Nation's Reportcard

[ksinger]

One of the problems as I said, is also deciding what a person ought to know for college.

Most people in a college degree plans have a mathematic requirement. Every semester, my hubby Mr Bcat, who as Asst Dpt Chair also advises student on courses. Many students do not come out of high school with even the basic understanding to pass the lowest college level math course a person can take to satisfy their degree plan- college algebra. There is a math for non science majors class that he always encourages students not going on in science to take, which covers more "life applications" like understanding loans and mortgages and fulfills their requirements. Still, they sign up for CA, and fail. And fail. And fail....over and over til they are forced to go back to remedial math (that you don't get any college credit for) and they fail, and they fail. And this is a whole topic unto itself - why are we as a society, admitting so many kids to college who are either not prepared, or never will be. Lots of ways to look at it. It certainly is discouraging and costly to those kids who are admitted and can't cut the muster. Cynical me would suggest there is an element of cash cow for the colleges to have a revolving door of kids having to take remedial classes. Big picture me says because we as a society are in a big bucket of denial about the economic and technological changes that mean many of our kids who can't/shouldn't do college (like the 70+% who have never in our entire history, been college educated), will never find good, livable jobs like those available in the 60s and 70s. Desperate parents and governments just won't stop trying though, and I get it, but it still doesn't address the real issues.

And the state (and I live in a SUPER conservative state, so I hope no one thinks to say to me that the "progressive" state and their mandates to the "progressive" universities is the problem as the article assumes- I assure you, the last time this state was blue was in the 70's).

So the state mandates to the colleges (and the college the hubbles teaches at is fairly progressive) what scores a student must have to take Coll Alg, and the hubbles has argued repeatedly that they have set the score too low, and people just are not prepared, and aren't going to pass. You'd be surprised how many college profs (at least that my husband has had dealings with) apparently have zero idea that the state mandates what is taught. My hubs was at an in-service where a college prof (women's studies) was ragging on the high school teachers the kids were so ignorant on any kinds of women's history. My husband pointed out that these topics were not part of our state's pass standards, and the guy looked blank and then asked what those were. He had zero clue.

And after 25 years teaching at college level...I know who I believe when it comes to him or them! The state controls funding. The state and state school board (let me remind you this is a very very very red state) also regulates the curriculum that must be covered. I can assure you it is not a "liberal" issue, but as I said, reactions to the way our society has developed. The solutions aren't WORKING, but that is beside the point here. My husband's father taught for 34 years, my husband for 20, starting about the time his father retired - he did his student teaching while his dad was still in the building where both of their careers have played out - teaching the same topic, history. So over 50 unbroken years of insider knowledge of how education works in my state. And my hubs would agree with yours on the above.


With further research (as opposed to researching just enough to be able to twist ideas to ride their obvious political hobby horse) they might have found that the issue is not the testing itself, but the testing as a be all and end all, to the point that it has become the basis for funding. Objective tests are fine, the show certain things that used correctly can be helpful. However, basing an education system around only those is how we got to where we are. Word.

 

[bunnycat]

One of the problems as I said, is also deciding what a person ought to know for college.

Most people in a college degree plans have a mathematic requirement. Every semester, my hubby Mr Bcat, who as Asst Dpt Chair also advises student on courses. Many students do not come out of high school with even the basic understanding to pass the lowest college level math course a person can take to satisfy their degree plan- college algebra. There is a math for non science majors class that he always encourages students not going on in science to take, which covers more "life applications" like understanding loans and mortgages and fulfills their requirements. Still, they sign up for CA, and fail. And fail. And fail....over and over til they are forced to go back to remedial math (that you don't get any college credit for) and they fail, and they fail. And this is a whole topic unto itself - why are we as a society, admitting so many kids to college who are either not prepared, or never will be. Lots of ways to look at it. It certainly is discouraging and costly to those kids who are admitted and can't cut the muster. Cynical me would suggest there is an element of cash cow for the colleges to have a revolving door of kids having to take remedial classes. Big picture me says because we as a society are in a big bucket of denial about the economic and technological changes that mean many of our kids who can't/shouldn't do college (like the 70+% who have never in our entire history, been college educated), will never find good, livable jobs like those available in the 60s and 70s. Desperate parents and governments just won't stop trying though, and I get it, but it still doesn't address the real issues.

And the state (and I live in a SUPER conservative state, so I hope no one thinks to say to me that the "progressive" state and their mandates to the "progressive" universities is the problem as the article assumes- I assure you, the last time this state was blue was in the 70's).

So the state mandates to the colleges (and the college the hubbles teaches at is fairly progressive) what scores a student must have to take Coll Alg, and the hubbles has argued repeatedly that they have set the score too low, and people just are not prepared, and aren't going to pass. You'd be surprised how many college profs (at least that my husband has had dealings with) apparently have zero idea that the state mandates what is taught. My hubs was at an in-service where a college prof (women's studies) was ragging on the high school teachers the kids were so ignorant on any kinds of women's history. My husband pointed out that these topics were not part of our state's pass standards, and the guy looked blank and then asked what those were. He had zero clue.

And after 25 years teaching at college level...I know who I believe when it comes to him or them! The state controls funding. The state and state school board (let me remind you this is a very very very red state) also regulates the curriculum that must be covered. I can assure you it is not a "liberal" issue, but as I said, reactions to the way our society has developed. The solutions aren't WORKING, but that is beside the point here. My husband's father taught for 34 years, my husband for 20, starting about the time his father retired - he did his student teaching while his dad was still in the building where both of their careers have played out - teaching the same topic, history. So over 50 unbroken years of insider knowledge of how education works in my state. And my hubs would agree with yours on the above.


With further research (as opposed to researching just enough to be able to twist ideas to ride their obvious political hobby horse) they might have found that the issue is not the testing itself, but the testing as a be all and end all, to the point that it has become the basis for funding. Objective tests are fine, the show certain things that used correctly can be helpful. However, basing an education system around only those is how we got to where we are. Word.

Yes- my hubs is in the very same boat. Father was a math professor. Son grew up to be math professor at the same college. Decades of insider information. Hubs dropped another one on me today. Apparently, the state legislature recently passed a law that professors have to make their curriculum accessible for online viewing. So that the state can monitor what is being taught in case it is too liberal. Not kidding. How 'bout that for invasion of privacy and eavesdropping. Pretty much like professors having to agree to be wire tapped to teach....

I'd have to check, but I don't think having to take these college courses over and over is a cash cow for the state. In fact, the state funds some of the remedial work and hubs feels like sometimes the students use that fact to not care if they fail because they can take it over and over to a certain point with a state voucher....Currently the state is cutting funding and with a new mandate leaving the colleges to scramble to combine coursework so that 2 of the remedial classes worth of work are being covered in one semester. I don't have to tell you how that's going to go, when they can't even pass ONE courses worth of work in a semester....

 

[Maria D]

As an aside, I happen to think these issues about the disconnect and decline of contemporary education have been going on a very long time, since well before they became a partisan pet topic.

For example, here is a high school math book from 1968 I have. It covers abstract ideas like group theory and set theory which are math structure concepts that underly our system of mathematics. (Or, if you like, a concrete look at how the human brain organizes numbers and how we compute and why it follows the logic it does.) Even back in the day, there was "concern" people weren't learning properly, or thoroughly, and this would have been considered an attempt at "common core" back then. It's abstract conceptual (intuitive) learning, but this book is based in applying logic definitions as opposed to "how do you feel about x or y?". FWIW- this clearly didn't work either....

bunnycat, do you teach high school math?

I read as far as this post in the thread, and haven't read any of the articles. Just want to pop in and say, I feel ya bunnycat!

I teach high school math; my educational background was in engineering. So I didn't get the theory training through my undergrad courses. I started teaching in my forties and have taken continuing ed classes and learned even more on my own. But all I had to do (beyond undergrad degree and certain coursework) to get a teaching credential for high school math in my state was get a score of 126 on an exam where the range/scale was, I kid you not 100-200. Seriously!! And even with that low bar we have a hard time filling math teacher positions in Maine.

Back in 1968, I'll bet a small percentage of high school students were learning algebra from that textbook. When I took calculus in high school ('70s), there was one section of about 20 students - about 4% of the graduating class. In the high school I teach at today, fully 25% of the graduating class will have *at least* the equivalent of one semester of college level calculus on their transcript with 2/3 of those having two semesters.

But here's the rub - there's still only about 4% of the class that actually understands calculus. It is so difficult for me to get through the curriculum because I'm constantly remediating on the algebra and trig that they never really "got." And fractions? We have students taking calculus that can't perform arithmetic at the 5th grade common core standard level. I tell my students that if they learn nothing else in my class, they will be high school graduates that can add -5 1/3 to 4 2/7 without a calculator! We (my colleagues and I) are asking basically the same question about our students that redwood is asking - how did they get to high school without being able to do elementary school math?

I've got a few theories. One of them is that the women's movement screwed everything up by giving smart STEM inclined women the opportunity to go out and make real money. There went that pool of potential elementary school teachers that not only weren't afraid of math but actually enjoyed it.

 

[bunnycat]

As an aside, I happen to think these issues about the disconnect and decline of contemporary education have been going on a very long time, since well before they became a partisan pet topic.

For example, here is a high school math book from 1968 I have. It covers abstract ideas like group theory and set theory which are math structure concepts that underly our system of mathematics. (Or, if you like, a concrete look at how the human brain organizes numbers and how we compute and why it follows the logic it does.) Even back in the day, there was "concern" people weren't learning properly, or thoroughly, and this would have been considered an attempt at "common core" back then. It's abstract conceptual (intuitive) learning, but this book is based in applying logic definitions as opposed to "how do you feel about x or y?". FWIW- this clearly didn't work either....

bunnycat, do you teach high school math?

I read as far as this post in the thread, and haven't read any of the articles. Just want to pop in and say, I feel ya bunnycat!

I teach high school math; my educational background was in engineering. So I didn't get the theory training through my undergrad courses. I started teaching in my forties and have taken continuing ed classes and learned even more on my own. But all I had to do (beyond undergrad degree and certain coursework) to get a teaching credential for high school math in my state was get a score of 126 on an exam where the range/scale was, I kid you not 100-200. Seriously!! And even with that low bar we have a hard time filling math teacher positions in Maine.

Back in 1968, I'll bet a small percentage of high school students were learning algebra from that textbook. When I took calculus in high school ('70s), there was one section of about 20 students - about 4% of the graduating class. In the high school I teach at today, fully 25% of the graduating class will have *at least* the equivalent of one semester of college level calculus on their transcript with 2/3 of those having two semesters.

But here's the rub - there's still only about 4% of the class that actually understands calculus. It is so difficult for me to get through the curriculum because I'm constantly remediating on the algebra and trig that they never really "got." And fractions? We have students taking calculus that can't perform arithmetic at the 5th grade common core standard level. I tell my students that if they learn nothing else in my class, they will be high school graduates that can add -5 1/3 to 4 2/7 without a calculator! We (my colleagues and I) are asking basically the same question about our students that redwood is asking - how did they get to high school without being able to do elementary school math?

I've got a few theories. One of them is that the women's movement screwed everything up by giving smart STEM inclined women the opportunity to go out and make real money. There went that pool of potential elementary school teachers that not only weren't afraid of math but actually enjoyed it.

Hey! I'm not a teacher myself, though I considered the field (did student teaching and mentoring for awhile with inner city kids) though I do have a math degree. (However, I do not have the 5 degrees that hubster does, with 3 of them post graduate....You can bet it keeps me on my tippy toes dealing with that kind of intellect ( ) on a daily basis.... )

That is indeed something I had not considered. Well, in a way I considered it when I said that the awful corner many teachers find themselves painted in to often drives the talented away. Poor pay. I mean, when I considered it 20 years ago at the end of my undergrad, starting salary here was under $30K, and I had $25K in loans....hmmmmm....add the incessant "teach the TEKS" or whatever the equivalent is now and the horrid ISD administration made me run as fast as I could away. I could see the writing on the wall for how I'd do. And you are right, other jobs pay so much better, well....a person has to live and pay bills and afford a house or at least rent.

Everyone has valid points. How did we come to this pass? I just don't know. I don't think there's any one real reason. And if I knew, and could prove it, I'd have a PhD, and I don't. Probably many things. People wanted kids to learn more complicated math. Some peope came up with ideas how to do that. The ideas didn't work. They try something else. And the cycle continues. I firmly disagree with the second author that there was some "liberal" agenda as a reason. While it is true that people who acquire more education *tend* to be more liberally minded, that's just because they have experiences that would make them consider lots of options. I doubt it was them saying "hee hee let's do this and piss off conservatives". Policies usually get implemented on both sides of the isle to try to help a problem. Later, you see what works and what doesn't. A lot of this began in the decades before extreme partisanship took over as it has now, as you know. The textbook I showed would have been an example of some of those attempts at including higher math in a lower school level.

One thing is sure though. You can't have it both ways. You can't simultaneously be mad that kids aren't learning enough to be competent in today's world and want them to learn what they need to know and also simultaneously have the attitude of "well, bank teller was good enough for your grandma, it should be good enough for you" (Sorry, family reference again. This is what my family told me to my face when I went to college.) So you choice is to learn and to function competently in today's world you need to learn a lot because it's a complicated world, much more so than 50 years ago, though some would prefer to return to it (heads up-we can't) and with that comes the risk of a person learning enough to develop ideas different from their family and risk estrangement. The other choice would seem to be learn just enough not to "leave the fold" and by implication remain under others control. A very few people I know do both. Their parents encourage learning, and yet the kids also remain "in the fold". Their parents also don't try to control the ones who don't think as they do (mostly on the religious aspect) and they accept them. I am not religious, but I respect them a lot for their positive parenting.

I'm still mulling over the second article (because it was irritating and it's also 20 years out of date) and trying to get to the meat of it and past the author's every other sentence disparagement of "progressives" and spending most of the article furthering that agenda rather than actually discussing objectively the pros and cons of "child-centered" learning (which I have mixed feeling on myself). Again, the article is 20 years old. And, as the librarians around the country are finding out and having to reassess the CRAAP test, currency is as important as authority and in today's internet world, anyone can write anything and call themselves an expert and kids cite it as a source of truth on a report.

(I am officially calling myself NOT an expert....)

 

[bunnycat]

I'll give you a quick analogy of learning.

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

 

[redwood66]

Thank you all. I appreciate the comments made here, especially that it did not devolve into something unpleasant. While we may not agree politically with everyone I think we all have the best for our children in mind.

 

[AGBF]

I was scrolling through this thread, in which I somehow managed not to become involved, and I saw this this book. I have a sick sense that it may have been my Algebra II book.

AGBF

 

[bunnycat]

I was scrolling through this thread, in which I somehow managed not to become involved, and I saw this this book. I have a sick sense that it may have been my Algebra II book.

AGBF

Lol...I loved this book. I found it in a bookstore when I was in college and began working my way through it. Had a lot of fun....

 

[bunnycat]

Thank you all. I appreciate the comments made here, especially that it did not devolve into something unpleasant. While we may not agree politically with everyone I think we all have the best for our children in mind.

Thank you for taking time out to read and discuss. I should have been less knee jerky in regards to the second article. I did take time to look through it again. Although I disagree, as you can tell, with the tone I don't disagree that there are issues with challenge based or child-centered learning but can't make any further comment as the article was from back the time period when I was in college and some of the teaching models were challenge based, though not quite in the way they put forth in the article and I don't really know what the most current teaching models are. Maybe Maria might know, since she currently teaches and so would have a lot more info on how this format may have changed or what parts might be working. It sounds though like they are frustrated as well. I think we all are.

 

[t-c]

But here's the rub - there's still only about 4% of the class that actually understands calculus. It is so difficult for me to get through the curriculum because I'm constantly remediating on the algebra and trig that they never really "got." And fractions? We have students taking calculus that can't perform arithmetic at the 5th grade common core standard level. I tell my students that if they learn nothing else in my class, they will be high school graduates that can add -5 1/3 to 4 2/7 without a calculator! We (my colleagues and I) are asking basically the same question about our students that redwood is asking - how did they get to high school without being able to do elementary school math?

I've got a few theories. One of them is that the women's movement screwed everything up by giving smart STEM inclined women the opportunity to go out and make real money. There went that pool of potential elementary school teachers that not only weren't afraid of math but actually enjoyed it.

I studied pure math and I don't know that I completely understood calculus And I've become horrible at basic arithmetic calculations (I like to think it's because my brain calculates faster than I write and it all goes to hell on paper - hah!), thankfully more advanced math used fewer numbers.

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

I think we require people to learn on a schedule and some (most?) just don't. I once had such difficulty with the concept of Sn1 and Sn2 reactions in Orgo but when I encountered it again later, it was so easy! Age of Innocence just annoyed me in 6th grade, but I appreciated it when I reread it in 11th grade. And I remember confessing to my Abstract Algebra professor that although I did well in the exam, I didn't think I was learning the material because it was all memorization and luck -- I packed all those definitions and theorems in my head and prayed that I used them correctly during the exam. He laughed and said it usually takes the 2 or 3 abstract algebra classes for the concepts to fully click.

So do (can) we give students the environment where things can click? Perhaps their brains aren't ready the first time they encounter new concepts; maybe it requires a second or third exposure. If teachers aren't well versed on the subject they are teaching, maybe they don't (know to) reach back to connect concepts learned earlier in the year or even the previous year. These connections, I feel, are necessary to see how ideas are built on other ideas. Having these connections make learning easier and more intuitive.

 

[bunnycat]

But here's the rub - there's still only about 4% of the class that actually understands calculus. It is so difficult for me to get through the curriculum because I'm constantly remediating on the algebra and trig that they never really "got." And fractions? We have students taking calculus that can't perform arithmetic at the 5th grade common core standard level. I tell my students that if they learn nothing else in my class, they will be high school graduates that can add -5 1/3 to 4 2/7 without a calculator! We (my colleagues and I) are asking basically the same question about our students that redwood is asking - how did they get to high school without being able to do elementary school math?

I've got a few theories. One of them is that the women's movement screwed everything up by giving smart STEM inclined women the opportunity to go out and make real money. There went that pool of potential elementary school teachers that not only weren't afraid of math but actually enjoyed it.

I studied pure math and I don't know that I completely understood calculus And I've become horrible at basic arithmetic calculations (I like to think it's because my brain calculates faster than I write and it all goes to hell on paper - hah!), thankfully more advanced math used fewer numbers.

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

I think we require people to learn on a schedule and some (most?) just don't. I once had such difficulty with the concept of Sn1 and Sn2 reactions in Orgo but when I encountered it again later, it was so easy! Age of Innocence just annoyed me in 6th grade, but I appreciated it when I reread it in 11th grade. And I remember confessing to my Abstract Algebra professor that although I did well in the exam, I didn't think I was learning the material because it was all memorization and luck -- I packed all those definitions and theorems in my head and prayed that I used them correctly during the exam. He laughed and said it usually takes the 2 or 3 abstract algebra classes for the concepts to fully click.

So do (can) we give students the environment where things can click? Perhaps their brains aren't ready the first time they encounter new concepts; maybe it requires a second or third exposure. If teachers aren't well versed on the subject they are teaching, maybe they don't (know to) reach back to connect concepts learned earlier in the year or even the previous year. These connections, I feel, are necessary to see how ideas are built on other ideas. Having these connections make learning easier and more intuitive.

I've certainly seen (and experienced!) in it adult education situations where a concept doesn't make it in to the grey matter on the first (or second or maybe even third or 4th) try. The trouble is, at some basic level, there has to be a consensus about what basics people should know and you need talented and competent people doing it. I've seen many many examples of teaching adults from a strictly concept base learning environment where it just becomes confusing, and they get bogged down with lack of direction, make no progress at all and give up. To me, moderation is the key, as in all things.

To argue the case of the articles red pointed out, I think it would have been better and more plausible if the author's cases had come from a standpoint that some rote learning, especially at an early age, may not be a bad thing and adding concept based learning in at appropriate times or as a portion of the more traditional approach might have been/would be a better option. Talking about a simple version of a concept without getting horridly in depth can still provide an opening to the "why-ers" like myself and spur them on while not getting so bogged down the people that do better with a more standard approach aren't left out and left feeling directionless. Still, you need someone who understands both of those modes to make it work.

I adored calculus....such a nerd....but then I adored Real Analysis and I had a fantastic professor for Differential Equations (who was a big researcher in Chaos theory). It's all in your professors, honestly, and their talent. I was very careful who I took classes from. I wasn't even planning on majoring in math. I just happened to take my "non science math" course and somehow got ahold of a great teacher who taught us how to do proofs and Complex numbers, and we read lots of philosophy of math, then I was hooked.

 

[Maria D]

I studied pure math and I don't know that I completely understood calculus And I've become horrible at basic arithmetic calculations (I like to think it's because my brain calculates faster than I write and it all goes to hell on paper - hah!), thankfully more advanced math used fewer numbers.

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

I think we require people to learn on a schedule and some (most?) just don't. I once had such difficulty with the concept of Sn1 and Sn2 reactions in Orgo but when I encountered it again later, it was so easy! Age of Innocence just annoyed me in 6th grade, but I appreciated it when I reread it in 11th grade. And I remember confessing to my Abstract Algebra professor that although I did well in the exam, I didn't think I was learning the material because it was all memorization and luck -- I packed all those definitions and theorems in my head and prayed that I used them correctly during the exam. He laughed and said it usually takes the 2 or 3 abstract algebra classes for the concepts to fully click.

So do (can) we give students the environment where things can click? Perhaps their brains aren't ready the first time they encounter new concepts; maybe it requires a second or third exposure. If teachers aren't well versed on the subject they are teaching, maybe they don't (know to) reach back to connect concepts learned earlier in the year or even the previous year. These connections, I feel, are necessary to see how ideas are built on other ideas. Having these connections make learning easier and more intuitive.

t-c, while you may now feel that since you were able to study pure (advanced) math with fewer numbers that versatility with basic arithmetic isn't essential, please know that in my state high school graduates that wish to take college level courses at a community college must take the Accu-placer test. In order to test out of a basic, cost-incurring but non-credit earning math class, they will need to answer questions such as these without a calculator:

* 3 1/3 – 2 2/5 =
* 2 1/2 + 4 2/3 =
* There are 3 people who work full-time and are to work on a project, but their total time on the project is to be equivalent to that of only one person working full-time. If one of the people is budged for 1/2 of his time to the project and a second person for 1/3 of her time, what part of the third worker's time should be budgeted for his project?

(These and more can be found at https://accuplacer.collegeboard.org/sites/default/files/accuplacer-sample-questions-for-students.pdf)

That last one is a fairly simple problem. 1/2 + 1/3 + ? = 1. But I know MANY high school juniors and seniors that won't even attempt it because it has the two things they fear most in math: problems with words and fractions. I contend that the reason they are so freaked out by it is because they can't do the first two problems either. In fact, because they don't have basic numeracy skills, many get the first one wrong even WITH a calculator. They will do 3 + 1/3 – 2 + 2/5.

My issue with so many students (now) taking Calculus isn't that I feel it's such an important subject that all must fully understand. (I actually do not.) I only came to understand it well through teaching it. My issue is that we are pushing kids along on this Calculus before graduation track to the detriment of their basic numeracy skills. So while I am teaching what's supposed to be high school math, the teaching/learning experience gets bogged down with an incredible amount of remediation of the basics. I agree with you that second and third exposures are needed for many concepts. If only we teachers were allowed that luxury!

Bunnycat, interesting analogy with Argentine tango dancing. I feel the basic problem is the refusal to admit that some people have an affinity for certain things and some don't. Sure, everyone can learn and improve. But some people take to things easier and more naturally than others. Some are going to learn to dance the tango well if you teach them rote patterns because they will discern the form through practice. Some will learn well the other way around. And some will be flat-footed clumsikins (me) who might be able to do a half-way respectable tango only after committing routines to memory and practicing really hard. If you don't believe that, then it's easy to think that all that needs to be done is find the perfect teaching style - then everyone will achieve mastery at the same time.

 

[bunnycat]

I studied pure math and I don't know that I completely understood calculus And I've become horrible at basic arithmetic calculations (I like to think it's because my brain calculates faster than I write and it all goes to hell on paper - hah!), thankfully more advanced math used fewer numbers.

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

I think we require people to learn on a schedule and some (most?) just don't. I once had such difficulty with the concept of Sn1 and Sn2 reactions in Orgo but when I encountered it again later, it was so easy! Age of Innocence just annoyed me in 6th grade, but I appreciated it when I reread it in 11th grade. And I remember confessing to my Abstract Algebra professor that although I did well in the exam, I didn't think I was learning the material because it was all memorization and luck -- I packed all those definitions and theorems in my head and prayed that I used them correctly during the exam. He laughed and said it usually takes the 2 or 3 abstract algebra classes for the concepts to fully click.

So do (can) we give students the environment where things can click? Perhaps their brains aren't ready the first time they encounter new concepts; maybe it requires a second or third exposure. If teachers aren't well versed on the subject they are teaching, maybe they don't (know to) reach back to connect concepts learned earlier in the year or even the previous year. These connections, I feel, are necessary to see how ideas are built on other ideas. Having these connections make learning easier and more intuitive.

t-c, while you may now feel that since you were able to study pure (advanced) math with fewer numbers that versatility with basic arithmetic isn't essential, please know that in my state high school graduates that wish to take college level courses at a community college must take the Accu-placer test. In order to test out of a basic, cost-incurring but non-credit earning math class, they will need to answer questions such as these without a calculator:

* 3 1/3 – 2 2/5 =
* 2 1/2 + 4 2/3 =
* There are 3 people who work full-time and are to work on a project, but their total time on the project is to be equivalent to that of only one person working full-time. If one of the people is budged for 1/2 of his time to the project and a second person for 1/3 of her time, what part of the third worker's time should be budgeted for his project?

(These and more can be found at https://accuplacer.collegeboard.org/sites/default/files/accuplacer-sample-questions-for-students.pdf)

That last one is a fairly simple problem. 1/2 + 1/3 + ? = 1. But I know MANY high school juniors and seniors that won't even attempt it because it has the two things they fear most in math: problems with words and fractions. I contend that the reason they are so freaked out by it is because they can't do the first two problems either. In fact, because they don't have basic numeracy skills, many get the first one wrong even WITH a calculator. They will do 3 + 1/3 – 2 + 2/5.

My issue with so many students (now) taking Calculus isn't that I feel it's such an important subject that all must fully understand. (I actually do not.) I only came to understand it well through teaching it. My issue is that we are pushing kids along on this Calculus before graduation track to the detriment of their basic numeracy skills. So while I am teaching what's supposed to be high school math, the teaching/learning experience gets bogged down with an incredible amount of remediation of the basics. I agree with you that second and third exposures are needed for many concepts. If only we teachers were allowed that luxury!

Bunnycat, interesting analogy with Argentine tango dancing. I feel the basic problem is the refusal to admit that some people have an affinity for certain things and some don't. Sure, everyone can learn and improve. But some people take to things easier and more naturally than others. Some are going to learn to dance the tango well if you teach them rote patterns because they will discern the form through practice. Some will learn well the other way around. And some will be flat-footed clumsikins (me) who might be able to do a half-way respectable tango only after committing routines to memory and practicing really hard. If you don't believe that, then it's easy to think that all that needs to be done is find the perfect teaching style - then everyone will achieve mastery at the same time.

Actually- I haven't seen anyone learn to dance really well from doing strictly one or the other type of learning (been dancing it for 12 years and teach occasionally). Sticking strictly to pattern cripples them (figuratively) as a dancer because they can't function fully in an improvisational format (which is must in social dancing in the type of extremely crowded situations you encounter in argentine tango) and most people need *some* sort of structure to learn so purely intuitive learning also doesn't work. They each have their place. The analogy I am trying to make is that I don't really think people's learning style changes much from subject to subject, or even child to adult. It's part of who you are and you need both, sometimes (or often) at different times. And no, not everyone will achieve mastery, in dance or in math and science. Many, many in tango find their "happy place" usually somewhere in a low to intermediate skill level (though psychology gets in the way and they often *think* they are Chicho Frumboli.....). Same with math. Hardly anyone will go on to advanced or more abstract levels, but basic skills ought to be learned.

For kids, learning to compute fractions to me is like needing to learn the rote basics (perhaps with some conceptual abstract assists) and then later or for those who go on, revisiting core ideas in a general context, or expanding the ideas.

 

[Maria D]

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

Bunnycat, interesting analogy with Argentine tango dancing. I feel the basic problem is the refusal to admit that some people have an affinity for certain things and some don't. Sure, everyone can learn and improve. But some people take to things easier and more naturally than others. Some are going to learn to dance the tango well if you teach them rote patterns because they will discern the form through practice. Some will learn well the other way around. And some will be flat-footed clumsikins (me) who might be able to do a half-way respectable tango only after committing routines to memory and practicing really hard. If you don't believe that, then it's easy to think that all that needs to be done is find the perfect teaching style - then everyone will achieve mastery at the same time.

Actually- I haven't seen anyone learn to dance really well from doing strictly one or the other type of learning (been dancing it for 12 years and teach occasionally). Sticking strictly to pattern cripples them (figuratively) as a dancer because they can't function fully in an improvisational format (which is must in social dancing in the type of extremely crowded situations you encounter in argentine tango) and most people need *some* sort of structure to learn so purely intuitive learning also doesn't work. They each have their place. The analogy I am trying to make is that I don't really think people's learning style changes much from subject to subject, or even child to adult. It's part of who you are and you need both, sometimes (or often) at different times. And no, not everyone will achieve mastery, in dance or in math and science. Many, many in tango find their "happy place" usually somewhere in a low to intermediate skill level (though psychology gets in the way and they often *think* they are Chicho Frumboli.....). Same with math. Hardly anyone will go on to advanced or more abstract levels, but basic skills ought to be learned.

For kids, learning to compute fractions to me is like needing to learn the rote basics (perhaps with some conceptual abstract assists) and then later or for those who go on, revisiting core ideas in a general context, or expanding the ideas.

Yes, you definitely need both (in math). I probably shouldn't have tried to work with the dancing analogy as I know nothing about teaching dancing.

Learning style is a topic fraught with controversy as well. I've had so much "professional development" on "differentiated instruction" that was supposed to train me on how to adapt lessons to the particular "learning style" of each learner. The auditory learner needs to hear it, the visual learner to see it, the kinesthetic learner needs hands-on manipulative...all this well meaning stuff that was never research based. It sounds like it makes so much sense but as it turns out, people actually don't have a "learning style." Different teaching/learning methods need to be used for different content. For example, if you are teaching geography, having some maps to look at is going to work a lot better than telling your supposedly "auditory learner" that Canada is north of the U.S. But for years we were subjected to all this training on how to identify and work with different learning styles - that don't actually exist! https://www.researchgate.net/profile/Cedar_Riener/publication/249039450_The_Myth_of_Learning_Styles/links/0046353c694205e957000000.pdf

 

[bunnycat]

I dance Argentine tango. Similar shifts in teaching have occurred in the dance world as in other realms. Some time ago, people got fed up with the fact that dance students were only learning "rote" routines and patterns and wanted to break the habit to understanding the "form" (the abstract, like math). So, people stopped teaching patterns, and began teaching only abstract form, thinking people would develop the patterns through exploration. It didn't work. Neither does the other way (pattern based rote learning). In reality, a person needs a bit of both and at different times. Pattern and technique at first (but taught with the full understanding that that isn't all there is to it) and then later, when they have assimilated some basic ideas, exploration and abstracts.

That's sort of a microcosm of how I feel people learn in general. I see nothing wrong with some rote learning in education. In fact, like learning some rote dance movement, it provides a solid base in the mind to build questions and exploration at a later time. Some people may never get to that time, and you have to accept that too.

Bunnycat, interesting analogy with Argentine tango dancing. I feel the basic problem is the refusal to admit that some people have an affinity for certain things and some don't. Sure, everyone can learn and improve. But some people take to things easier and more naturally than others. Some are going to learn to dance the tango well if you teach them rote patterns because they will discern the form through practice. Some will learn well the other way around. And some will be flat-footed clumsikins (me) who might be able to do a half-way respectable tango only after committing routines to memory and practicing really hard. If you don't believe that, then it's easy to think that all that needs to be done is find the perfect teaching style - then everyone will achieve mastery at the same time.

Actually- I haven't seen anyone learn to dance really well from doing strictly one or the other type of learning (been dancing it for 12 years and teach occasionally). Sticking strictly to pattern cripples them (figuratively) as a dancer because they can't function fully in an improvisational format (which is must in social dancing in the type of extremely crowded situations you encounter in argentine tango) and most people need *some* sort of structure to learn so purely intuitive learning also doesn't work. They each have their place. The analogy I am trying to make is that I don't really think people's learning style changes much from subject to subject, or even child to adult. It's part of who you are and you need both, sometimes (or often) at different times. And no, not everyone will achieve mastery, in dance or in math and science. Many, many in tango find their "happy place" usually somewhere in a low to intermediate skill level (though psychology gets in the way and they often *think* they are Chicho Frumboli.....). Same with math. Hardly anyone will go on to advanced or more abstract levels, but basic skills ought to be learned.

For kids, learning to compute fractions to me is like needing to learn the rote basics (perhaps with some conceptual abstract assists) and then later or for those who go on, revisiting core ideas in a general context, or expanding the ideas.

Yes, you definitely need both (in math). I probably shouldn't have tried to work with the dancing analogy as I know nothing about teaching dancing.

Learning style is a topic fraught with controversy as well. I've had so much "professional development" on "differentiated instruction" that was supposed to train me on how to adapt lessons to the particular "learning style" of each learner. The auditory learner needs to hear it, the visual learner to see it, the kinesthetic learner needs hands-on manipulative...all this well meaning stuff that was never research based. It sounds like it makes so much sense but as it turns out, people actually don't have a "learning style." Different teaching/learning methods need to be used for different content. For example, if you are teaching geography, having some maps to look at is going to work a lot better than telling your supposedly "auditory learner" that Canada is north of the U.S. But for years we were subjected to all this training on how to identify and work with different learning styles - that don't actually exist! https://www.researchgate.net/profile/Cedar_Riener/publication/249039450_The_Myth_of_Learning_Styles/links/0046353c694205e957000000.pdf

Yeah- that doesn't surprise me at all. People get ideas in order to try to fix a problem. It works. It bombs. It does something in between....and the cycle continues. And boy, do y'all deserve to get paid better....

 

[t-c]

t-c, while you may now feel that since you were able to study pure (advanced) math with fewer numbers that versatility with basic arithmetic isn't essential

I didn't say arithmetic isn't essential, just that I'm terrible at it. But then that's "terrible" by my standards. I see now that my lighthearted tone wasn't communicated clearly. I assure you that I can do arithmetic (although I admit I don't take as much care as I should) -- and I offer my 4.0 gpa in math in college as proof

Learning style is a topic fraught with controversy as well. I've had so much "professional development" on "differentiated instruction" that was supposed to train me on how to adapt lessons to the particular "learning style" of each learner. The auditory learner needs to hear it, the visual learner to see it, the kinesthetic learner needs hands-on manipulative...all this well meaning stuff that was never research based. It sounds like it makes so much sense but as it turns out, people actually don't have a "learning style." Different teaching/learning methods need to be used for different content. For example, if you are teaching geography, having some maps to look at is going to work a lot better than telling your supposedly "auditory learner" that Canada is north of the U.S. But for years we were subjected to all this training on how to identify and work with different learning styles - that don't actually exist! https://www.researchgate.net/profile/Cedar_Riener/publication/249039450_The_Myth_of_Learning_Styles/links/0046353c694205e957000000.pdf

Too bad students aren't taught to teach themselves early on. I did a few independent study courses in high school -- subjects that I felt I didn't get enough instruction on (ironically Algebra 2/Trig was one) -- and it served me well at uni when we were responsible for learning the material ourselves. Although I find that college students seem to expect to be spoonfed information -- they demand to know exactly what will be in the exams! Totally ridiculous, I thought, because if they paid any attention in the lectures and discussion sections, they would know which concepts were deemed important.