Parents' frustration with distance learning -- "Common Core Math Methods"

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  • #51
mathwonk
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I agree with you 100% about the constructive proofs being more persuasive. It seems there are 2 ways of doing calculus rigorously, either assuming the reals are represented by infinite decimals, or just assuming they form an ordered field with the least upper bound property. This second approach is much more abstract but is apparently favored by the experts as being cleaner and slicker for giving proofs. We ordinary people however who like to see real numbers concretely are more persuaded by the actual process of constructing a solution to our problem. The contrast is between exhibiting a concrete reasonably familiar model for the real numbers, as opposed to just stating axioms they should satisfy, without convincing us these axioms are reasonable. I.e. the abstract approach does not even consider the question of whether the reals actually exist. The tedious method of Dedekind cuts to construct them (as in Rudin) is also quite painful in my opinion. This is the contrast betwen representing a number c either in terms of the decimal expansion for c, or in terms of the entire (rational) part of the real axis lying to the left of c!

The most popular books for rigorous calculus, Spivak and Apostol, (as well as the less available but excellent books by Lang, Analysis I, and by Kitchen, Calculus of one variable), do things the abstract axiomatic way, and so did my intro college course, but the remarkable book by Courant does things using infinite decimals. Fortunately Courant was recommended reading for my course. To be sure Courant quickly ramps up and uses infinite decimals to prove an abstract property, the principle of the point of accumulation (every infinite subset of a closed bounded set must accumulate, or bunch up, about at least one point). Even this principle however is more concrete to me than a bare axiom, especially since it can be easily proved by subdividing intervals as in the proofs above.

It seems we are mostly given two types of presentations nowadays, either no proof at all of basic facts about limits in calculus, or abstract proofs based on axioms for the reals. I think many more people could be introduced to rigorous calculus if the constructive infinite decimal approach were more common.

I wonder what the common core approach to calculus is?

Edit: Well I just read the calculus chapter of the 2013 California standards for high school math. The first sentence is :

"When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses."

I think this is neither entirely meaningful, nor convincing. I.e. entry level (i.e. freshman) calculus courses throughout the country vary from completely intuitive ones to math 55 at Harvard, the "hardest course in America", apparently taken mainly by people who have already been Olympiad fellows. Moreover the whole advantage to taking a hard subject early is to have an easier, slower, introduction to it that will help later when a more rigorous version is encountered. In spite of the language in this chapter, my experience is that high schools tend to spend roughly twice as much time on an intro calc course as does a college course, even a non honors one. So do they mean "at the same depth", but twice as slowly?

The description given suggests the real model for the CA course is the AP test. To be fair, it also matches closely what is usually offered to non honors students in most non elite colleges, namely a course that states but does not prove the fundamental theorems underlying the calculus, the intermediate value and extreme value theorems, which we have just seen can be rigorously proved in a way that even high schoolers could easily grasp.

One thing which is taken for granted, but seems to me questionable, is the idea that a calculus course can be given at the same level as a college course when the high school teacher giving it has usually nowhere near the same training in math as a college teacher. I.e. even in top ranked private high schools of my acquaintance, calculus could be taught by anyone who had taken calculus in college, whereas in college it is often taught either by a graduate PhD student or, frequently, a professor with mathematical research experience. A quick look at my vita shows over 40 first year calculus courses taught in the 30 years after receiving the PhD, and it is incomplete.
 
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  • #52
mathwonk
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I looked at the article and admit I find it somewhat depressing that parents are struggling with understanding that a number like 43 means we have 43 separate items that are grouped into 4 sets with 10 things in each set plus 3 remaining single times. They are just asking us to realize that the '4' stands for 4 tens and the '3' stands for 3 ones. In particular if we add 7 more we get enough single items to make another ten, so it becomes 5 tens and no ones, or 50. But people are complaining "don't tell me that! just let me write 7+3 as 10 and the '1' adds to the '4'! That's all I need to know how to do! I don't want to understand why it works."

Believe me, I had to teach this class, and it was brutal. Some of those students, all of them future teachers, were pretty resistant (not all by any means), and it showed on your class evaluations. At the end I got the worst evaluations I had ever had. One of my friends who had taught it the semester before asked to see mine with a smile of anticipation on his face. When he read my terrible evaluations, he looked at me and said "actually, those aren't so bad." I felt sorry for him.
 
  • #53
mathwonk
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Another place I encountered a difference of approach was in multiplication. My class had only been told that multiplication is repeated addition, so they had trouble grasping how to multiply numbers that are not integers. I.e. is sqrt(2)x pi equal to sqrt(2) added to itself pi times? I myself always looked at multiplication as area, i.e. 2x3 is the area of a rectangle with base 2 and height 3, so sqrt(2)xpi is the area of a rectangle with base sqrt(2) and height pi. This seemed foreign to even the professors they had studied with. But I just saw this approach to multiplication yesterday on the internet in a Montessori class for kindergarteners, using rods. What gives?

One way to deal with equal ratios is in terms of area, since a/b = c/d is the same as ad = bc, or the area of the rectangle with sides a,d has same area as that with sides b,c. This principle appears already in Euclid, in Prop. III.35, where he shows that two triangles formed in a circle by intersecting two secants, hence with equal corresponding angles, have equal ratios in this sense of area. In my opinion, the damage that has been done by not teaching from Euclid for the last 100 years is really enormous.

Here is another little tidbit from geometry. Some modern books on geometry illustrate the fact that two triangles with the same corresponding sides are congruent by making a triangle from three straws with a thread running through them, and observing that the triangle is rigid, in the sense that you cannot change the angles without breaking the straws or the string.

This nice and hands on, but ignores the fact that rigidity of a figure only shows that there is no family of congruent figures all nearby, to which it can be continuously deformed. It does not show there might not be another congruent figure some distance away which can only be reached by a discrete motion, breaking the figure and reassembling it. E.g. if this principle were enough, then SSA would be sufficient for two triangles to be congruent, since fixing two adjacent sides and an angle not contained between them, allows exactly two different triangles in general (as long as the angle is not 90 degrees). They are not congruent, but neither can be continuously deformed into the other.
 
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  • #54
Averagesupernova
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That's all I need to know how to do! I don't want to understand why it works."
Unfortunately that is a common theme im many areas of people's lives these days. I will admit there are many times this is acceptable. After all, we can't be experts at everything. But to be unwilling to accept or learn something as simple as the math example @mathwonk gave depresses me.
 
  • #55
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That's all I need to know how to do! I don't want to understand why it works."
I have had a couple opportunities to "help" children of friends with their homework. And it was not easy to figure out how they had been taught to do the multiplication and the division. The "format" was just so different than the traditional algorithms I learned in grade school ("borrow from the nine," and "carry the two"). Once I deciphered the new method, it is of course obvious what they are doing. I think most people here on PF get it, and see how the new approach could be less mysterious than the old, and is probably "deeper" somehow.

But - the kids I dealt with did not look at this as anything other than an algorithm. They had been taught to draw out these groups of boxes and what goes in each box. Very mechanical, no less so than the traditional method. "That's all I need to know." Grade school "math" homework is still seen as a chore.
 
  • #56
mathwonk
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Probably these kids are smarter than I am in a sense, with their agile young brains and good memories, but I admit that now in my old age, I sometimes forget the algorithm for subtracting and borrowing, and it helps me to rethink that when I borrow one from the tens column, that is really a batch of ten ones. More meaningful for me, it is like opening a sixpack, and having one less one carton, but six more bottles.

When I taught it, I tried to make games like that, where kids would write 43 to represent 4 six-packs and 3 extra bottles of soft drink. Or if a case held 4 six packs, and we were also using cases, then we would write 103 for one case, no loose six-packs, and 3 extra bottles.

But it was a struggle, between - "lets understand this, it can be fun", and -" lets just get through this, I hate it". And those were future teachers. The challenge is keep the fun in.
 
  • #57
gmax137
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More meaningful for me, it is like opening a sixpack, and having one less one carton, but six more bottles.
I don't remember Miss Runcible explaining "borrow the one" that way. Who says math can't be fun?
 
  • #58
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I always wonder what it means to "do calculus", in say the 9th grade.

That I would call real analysis - not calculus which is more intuitive in its approach. The intuition is the intuitive idea of limit - although some approaches ask students to think of dy and dx as numbers so small, that for all practical purposes are zero, but are not zero. And most certainly dy or dx squared can be neglected.

Can you say a bit about what a 10th grader in Australia would learn about calculus? E.g. would he/she see the mean value theorem proved assuming a continuous function has a maximum on a closed bounded interval? Would he/she see proved that (more difficult) preliminary result? Or would one just rely on a picture as in Thompson, to conclude that the slope is zero at a maximum?

Ok - first until very recently here in Queensland Australia we stared grade 1 at 5 so 10th grade in the US (and now here) would correspond to 11th grade when I did it. Also calculus is formally taught here in Aus in grade 11 and 12 - we combine calculus and pre-calculus together. We do not cover the intermediate value theorem etc - but stop at L-Hopital. It is very similar to IB HL math (have a look at the table of contents):
https://www.haesemathematics.com/books/mathematics-analysis-and-approaches-hl

Many schools have for good students an accelerated math program where they start grade 11 and 12 math in grade 10, leaving year 12 for university math subjects taught at your school. That would be starting in grade nine in the US system. That came in a bit after I finished so an accelerated program did not apply to me. Hence I can speak from experience on what the equivalent of a 10th grader would learn about calculus. It was all done on the intuitive idea of a limit eg the idea of instantaneous speed as you make the time period smaller and smaller so it effectively becomes zero. No real analysis - that was left to university first year. People hated it (I loved it personally) so was dropped as a requirement, hence some more applied math types never even took it - but did get a sort of an idea about things like the GLB axiom etc. Here is a typical first year university calculus subject here in Aus for those that did calculus at HS:
https://handbook.unimelb.edu.au/subjects/mast10006

If you were to stop at grade 10 you would probably learn something similar to IB SL:
https://www.haesemathematics.com/books/mathematics-analysis-and-approaches-sl

But good students would complete the full HL syllabus starting grade 9.

I quite like the idea, by the way, that in Australia one need not complete years and years of college to get a useful practical degree and job related training.

So does the government - it is worried far too many people think going to uni is the only route to a successful career. I think they want everyone who can to eventually reach university level in their knowledge - but you can do that while working and taking courses along the way to increase your knowledge. Also you often learn a huge amount on the job in many professions/careers. For example you can do a number of what are called Graduate Certificates with just a Diploma (equivalent to an associate degree in the US) and some work experience eg:
https://www.griffith.edu.au/study/degrees/graduate-certificate-in-finance-3266#entry-requirements

Upon completion you get credit for those subjects and then complete a Masters (if you wish).

Thanks
Bill
 
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  • #59
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12x12, late '60s... maybe the excuse for stopping at 10x10 is that "dozen" is now considered archaic ?

Meh, make them go to 16x16 to be modern.
Yes, I learned them in the sixties I think circa 1962 4th grade, when my brother was born as I recall reciting them to my mom in the hospital.

The push to teach about money and the metric system may have influenced teachers to stop at 10x10. I do remember special cases like 12x12 is 144 or things x 10 as extensions of the table.

Another influence was the so-called Sputnik Crisis where the US needed more engineers to compete with the Russians And so developed the New Math curriculum featuring Set Theory, module numbers...

https://en.wikipedia.org/wiki/New_Math
 
  • #60
PAllen
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My wife teaches 3,4,5 year olds. She needs to know calculus why? My wife has noted a common issue is that teachers teach how they were taught. This leads to generational stagnation and stubborness. Why we are teaching the same we did 50 years ago blows my mind.
Well, one answer is "why not?" if it works. Is there some reason to think the today's children learn differently than children 100 years ago? I recall the experience of looking at a calculus textbook in French from circa 1725 (in a rare book library) and noting that despite not knowing word of French, it was easy to follow because the order of presentation and notation were already similar to what I learned. My thought was not "oh how terrible", but "wow, so much of how to teach this was worked out in a matter of decades from first discovery".
 
  • #61
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Well, one answer is "why not?" if it works. Is there some reason to think the today's children learn differently than children 100 years ago?
"If it works" is the issue. We know more now about how people learn compared to what we knew 100 years ago, and it makes sense to change how students are taught in light of this new knowledge. For example, studies consistently show that the most relevant factor in improving student learning in introductory astronomy is the amount of interactive learning in class. So rather than having a class that consists of the professor lecturing for 50 minutes, you might have a class consisting of sequences of a short lecture focused on a particular topic and then an activity or two where students work with the ideas they were just introduced to. Yet many professors still stick with straight lecture, where students take a passive role, because that's what they're familiar with, i.e., that's how they were taught.
 
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  • #62
Office_Shredder
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Yet many professors still stick with straight lecture, where students take a passive role, because that's what they're familiar with, i.e., that's how they were taught.

In fairness, it's not just something they're familiar with, it's something that they were able to use to master the subject. They know for a fact that a student can become a world expert in the material from this teaching method.
 
  • #63
berkeman
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Well, one answer is "why not?" if it works. Is there some reason to think the today's children learn differently than children 100 years ago?
I spent a lot of time on my smartphone in the late 1970s when I was graduating high school and going through undergrad and doing my MSEE. Oh wait...

I very much approach my work differently currently with instant access to the Internet, and have recycled all of my databooks and most of my textbooks, and instead look up the information I need in less than 5 seconds by pulling my cellphone off my hip.

Yes, today's children are learning differently, and I don't think it's a bad thing. As long as they truly learn and learn how to use Internet resources to find information more quickly.

"If it works" is the issue. We know more now about how people learn compared to what we knew 100 years ago, and it makes sense to change how students are taught in light of this new knowledge. For example, studies consistently show that the most relevant factor in improving student learning in introductory astronomy is the amount of interactive learning in class.
And we have found through the last school cycle how significant the impact has been on many students to trying to adapt to distance learning. And even for the teachers -- locally I've seen news reports by teachers who are trying as hard as they can to get back to in-person learning because they've seen the shortcomings in their students when they came back to the classrooms lately.
 
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"If it works" is the issue. We know more now about how people learn compared to what we knew 100 years ago, and it makes sense to change how students are taught in light of this new knowledge. For example, studies consistently show that the most relevant factor in improving student learning in introductory astronomy is the amount of interactive learning in class. So rather than having a class that consists of the professor lecturing for 50 minutes, you might have a class consisting of sequences of a short lecture focused on a particular topic and then an activity or two where students work with the ideas they were just introduced to. Yet many professors still stick with straight lecture, where students take a passive role, because that's what they're familiar with, i.e., that's how they were taught.
Absolutely, and traditional schooling did not work for me. I only clicked into gear once I got to college. Having kids sit in desks, quiet, raise their hand to say something, then do 50 home sheets at home is not developmentally appropriate, inventive, or effective. I once had an algebra class where the teacher worked problems on an overhead projector for 45min in a monotone voice every single day. Is that the best we got?
 
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  • #65
Astronuc
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I once had an algebra class where the teacher worked problems on an overhead projector for 45min in a monotone voice every single day. Is that the best we got?
I certainly had those kinds of teachers, but I had more animated teachers in my honors classes who more or less left it up to us to teach ourselves. In math and science classes, we were given examples of types of problems, and then we were left to generalize and solve more complex problems.

I hired a student at my office and he went on to major in math at Harvard. Some of his work in high school was more advanced than I remember, but then he was one of a handful of students. As I recall, he was valedictorian, or co-valedictorian. My kids went to the same high school, but they were not exposed to the same rigorous math program, and in fact my kids struggled with math, which seemed less advanced than what I studied in high school. Certainly, education, like health care, is inconsistently provided across the nation. Far too many children/youth get left behind.

or along the lines of say George Thomas's early books, where some rigor is given but as I recall real numbers are a bit cavalierly treated and basic existence theorems are not proved, but many useful practical methods are taught;
I used this text in high school. I actually purchased a copy probably around 10th grade because I was interested in learning calculus, and at the time, it was not taught at the high school I was attending. I changed high schools between 10 and 11th grades, and it turned out that calculus was taught at the second high school, and the program used Thomas, but are different edition.

https://en.wikipedia.org/wiki/George_B._Thomas

I also had a copy of the CRC Standard Mathematical Tables and Formulae, from about 1972-3. I was intrigued by the tables of integrals and the geometry and trigonometry.

watching TV at home one night when the "Beverley Hillbillies" TV show
This reminded me of Jethro doing his ciphering or go's-intos. :oldbiggrin:
 
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  • #66
PeroK
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I once had an algebra class where the teacher worked problems on an overhead projector for 45min in a monotone voice every single day. Is that the best we got?
That's bad teaching in anybody's book. It's not necessarily a result of educational policy.
 
  • #67
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In fairness, it's not just something they're familiar with, it's something that they were able to use to master the subject. They know for a fact that a student can become a world expert in the material from this teaching method.
It would be more accurate to say they know that some students like them can become experts. But what about everybody else?
 
  • #68
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It would be more accurate to say they know that some students like them can become experts. But what about everybody else?

That's why I said *a* student, not *all* students.

My point here is you're not asking this professor to do something different from what they have simply seen someone do before, you're telling them that the thing they went through that was very successful for them, is not actually good. They have a deeply personal experience of this old system working, and zero experience of the new thing working. It is not a trivial thing as a human, to make that adjustment.
 
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That's why I said *a* student, not *all* students.

My point here is you're not asking this professor to do something different from what they have simply seen someone do before, you're telling them that the thing they went through that was very successful for them, is not actually good. They have a deeply personal experience of this old system working, and zero experience of the new thing working. It is not a trivial thing as a human, to make that adjustment.
I taught statistics. I thought the methods were archaic, relics of the precomputer age that survived due to laziness and tradition. The students hated learning this meaningless stuff.

I guess the idea is to test the ability of the student to memorize and apply arbitrary complicated routines. That;s what they'll be doing if they can get a white color job. In that way it makes sense.
 
  • #70
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teachers teach how they were taught. This leads to generational stagnation and stubborness. Why we are teaching the same we did 50 years ago blows my mind.
Some basic things simply don't change - ever. Certainly, if something works, keep using it. If there is a better method that teaches more efficiently, then by all means, use that method.

I'm trying to remember how I learned math along the way, and why 50 years later, there seems to be little progress with learning in grades 4-12, i.e., the majority of kids still struggle with math, and many do not learn calculus in high school. There are the odd few % who do however.

I've interviewed college seniors who couldn't write/communicate much better than 8th graders, or whose math abilities were rather limited. At the same time, I've worked with high school seniors and undergrads who were brilliant programmers and problem solvers.

When I was in high school learning geometry and trigonometry, I was wondering why were were not taught it in elementary school, and similarly about matrices and matrix algebra. I was exposed to matrices in 5th grade, but were really didn't make the connection with simultaneous equations, and we were limited to 2x2 and 3x3, and basic properties. Then I didn't do matrices until later in high school (again limited) and university.

In high school, I learned dot product and cross product, but not inner and outer products, so when I got to university, I had to learn new terminology. Most high school students didn't even get the basic exposure I did, not until university. So teachers (are people) and teaching are not consistent.
 
  • #71
mathwonk
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Please take this with a grain of salt, as everyone's experience and opinion of teaching and learning differ, but in my opinion, (at the moment), the primary problem with teaching is how to teach more than one person at a time. I.e. every student's background information, motivation, and speed of learning is different, so it is very challenging to keep the attention of a class of more than one, and present useful material without going too slow or too fast. If you are the fastest learner in a class of 35, or the most conscientious, it is quite likely you will almost always be bored and wonder why more challenging content is not offered.

Even with one student it is not trivial to teach significant information. Just think of this forum, where each student has the freedom to start his own thread devoted to his own specific question, and the answers are directed precisely at him/her alone. Experts here, even in combination, still often struggle to make themselves clear and eradicate misunderstandings.
Now place yourself in a high school classroom with 40 kids, some or most often ill prepared and uninterested, and try to design an effective program that will satisfy most of them, hopefully including the most gifted.

Perhaps for this reason the most accomplished young students i have met were mostly home schooled, essentially individually. This unfortunately may deny them the socialization benefits and comradeship of a standard school atmosphere.
What to do? I do not know the answer, after some 50+ years of teaching, with my most successful experiences limited to very small groups of highly gifted and motivated students.

Of course I may be unusually challenged, as I can share one teaching technique I have used that absolutely guarantees failure: once while I was giving an explanation of a clever trick for proving Taylor's theorem in an honors class, (the argument from Courant, and reproduced in Spivak), a very bright and motivated student in the front row leapt ahead mentally and shouted out the point of the still incompletely explained trick. Delighted, and somewhat embarrassed to continue, since I concluded the point was now obvious, I complimented him and stopped the explanation right there, thus guaranteeing that exactly one student in the class would understand it. Never do this. Plod right ahead with the explanation in full. The silent majority will thank you, as no doubt everyone else here already knows.
 
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  • #72
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Some basic things simply don't change - ever. Certainly, if something works, keep using it. If there is a better method that teaches more efficiently, then by all means, use that method.
Completely agree.
 

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