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Math Education, elementary

#21 User is offline   Elianna 

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Posted 2016-October-26, 21:14

View PostKaitlyn S, on 2016-October-26, 19:19, said:

I would be interested in hearing the different ways students come up to do this.


Some common responses are usually versions of 180 + 18 + 18, or 240 - 12 - 12. A few students still do the algorithm (ignoring instructions). A few students might do 180 + 36 but not realize it was the algorithm. Since I make all students wait until all students indicate they have come up with an answer, this causes my more inventive students to come up with less obvious ways like: 225 - 45 + 36 (15^2 - 3*15 + 3*18).
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#22 User is offline   Elianna 

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Posted 2016-October-26, 21:23

View Postkenberg, on 2016-October-26, 20:49, said:

I was hoping that you would reply..Your estimates surprise me in two ways. I am skeptical that the numbers are that large. But also, problem 1 is first because I thought it would be the easiest. You can visualize a clock and see that in half an hour it will be 10:05 and then add an hour.

For the second one would it matter if I said 100 days instead of 37? I had in mind that they would realize the relevance of 37 being 2 more than a multiple of 7.

For the third, yes, I could tell them how many days. Here I can well imagine using pencil and paper. It could be tough to get it all organized in the head.

I see these questions as being different from finding alternative ways of multiplying 18 X 12. Asking them to do this multiplication is telling them what calculation to perform, even if the detailed method is not stipulated. It has no specific real world connection.

Another story. I have told this before also, but I swear it is true and it will explain why I am skeptical of the 90%


I was in a generic McD type of place. One line over the amount due, let's say $16.27, was announced. The customer gave the cashier some money and received change.

Customer, No, I gave you a $100 bill, not a $20.

Cashier, checking the drawer. Oh! I am sorry. Yes you did.

Total crisis. The cashier had rung it up as 20, the register said the customer was to get $3.73 in change. Nobody knew how to reset the cash register so that 100 could be entered. The cashier talked to an older cashier. No idea. They went over to talk to the guy doing the french fires. Nope.

I intervened. Everyone agreed that 100 was 80 more than 20. I explained that since the customer had given the cashier 80 more than the 20 she rang up, the cashier should give the customer 80 more in change than what the cash register said. Everyone agreed that this sounded fair, it was done, and they could go on to the next customer.

There were either three or four people behind the counter, and one customer, all trying to think of how to figure out how much change was right.

This seems pretty basic. I would love to believe 90%, or even 75%, with my questions but I am not so sure.

I thank you for your estimate, and I would like to hear from other teachers. Or from other parents, except that I have found that everyone's child is at the top of his/her class and very advanced. :)

I may do some experimenting with some random 12 year olds.


Added: I assume that if, with the clock problem, I asked what time it would be after five and a half hours the number of correct answers would go down?


I estimated less on the clock problem because reading a clock is not really covered in common core. They've basically cut out analog clocks so students won't be familiar with it.

However, your other two seem specifically examples of counting up, which is emphasized as a problem solving method in elementary school math, so that's why I estimated those higher.

Your example with change is exactly what common core is attempting to address, and I would be shocked if a student who had common core throughout elementary school could not do it. Right now, my students have at earliest been exposed to common core starting at 9th grade, but to be truthful, my course is the first common core course they've had.

ETA: The other big point in common core is to rely heavily on "real-world problems" and modeling, so that's why I think that students will automatically know that they need to add, because that's how Math is supposed to be presented to them.

I also will add that your supposed 12th graders will not necessarily have had a true common core math class that challenges them in the way they're supposed to be challenged and works on emphasizing the Standard Math Practices. Another big challenge in Elementary Education is that the teachers usually don't understand Math very well, and therefore don't know where they're supposed to push kids to think for themselves, and don't know how to evaluate correct (but different) answers.
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#23 User is offline   Kaitlyn S 

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Posted 2016-October-26, 23:01

Elianna, I presume you are dealing with gifted students. When I was in school with ordinary students, I'd be surprised if many of them even knew what 18 was. It's hard to believe that anybody that isn't gifted would come up with a start of 15 squared. If these are ordinary students, then you and your predecessors are doing a tremendous job.
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#24 User is offline   cherdano 

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Posted 2016-October-27, 00:46

View PostKaitlyn S, on 2016-October-26, 23:01, said:

When I was in school with ordinary students, I'd be surprised if many of them even knew what 18 was.


That's just absurd.
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#25 User is offline   hrothgar 

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Posted 2016-October-27, 00:52

View PostKaitlyn S, on 2016-October-26, 19:08, said:

I agree. That is why many are either home schooling or using non-public education.


That how many people justify it.

Lee Atwater famous said:

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You start out in 1954 by saying, “Nigger, nigger, nigger.” By 1968 you can't say “nigger” — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now [that] you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is [that] blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that. But I'm saying that if it is getting that abstract, and that coded, that we are doing away with the racial problem one way or the other. You follow me — because obviously sitting around saying, “We want to cut this,” is much more abstract than even the busing thing, and a hell of a lot more abstract than “Nigger, nigger.”


In a similar vein, there is a pretty straight line from Segregated Schools --> Segregation Academies --> Charter Schools / Home Schooling
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#26 User is offline   WellSpyder 

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Posted 2016-October-27, 03:39

View PostElianna, on 2016-October-26, 21:14, said:

Some common responses are usually versions of 180 + 18 + 18, or 240 - 12 - 12. A few students still do the algorithm (ignoring instructions). A few students might do 180 + 36 but not realize it was the algorithm. Since I make all students wait until all students indicate they have come up with an answer, this causes my more inventive students to come up with less obvious ways like: 225 - 45 + 36 (15^2 - 3*15 + 3*18).

I went for 12^12 + 6*12 = 144 + 0.5*144. Not trying to come up with something different, just the easiest way that came to mind to tackle the problem. (I'm really not sure why I did 6*12 by division rather than multiplication! Nor why I used something that looks like base 12 arithmetic rather than the base 10 approaches you mention....)
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#27 User is offline   Zelandakh 

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Posted 2016-October-27, 05:51

View Postkenberg, on 2016-October-26, 20:49, said:

I was hoping that you would reply..Your estimates surprise me in two ways. I am skeptical that the numbers are that large. But also, problem 1 is first because I thought it would be the easiest. You can visualize a clock and see that in half an hour it will be 10:05 and then add an hour.

I was hoping Elianna would reply too and was similarly surprised by her answers and in the same way. I also considered #1 to be the easiest and thought the percentages would be significantly lower. If 90% of 12 year olds can answer #2 and #3 correctly then any talk about dropping standards is to be taken with a pinch of salt.

The push to teach maths concepts rather than learning rote methods is one that has gathered pace in recent years internationally. There are also often many more classes on discovering patterns and relationships than people of my (or Ken's) generation would recognise. That said, just recently there has been a renewed interest in a more traditional approach in some places, driven primarily from the excellent performance of China's education system.

Finally, I absolutely do use the 23+24 approach for mental arithmetic and my generation learned both ways, pen and paper subtraction and mental arithmetic short-cuts. The 18x12 I would do the "normal" way though, with either 120+96 or 180+36. The direct alternatives involving (20-2), (10+2) or both are not faster for me, although those that use this method will tell you that it is with a little practice and, for example, I do like the "difference of 2 squares" method where one calculates 99*101 as 100^2-1 and 98*102 as 100^2-4.

As for calculators, the one piece of advice I give everyone is always to do a rough calculation in your head at the same time. If the result you get is a factor of 10 or more out from that then re-enter the numbers and check your working. Aside from that, they help to save lots of the drudgery. Understanding is certainly much more important!
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#28 User is offline   billw55 

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Posted 2016-October-27, 06:22

View PostElianna, on 2016-October-26, 18:56, said:

Students complain about the amount of explaining I require them to do. In other words, it's not enough for them to be able to solve for x in 3x + 15 =75; They must be able to explain why each step is mathematically valid, and be able to critique other methods of solving (as in recognizing correctness/noncorrectness) and be able to explain to others faults in their reasoning. Further, they must demonstrate ability to use this in modeling. So for example, I might require them to create a word problem that this equation would solve (and of course the classical given situations, come up with equations to help solve them). (I actually don't teach this math, I just wanted to use an easily understood example.)

But now you have spent perhaps half an hour on 3x+15=75. In my experience one of teachers' top complaints is not enough instruction time. Certainly they cite this when opposing standardized testing. So is this a wise use of time? As an alternative, how about using that half hour to drill them on perhaps two dozen such equations? Which approach would actually get them proficient at basic algebra faster? I'm not arguing one way or the other. I'm open to discussion. But I know how I learned, and it was good enough to get me through 30 credits of math in college.
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#29 User is offline   kenberg 

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Posted 2016-October-27, 06:51

More thoughts about Elianna's thoughts.

With the clock problem it had slipped my mind that clocks with faces have become passe. Becky and I sometimes joke about it, wondering (and I really am not sure of the answer here either) if kids know what it means to traverse a circular track counter-clockwise. I can see why they might not. So, at the very least, my idea that you "just visualize the clock to see that if it is 9:35 now then it will be 10:05 in a half hour" might not be as obvious as I thought.
But it still seems natural to do this by first seeing what time it will be half an hour from now and then adding on the number of hours. Half an hour after 9:35 it will be 10:05, and an hour later it will be 11:05. If, instead I asked for the time five and a half hours later then they would have to deal with "what time is it five hours after 10 o'clock?" A little trickier.

The possibility of kids growing into adulthood without being able to do such things is a little unsettling. And my guess is that if they can't do this when they are 12, they won't be able to do it when they are 20 either.

My guess, just a guess, about Common Core is that like practically everything else it depends on how it is implemented. Most things can be done well, most things can be done poorly.


A few words about home schooling.
First, just emotionally, I come from the totally opposite extreme. I regard my childhood as being happy overall. But my mother tried to teach me to play the piano. That was not one of the happy parts. Parental teaching does not always go well. Not at all.
Second, it is often not practical. Take a divorced mother with four kids and a full time job. I know one. She is going to home school?
Third, I have seen it close up. Sometimes it works, sometimes it is a disaster.
Conclusion: If someone wants to home school their kids, let them. Well, even parental rights can be overridden in extreme cases, but mostly we let them. But we need to go forward creating a good system for those who do not home school.


I am thinking to put some effort into my vote for the school board (retirement is great!). I have strong beliefs ni the absolutely crucial nature of a good school system I am much less clear as to how to bring this about.
Ken
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#30 User is offline   PassedOut 

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Posted 2016-October-27, 08:17

View Postkenberg, on 2016-October-27, 06:51, said:

A few words about home schooling.

First, just emotionally, I come from the totally opposite extreme. I regard my childhood as being happy overall. But my mother tried to teach me to play the piano. That was not one of the happy parts. Parental teaching does not always go well. Not at all.

Second, it is often not practical. Take a divorced mother with four kids and a full time job. I know one. She is going to home school?

Third, I have seen it close up. Sometimes it works, sometimes it is a disaster.
Conclusion: If someone wants to home school their kids, let them. Well, even parental rights can be overridden in extreme cases, but mostly we let them. But we need to go forward creating a good system for those who do not home school.

I agree completely with the need for excellent public schools. Growing up, I went to three different elementary schools and three different (9-12) high schools. The differences in quality among those schools was pretty surprising to me (and to my parents). The kids were great, but I have to say that my last two years in high school were much too easy, although the socializing was a joy. When I entered college, I had to buckle down right away to overtake the students who had had a stronger education all along.

That said, I do have personal experience with home schooling, and it was positive.

Our sons were all born during the 1980s and we sent them to Montessori school from age three through twelve (Atlanta has a good Montessori elementary school, with a lot of parental support). By the time they finished there, they were all voracious readers and were experienced in using their own computers. It turned out that the public schools were slowing them down. After a year, we moved back north to be nearer our aging parents and the rest of our families, and -- after a couple of long discussions, including with our sons -- decided to move forward with home schooling. We were fortunate to have a local college available willing to enroll our sons in courses that we weren't prepared to offer, such as laboratory sciences and life drawing, but our sons didn't need any pushing from us to read up on and master the other academic subjects. And all three had an easy transition to college.

For the good of the country though, this can't be the direction to go. Every kid needs a shot at reaching his or her full potential regardless of the resources of his or her parents, and the US needs that also to meet the challenges ahead. It seems to me that the Common Core is a very reasonable way to address this: What approach could work better?

And you can count me with those who feel that teaching kids creationism instead of science is a form of child abuse.
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#31 User is offline   Winstonm 

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Posted 2016-October-27, 08:40

Perhaps the Finnish education model should be our aim?


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There are no mandated standardized tests in Finland, apart from one exam at the end of students’ senior year in high school. There are no rankings, no comparisons or competition between students, schools or regions. Finland’s schools are publicly funded. The people in the government agencies running them, from national officials to local authorities, are educators, not business people, military leaders or career politicians. Every school has the same national goals and draws from the same pool of university-trained educators. The result is that a Finnish child has a good shot at getting the same quality education no matter whether he or she lives in a rural village or a university town. The differences between weakest and strongest students are the smallest in the world, according to the most recent survey by the Organization for Economic Co-operation and Development (OECD). “Equality is the most important word in Finnish education. All political parties on the right and left agree on this,” said Olli Luukkainen, president of Finland’s powerful teachers union.

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#32 User is offline   Winstonm 

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Posted 2016-October-27, 08:48

View PostPassedOut, on 2016-October-27, 08:17, said:


And you can count me with those who feel that teaching kids creationism instead of science is a form of child abuse.


I have to go one step further here, and it comes from my personal experience: teaching children that religious beliefs are facts rather than faith is child abuse.
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#33 User is offline   barmar 

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Posted 2016-October-27, 08:52

View PostKaitlyn S, on 2016-October-26, 19:58, said:

Totally agree. Also, if they don't understand why they are doing what they are doing, it will hurt them later in life when they need to understand the whys of more complicated things. Essentially it sounds like we want to teach children to think. Thinking is work and is hard, so the lazy child is going to try to avoid it. I would be in favor of a program that can get children to think to the best of their ability.

One of the techniques that is becoming popular is "active learning", which replaces much of the lecturing and textbook reading with group activities that require the students to solve problems. Children are naturally good at this, it's basically how they learn most of the games they play (no kid ever sat in a lecture learning how to play sports, and they pick up card and board games with only a minimal amount of explanation of the basic rules). And when they learn through an active process, they understand and retain it better.

#34 User is offline   billw55 

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Posted 2016-October-27, 09:24

While I am certainly opposed to teaching creationism in public schools, I think calling it child abuse is a large exaggeration.
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#35 User is offline   PassedOut 

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Posted 2016-October-27, 09:26

View PostWinstonm, on 2016-October-27, 08:40, said:

Perhaps the Finnish education model should be our aim?

Actually, that would be great, but it seems to me that a lot of folks in the US would freak out at the very idea. Perhaps the politics here influence me to set my sights too low.
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#36 User is offline   mycroft 

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Posted 2016-October-27, 09:55

  • Lunchtime. Frankly, my biggest problem with "the younguns" is that they don't realize that the difference between 1205 and 1207 is frequently irrelevant. One thing we lost with digital watches (which are still a neat idea).
  • The Day of Struggle against the Great Clinton Conspiracy. (sorry, cheating - it's after November 9. Every day is the Day of Struggle against the GCC. (Sorry, cheating. Every day since at least 2002 has been the Day of Struggle against the GCC, at least on BBF.).).
  • Daddy, I want to go see Santa! (and the corollary, "10 days fewer than that").

In re: Elianna, I was going to start by being a smartarse and saying "12x18". Or less so, "9x24". But seriously:
18  12
 9  24
 3  72
 1 216
(with a question as to why I did it that way instead of going the other way and only having to do the x3 once)
And quite seriously, I have done multiplication like this by:
2.2.3.2.3.3 = 3.3.3.2.2.2 = 27, 54, 108, 216

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#37 User is offline   Winstonm 

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Posted 2016-October-27, 10:19

View Postbillw55, on 2016-October-27, 09:24, said:

While I am certainly opposed to teaching creationism in public schools, I think calling it child abuse is a large exaggeration.


What if a Taliban fighter was teaching his beliefs to your son, then would it be child abuse?
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#38 User is offline   Winstonm 

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Posted 2016-October-27, 10:21

View PostPassedOut, on 2016-October-27, 09:26, said:

Actually, that would be great, but it seems to me that a lot of folks in the US would freak out at the very idea. Perhaps the politics here influence me to set my sights too low.


From what I have learned, the Finnish have the best public schools in the world. The do not test. They do not have homework. And, if memory serves, it is illegal to have more than 20 hours a week of class time.

Obviously, they seem to know something we don't.

Edit: It is not illegal but teachers only spend 4 hours per day in the classroom, with 2 hours a day in preparation.
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#39 User is offline   akwoo 

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Posted 2016-October-27, 10:22

Re: Finland.

I am sure that the reason that the Finnish education works so well is that teaching is a highly regarded, well paid, and selective profession in Finland. In fact, as I understand it, the Education majors are the hardest to get admitted to, and, as a minimum requirement, you must score in the top third of national exams to be allowed to study Education (and then later become a teacher).
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#40 User is offline   Kaitlyn S 

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Posted 2016-October-27, 10:27

What seems interesting to me is that we are discussing teaching children as if it is a given that you will be able to spend your time teaching.

I have several friends and acquaintances that are teachers and every single one of them tells me that the teacher has to spend more time on discipline problems than actually teaching; they are not allowed to do anything about a disruptive child that will cause him not to be disruptive (they are allowed to yell at him but anything else would be grounds for disciplinary action against the teacher and/or a lawsuit from the parents), and yelling rarely does any good, and the disruptive children simply ignore requests for time-out or to go to the principal's office; and there are usually multiple disruptive children per class not just one.

Teachers who have taught college level courses where the child or his parents are paying for it actually get to teach. Their experience is that they must teach the material that should have been taught in high school but nobody got to learn anything in high school because of all the disruption from misbehaving students.

In another discussion board, I suggested that a solution was to segregate based on behavior so that the 80% or so of children that want to behave and learn will all be in the same schools and the students will learn. Of course, since most internet discussion boards are mostly populated with liberals, I was chastised for this suggestion, but not one single poster disputed my depiction of what goes on in a high school classroom.
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