the bar diagram helps show that 4*(Paul's age in 3 years) minus 3 is 5*Paul's current age. 

Ugh, that’s what I get for rewriting a solution I did with fractions to try to simplify it. I got that 9/4 is a twentieth of his Dad’s current age and kept going from there. Let me fix.

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1 hour ago, wendyroo said:

Except Paul’s current age is 9.

If Paul was currently 12, then Dad would be 60.  And in 3 years Paul would be 15 and Dad would be 63, which is not 4 * 15. 

Using the bar diagram helps show that 4*(Paul's age in 3 years) minus 3 is 5*Paul's current age. 

Ok, fixed it. That’s embarrassing :-P.

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I met this sort of question twice during my education. The later one was indeed taught algebraically:

 

5P = D

4(P+3)= D+3

4P + 12 = D + 3 (multiply out the binomial)

4P + 9 = D (put all the numbers on the same side)

9 = D/4P (move the P to the other side of the equation)

9 = 5P/4P (D was defined at the beginning; we are replacing to have all the algebra as P)

9 = P (divide everything by 5)

(Hint: I'm fairly sure CLE 6 wasn't going for this as a solution method 😉 - this is just so that anyone who is planning to teach algebraically can see how that works).

 

However, the earlier one was taught as estimation and how to approach a correct answer. The idea is to plug in a number - any number that seems viable - see what happens, and then try a bigger/smaller number until the correct answer is obtained.

For example, let's try "Paul is 10". 10 is a good place to start because it's a common age for a child, and is also a round figure.

If Paul is 10, then Dad is 50. Which would mean Paul is 13 in three years' time, so Dad should be (13*4) = 52.

However, 52-50 = 2, not three. (Unless Dad is in a spacecraft travelling at near-light-speed, and I doubt a complication that difficult would be in CLE 6). So maybe the estimate we used for Paul was too young. Not much too young, though.

Let's try "Paul is 12" (most children above the age of 10 don't use halves in their age descriptions, and if this turns out to be too old, we can be almost certain that 11 is therefore the correct answer... ...though given this is maths, we'll test that anyway if necessary).

If Paul is 12, then Dad is currently 60. Which would mean Paul is 15 in three years' time, so Dad should be (15*4) = 60.

However, 60-60 = 0, not three. So we've just gone the wrong way. (It's not always obvious which direction of travel is best when learning to estimate unfamiliar types of equation, so it's useful to learn when to try estimating the other way).

Let's try "Paul is 9". Since children under 10 do sometimes call themselves things like "9 1/2", it's worth considering that as a possible answer.
If Paul is 9, then Dad is currently 45. Which would mean Paul is 12 in three years' time, so Dad should be (12*4) = 48.

48-45 = 3, which is the answer for which we searched.
 

Why consider teaching it this way? Because some maths is very complicated, and it's good sometimes to have strategies for deciding what sort of answers are plausible. Teaching estimation-based approaches to getting the correct answer helps a student get a sense of what sorts of answer might work, which gives them more options when faced with an overwhelming question. It's easier to teach it when there is a definite, known (to the teacher) answer to aim for that is simple, rather than waiting for when fractions and decimals complicate matters.

 

I do like the bar approach @wendyrooused and the "talk it out" method @Not_a_numberused. This goes to show that there are several good methods of finding an answer here, and the important thing is to use one that makes sense.

Edited by ieta_cassiopeia
Word added to make reasoning process make sense.
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13 minutes ago, ieta_cassiopeia said:

Why consider teaching it this way? Because some maths is very complicated, and it's good sometimes to have strategies for deciding what sort of answers are plausible. Teaching estimation-based approaches to getting the correct answer helps a student get a sense of what sorts of answer might work, which gives them more options when faced with an overwhelming question. It's easier to teach it when there is a definite, known (to the teacher) answer to aim for that is simple, rather than waiting for when fractions and decimals complicate matters.

Yes. Very much this. I’ve seen lots of kids who really have no feeling for whether they did it right or not. That’s why I would focus on making this make sense to a kid.

I worry about this student jumping to bar diagrams before the question is really internalized. That’s why I suggest talking it out and following the student’s lead.

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I do think guess and check is a very viable strategy for this problem.

In actuality, my kids would probably pull out the Hands on Equations manipulatives to solve this problem.  We use bar diagrams some in the early grades - my 2nd grader is finishing up Math Mammoth 4 and still uses bar diagrams occasionally - but my kiddos start Hands on Equations around 3rd grade and switch to that method.

Hands on Equations is really just the algebraic method expressed visually.
We always start by labeling our quantities on a piece of paper, so we would have a pawn labeled Paul's current age and 5 pawns labeled Dad's current age.

We would then set up a balance scale to represent "In 3 years, Paul will be 1/4 his dad's age" except rewritten as "In 3 years, Paul's dad will be 4 times as old as Paul."  On one side would be 5 pawns (Dad's current age) plus a 3 to show how old he will be in 3 years.  On the other side we would have four groups of a pawn and a 3 to show four times how old Paul will be in 3 years.

So our scale would look like:
X X X X X 3  ^ X 3 X 3 X 3 X 3

The child would remove one 3 and 4 pawns from each side to see that one pawn (which we have labeled Paul's current age) balances with three 3's aka 9.

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I’m going to ask DD8 to do it and see what happens 🙂 . 

Just asked: 

She says that Paul’s age is x, and Paul’s dad age is 5x, so then 5x+ 3 is 4(x+3). She then set those equal and got 5x+3= 4x+ 12, 5x= 4x+ 9 and then she said x=9.

Except she only wrote down the first equation and did the rest in her head, argh! So it took a while. I guess we better drill these, too. 

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1 hour ago, blessedmom3 said:

Notanumber,

just curious, what level is your 8yo working on? She must be really smart to be able to solve this! At 8, most kids barely learn their facts. 

Yeah, she's the kid of two mathematicians and she's very accelerated. She's probably around late middle school/early high school level, depending on the specific topic. (I make up her curriculum, so we haven't gone in anything like the standard order. I'm lucky enough to know enough math to be able to follow her interests.) As a concrete measurement, I gave her the AoPS "Are you ready?" test for Intro Algebra, and she placed fairly easily into that. 

On an unrelated note, @wendyroo, my math struggles have continued today, lol. I multiplied 40*250 and got 4000, for one thing. Not my day! 

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2 hours ago, Not_a_number said:

On an unrelated note, @wendyroo, my math struggles have continued today, lol. I multiplied 40*250 and got 4000, for one thing. Not my day! 

I had a rough math day too!! 

Peter got stuck on a practice SAT question asking for the period of the function f(x) = csc(4x).

He's never really studied trig, though he knows and understands SOH-CAH-TOA.  He knows that csc is the reciprocal of sine.  He knows that sine graphs as a wave and that the period and the amplitude can change based on...stuff.  But that is about it.

Note: My certainty of the correct answer was shaky.  I was 90% sure I knew the period of sin(4x), and 75% sure the period of csc(4x) was the same.

Undeterred, I confidently told him we could approach the question as a guess and check since we were presented with the correct answer...along with 4 incorrect.  I want his first instinct to be trying something with what he does know rather than dwelling on what he doesn't know about any given problem.

Crash and Burn!!  We created our little table, filled in the values for each answer choice, and none of them were even close!!  Gah.  The lesson quickly pivoted to the important, in an SAT situation, of using your time wisely.  Give a problem a good initial try, but know when to hold 'em and know when to fold 'em; know when to walk about and know when to eliminate answer choices that are clearly wrong and randomly choose between the remaining.  😄

 

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1 minute ago, wendyroo said:

I had a rough math day too!! 

Peter got stuck on a practice SAT question asking for the period of the function f(x) = csc(4x).

He's never really studied trig, though he knows and understands SOH-CAH-TOA.  He knows that csc is the reciprocal of sine.  He knows that sine graphs as a wave and that the period and the amplitude can change based on...stuff.  But that is about it.

Note: My certainty of the correct answer was shaky.  I was 90% sure I knew the period of sin(4x), and 75% sure the period of csc(4x) was the same.

Undeterred, I confidently told him we could approach the question as a guess and check since we were presented with the correct answer...along with 4 incorrect.  I want his first instinct to be trying something with what he does know rather than dwelling on what he doesn't know about any given problem.

Crash and Burn!!  We created our little table, filled in the values for each answer choice, and none of them were even close!!  Gah.  The lesson quickly pivoted to the important, in an SAT situation, of using your time wisely.  Give a problem a good initial try, but know when to hold 'em and know when to fold 'em; know when to walk about and know when to eliminate answer choices that are clearly wrong and randomly choose between the remaining.  😄

 

I'm afraid to say anything given how my day is going, but I think the period is pi/2 😉 

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1 minute ago, Not_a_number said:

I'm afraid to say anything given how my day is going, but I think the period is pi/2 😉 

Yeah, it turns out my initial instinct was right...but as you and I both know, it is very hard to conceptually teach a topic that you yourself don't truly understand.

I wasn't willing to burden him with my very shaky procedural knowledge about being kinda, sorta sure that the period of csc (4x) should, I think, be the same as sin (4x)...which I could in no way defend beyond a faint echo of a memory of trig lessons long ago.

I really, really wanted to be able to solve the problem by building on the information he already had.  It seemed reasonable, logical, feasible - but no!!  The math winds were not in my favor and I think the poor kid is more baffled than he started.  🙃

Oh, well.  Today Audrey accurately told her speech therapist that she is 4 and 11/12 years old.  And when pressed she explained that that means she has been alive for 4 whole years and all but one little piece of another year.  I saw some panic on the speech therapist's face at the mention of a fraction, and I got the impression that Audrey may actually have a better conceptual understanding of what it means than the therapist.  I'm going to count that as a win and call the day's math a wash on average!

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17 minutes ago, wendyroo said:

Yeah, it turns out my initial instinct was right...but as you and I both know, it is very hard to conceptually teach a topic that you yourself don't truly understand.

I wasn't willing to burden him with my very shaky procedural knowledge about being kinda, sorta sure that the period of csc (4x) should, I think, be the same as sin (4x)...which I could in no way defend beyond a faint echo of a memory of trig lessons long ago.

I really, really wanted to be able to solve the problem by building on the information he already had.  It seemed reasonable, logical, feasible - but no!!  The math winds were not in my favor and I think the poor kid is more baffled than he started.  🙃

Oh, well.  Today Audrey accurately told her speech therapist that she is 4 and 11/12 years old.  And when pressed she explained that that means she has been alive for 4 whole years and all but one little piece of another year.  I saw some panic on the speech therapist's face at the mention of a fraction, and I got the impression that Audrey may actually have a better conceptual understanding of what it means than the therapist.  I'm going to count that as a win and call the day's math a wash on average!

Hah, I bet Audrey really does have a better handle on fractions than the speech therapist!

Let me know if you want a quick run-through of why you were right for that trig question 🙂 .

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Reporting back:
I had my 9 year old complete the original problem using Hands on Equations pawns.  He cruised through it up until the very end when he subtracted 3 from 12 and got 8.  Oops.  He caught his own mistake because Hands on Equations teaches plugging your answers into the original equation to check your work.  Nope, 11 is not a quarter of 43.  He caught his mistake on the second go through and came up with the correct answer.

This is one of the exact types of problems in the HOE word problem book which DS has been completing for the last year and is almost done with, so I'm not surprised he knew how to handle it.  Knowing DS, I'm also not surprised that he tripped himself up on the simplest aspect of the whole process.  😄

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13 hours ago, wendyroo said:

I saw some panic on the speech therapist's face at the mention of a fraction, and I got the impression that Audrey may actually have a better conceptual understanding of what it means than the therapist. 

Most people's math ability tops out at easy fractions, decimals, and percents--so, like fifth grade level.

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38 minutes ago, EKS said:

Most people's math ability tops out at easy fractions, decimals, and percents--so, like fifth grade level.

I agree that matches my experience with the general public.

Do you think it is mostly nurture or nature?  Do you think it is possible to design an educational system that allows almost all students to thoroughly master arithmetic and to understand and competently use basic algebra, geometry, probability and stats?

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Just now, wendyroo said:

Do you think it is mostly nurture or nature?  Do you think it is possible to design an educational system that allows almost all students to thoroughly master arithmetic and to understand and competently use basic algebra, geometry, probability and stats?

This is actually a question I've been trying to answer for some time.  I guess one data point is math achievement in other countries, where they do approach achieving this goal.  At least I think they do.  

But then you look at the writings of some teachers in the trenches here in the US, and their experience is that most students are constitutionally unable to understand basic algebra.  Is that because they haven't been taught properly all along?  I don't know the answer to that.

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4 minutes ago, EKS said:

This is actually a question I've been trying to answer for some time.  I guess one data point is math achievement in other countries, where they do approach achieving this goal.  At least I think they do.  

But then you look at the writings of some teachers in the trenches here in the US, and their experience is that most students are constitutionally unable to understand basic algebra.  Is that because they haven't been taught properly all along?  I don't know the answer to that.

Yeah, I like to think that later deficits are simply cumulative poor teaching, that any child of fairly normal intelligence given a strong math education from an early enough age would be able to meet those modest goals.  But if that is the case then it is certainly a damning indictment of our current educational system because we are falling far short of that benchmark.

My local, solidly upper-middle-class, well-ranked school district has an average math proficiency score of 31%.  Over two thirds of the students cannot even hit the fairly low "proficient" level.  😞

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3 hours ago, EKS said:

This is actually a question I've been trying to answer for some time.  I guess one data point is math achievement in other countries, where they do approach achieving this goal.  At least I think they do.  

But then you look at the writings of some teachers in the trenches here in the US, and their experience is that most students are constitutionally unable to understand basic algebra.  Is that because they haven't been taught properly all along?  I don't know the answer to that.

It's because they haven't been taught properly all along. I have theories. 

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4 hours ago, Not_a_number said:

It's because they haven't been taught properly all along. I have theories. 

What I don't understand is why teaching elementary math so that kids are ready to deal with algebra has to be so difficult.  If you have the endpoint in sight, it's not rocket science.  At least it seemed that way to me.  And I had a terrible elementary and secondary math education as a kid.

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3 minutes ago, EKS said:

What I don't understand is why teaching elementary math so that kids are ready to deal with algebra has to be so difficult.  If you have the endpoint in sight, it's not rocket science.  At least it seemed that way to me.  And I had a terrible elementary and secondary math education as a kid.

Hmmmm. What would you say were the things you had to do? 

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28 minutes ago, Not_a_number said:

Hmmmm. What would you say were the things you had to do? 

This is actually a good question.  First, I had some clue about where we were going even in the beginning, though this got a whole lot better after I spent the time to relearn algebra and geometry.  I also had the advantage of actually interacting with my students every day, so I could make absolutely sure they understood what I was teaching them.  Because of this I could also fill gaps immediately when I uncovered them.  And after a problematic start with Saxon, I used good resources and supplements to help me stay on track, and I made sure that I understood how to teach from those resources effectively.

So how do you do those things in a classroom environment?  Ed schools should ensure that elementary teachers have a profound understanding not only of the math they are teaching, but also of algebra and geometry (at a minimum).  I would argue that even better would be having teachers who have been specially trained in math education (including taking real college math courses) do the teaching.  Since I have never taught a group before, I don't know how easy it is to assess kids' understanding individually, but I know that this was a huge key to our success here.  And then schools should adopt materials that work (Singapore was able to do it), provide training in their implementation, and the ensure that teachers are actually using them (rather than pulling random worksheets off the internet).

Again, other countries seem to be able to do it.  I don't buy the excuse that the US is "too diverse" to ensure decent math achievement.  If you start with properly trained teachers, develop a coherent program from the ground up, build in lots of places to check in with students' understanding, and provide room in the curriculum for reteaching, I can't imagine why it wouldn't work.

But maybe I'm wrong.

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On 9/26/2020 at 12:00 AM, Not_a_number said:

I also tend to think there’s a LOT of value in kids doing questions like this by guess and check. It gives them intuitions about how things work.

I just used guess and check.  Do it with Paul is 5 and 10.  10 is closer to working.  Try 8 Nope, try 9.  Bingo.

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1 hour ago, EKS said:

This is actually a good question.  First, I had some clue about where we were going even in the beginning, though this got a whole lot better after I spent the time to relearn algebra and geometry.  I also had the advantage of actually interacting with my students every day, so I could make absolutely sure they understood what I was teaching them.  Because of this I could also fill gaps immediately when I uncovered them.  And after a problematic start with Saxon, I used good resources and supplements to help me stay on track, and I made sure that I understood how to teach from those resources effectively.

So how do you do those things in a classroom environment?  Ed schools should ensure that elementary teachers have a profound understanding not only of the math they are teaching, but also of algebra and geometry (at a minimum).  I would argue that even better would be having teachers who have been specially trained in math education (including taking real college math courses) do the teaching.  Since I have never taught a group before, I don't know how easy it is to assess kids' understanding individually, but I know that this was a huge key to our success here.  And then schools should adopt materials that work (Singapore was able to do it), provide training in their implementation, and the ensure that teachers are actually using them (rather than pulling random worksheets off the internet).

Again, other countries seem to be able to do it.  I don't buy the excuse that the US is "too diverse" to ensure decent math achievement.  If you start with properly trained teachers, develop a coherent program from the ground up, build in lots of places to check in with students' understanding, and provide room in the curriculum for reteaching, I can't imagine why it wouldn't work.

But maybe I'm wrong.

New Zealand can't do it.  Half out primary teachers seem scared of maths.  The first time they get someone who has actually studied maths at degree level is high school and it is likely to be senior high school before the get a teacher with a maths degree.

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On 9/26/2020 at 1:21 AM, Not_a_number said:

I’m going to ask DD8 to do it and see what happens 🙂 . 

Just asked: 

She says that Paul’s age is x, and Paul’s dad age is 5x, so then 5x+ 3 is 4(x+3). She then set those equal and got 5x+3= 4x+ 12, 5x= 4x+ 9 and then she said x=9.

Except she only wrote down the first equation and did the rest in her head, argh! So it took a while. I guess we better drill these, too. 

If you use algebra like this it is a lot easier.

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8 hours ago, kiwik said:

New Zealand can't do it.  Half out primary teachers seem scared of maths.  The first time they get someone who has actually studied maths at degree level is high school and it is likely to be senior high school before the get a teacher with a maths degree.

I don't think that elementary educators need a math degree.  But I would think that there could be a specialist position in the elementary schools for a math teacher who has the interest and training to undertake such a thing.  The training wouldn't involve a math degree, but it would involve real college math courses that go beyond "how to teach elementary math."

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7 minutes ago, EKS said:

Also I'm surprised it's only half.

Yep, here it is actually a bit startling to find an elementary teacher who is truly comfortable with math.  I would guess that far fewer than 10% would be able to answer the original problem about Paul's age.  I would guess fewer than 25% would be able to convincingly explain why the double digit multiplication algorithm works...not how to do it, but why each step is done as it is.  Actually, I'm not really convinced all the elementary teachers would be able to accurately use the double digit multiplication algorithm!

I am still always astonished how many adults will tell kids that they, the adults, don't understand simple math.  They laugh it off like it like a joke.  "Ha! Ha! I don't know how to add fractions.  Ha! Ha!"  No.  Basic math illiteracy is not funny or cute, and it should not be normalized to children.  It would be like a teacher laughing off not being able to read.  🤨

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On 9/26/2020 at 4:48 PM, EKS said:

Most people's math ability tops out at easy fractions, decimals, and percents--so, like fifth grade level.

I have a theory on this, because it happens in the UK as well, including among adults who took and passed university/college-entrance-level maths that subsequently don't use high-level maths again for a few years.

 

Fifth-grade maths is usually reinforced in schools by a year of doing not a lot except reinforcing that level of maths. There might be time spent working on mental maths, or basic geography geometry/statistics, but it's often the focus of hitting standardised tests at the end of 6th grade using 5th-grade maths (which are a thing in the UK (SATS) and in some US states) and it's also the level of day-to-day maths that is typically encountered by people every single day. Yes, lots of people will encounter more advanced maths through their work, hobbies or on specific occasions that call for it such as buying a home. However, someone who has mastered 5th-grade maths is probably not going to need assistance to live independently (at least for their maths skills directly - students who cannot convince anyone to employ them will indirectly suffer for low maths attainment, but that's because of a lack of paperwork; someone who can cram their way through a maths qualification would be able to get the job without learning a jot of maths in a way that will be remembered afterwards).

 

It's possible to bluff one's way through the semi-regular intermediate-level maths (basic statistics, pre-algebra, early algebra, early geometry) that is encountered every so often. Mistakes get made by those who don't know, sometimes they have consequences, but rarely do those mistakes result in the immediate loss of major opportunities or creation of inconveniences during life (as would happen to people who cannot divide a recipe, use a calendar or read short general-adult-audience non-fiction). Adults who don't need to bluff fare better, but adults who rely on bluffing or avoidance at that level get by anyhow and set examples accordingly. It shouldn't happen in education, but someone not called upon to use, say, adding fractions, outside the classroom is at risk of underappreciating the importance of doing so. Communicating to to one's students that what they are learning is unnecessary goes against everything a teacher should do, of course. However, children are savvy and surprising, often leading teachers to forget themselves, or otherwise reveal their ignorance.

 

To learn maths, it either needs to be mastered and occasionally reinforced, or learned "well-enough" and get a lot of accurate reinforcement (environmental spiral). Despite many attempts, schools have struggled to get mastery-based teaching done. (The UK's most recent attempt at this was to start approving only mastery-based textbooks* for primary school that meet certain other criteria; using an approved textbook isn't compulsory for schools but is recommended. Though as it's only been going for 3 years, it may be a while before we have definitive results). If schools are typically only teaching "well-enough", then a lot of reinforcement afterwards will be needed. Well... ...schools are under pressure to rush through syllabi, and neither complex fractions/decimals/percents nor algebra are reinforced outside school in the same way as those early addition/subtraction/multiplication/division skills.

As far as I know, neither the UK nor the USA has a level in schools after fifth-grade maths that is primarily about reinforcing what is already learned. Homeschoolers and students in schools that manage to learn to mastery can get round this to varying extents by mastery (which then requires less follow-on reinforcement), but if a school cannot teach to mastery for any reason, the constant pushing without consolidation leads to students forgetting the maths soon after it is apparent that classes can be passed through cramming and/or bluffing rather than true learning.


Maths that is not reinforced is usually lost eventually. Also, it's not clear how much time is spent learning maths in college/university for potential teachers, let alone how that is done. (My impression is that "little and rushed" is the norm, in line with how it is expected to be taught in schools).

None of this is intended to explain maths that isn't really learned in the first place (previous posters have done a better job than me in explaining that common issue), rather how it becomes almost ubitiquous among adults who were taught it properly (which happens in some places) but then don't have reason to depend upon non-elementary maths on a regular basis.

 

* - In case you are wondering, the UK version of Singapore Maths (called Maths - No Problem!) and Power Maths (not to be confused with Math Power or Power Basics) are currently approved.

Edited by ieta_cassiopeia
Corrected a blooper - maths involves studying geometry more often than geography
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1 minute ago, ieta_cassiopeia said:

Fifth-grade maths is usually reinforced in schools by a year of doing not a lot except reinforcing that level of maths.

Yes--I've thought this as well, that fifth grade math is the last thing that is used again and again--in school and on into adult life--and that is the reason that this is where adults--even otherwise intelligent and educated adults--top out.   

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1 hour ago, EKS said:

Yes--I've thought this as well, that fifth grade math is the last thing that is used again and again--in school and on into adult life--and that is the reason that this is where adults--even otherwise intelligent and educated adults--top out.   

Except of course most people don't actually understand fractions or division. 

I remember teaching my kids about asymptotes in class and trying to explain WHY the function was going off to infinity. Well, that requires people to do a calculation like 5/0.0001 without thinking about it and to realize that this is a big number. This needs to be fully internalized and require no calculations. 

I'd say about 10% of my class was there. Most of them needed to actually do the calculation. Their mental model wasn't robust. 

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