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

[berkeman]

Interesting article. At first I thought the article was saying that for many parents, helping their kids with their math homework was the hardest because maybe they weren't all that great at math themselves.

But the real reason is that the "Common Core" math methods used in primary school these days involve new tricks ("grouping") to solve the problems. The methods that we are all used to have been replaced, so if you're going to help your kids with their work, you'll need to learn the new paradigm. Sigh.

https://www.cnn.com/2020/09/08/us/distance-learning-problems-parents-trnd/index.html

(CNN)It's a truly humbling moment when your child asks you to help diagram a sentence or solve a grade-level math problem and you, a functioning adult with a diploma and years of experience, draw a complete blank.

Anyone with school-aged children can probably relate. And as many schools start the year with virtual learning, parents are trying to summon even more of that long-forgotten knowledge.
Helping your child navigate Zoom tech support can be daunting. So can balancing work and household duties with making sure your children are engaged and learning.

But the single biggest challenge, many parents say, are the math topics taught through Common Core -- a standardized teaching method rolled out in 2010.

 

[jedishrfu]

Our parents had the same frustration as did I with my sons.

The usual complaint when helping with a math problem was "No that's not how the teacher did it." and when asked how the teacher did it " I don't remember" and finally after several fruitless minutes "Oh now I get it."

During a school open house, the math dept head talked about their teaching methods. I asked about why kids can't bring their tests home. The response was that's against school policy, we only have a limited number of test problems and we don't want the students passing them on to other students.

However, you as a parent are welcome to stop by to view the test questions provided the teacher is available and you've made an appointment and your boss let's you leave work early so you can get there.

My old HS math teacher said those teachers were just too lazy to make up new problems for their tests and makeup tests.

The end result was the graded homework was too late to fix mistakes that then cascaded onto quizzes and tests. Your kid does poorly and you are helpless to help them short of getting them a regular tutor to fill in for the missing feedback.

We did in fact get a math tutor and he was great. He had several students come in one by one. While waiting for your tutoring time, he would give you a quiz to work on. Tutoring time was spent reviewing the quiz and any questions you had and things progressed from there. He was an amazing tutor.

However, nowadays we also have Khan Academy and that's a great step in the right direction.

 

[Mark44]

But the real reason is that the "Common Core" math methods used in primary school these days involve new tricks ("grouping") to solve the problems.

A disclaimer: 'm not very familiar with what is in Common Core math at the elementary level; my math teaching experience is lower-division college level, and mostly ended back in '97. Also, it's been many years since my son was in the primary grades.
However, I have seen some of the new techniques that are used for teaching division, and I'm very skeptical about it, as well as many of the other concepts involving arithmetic.

The usual complaint when helping with a math problem was "No that's not how the teacher did it."

An important thing about elementary grade school teachers is that they very seldom have strong math skills. The chart here, https://www.businessinsider.com/heres-the-average-sat-score-for-every-college-major-2014-10https://www.businessinsider.com/heres-the-average-sat-score-for-every-college-major-2014-10, shows that Education majors are in the bottom quintile of SAT Math scores, with an average score more than 100 points below those of the highest scoring majors.

I'm also very skeptical about people who get PhD's in Education. When I was still teaching, one of my fellow instructors attended a program at the local university, and was granted a PhD in Education. One of the Chem instructors asked her, "Why didn't you opt for a real degree?"

My old HS math teacher said those teachers were just too lazy to make up new problems for their tests and makeup tests.

That would be my guess as well.

 

[jedishrfu]

Most of my math tutoring with my son was when he was in high-school where I encountered the attitude I mentioned above which extended into other subjects as well. My wife covered the grade school stuff since her schooling in Taiwan was superior to what we learned arithmetic-wise. Basically, her mental math is beyond my ability to compete with (hehe but I know Calculus).

One interesting thing to note is that US teachers stop at 10x10 multiplication tables whereas in other countries they extend it to 12x12 or even further to 20x20. Granted its repetitious to learn 20x20 tables but it does increase your ability as a kid to memorize things setting a higher bar which they can meet and it gives you a lifelong boost in that it encompasses so much of what we do with respect to mental math.

In counterpoint to 10x10 and beyond:

https://blog.wolfram.com/2013/06/26/is-there-any-point-to-the-12-times-table/

Similarly the Japanese limit times tables to 9x9 but with extensive memory drills, mnemonic schemes and speed competitions:

https://www.japantimes.co.jp/life/2002/10/11/lifestyle/chant-away-to-calculation-competence/

Of course, one can't neglect some common basic knowledge:

A Sufi Story from the Middle East

A scholar asked a boatman to row him across the river. The journey was long and slow. The scholar was bored. "Boatman," he called out, "Let's have a conversation." Suggesting a topic of special interest to himself, he asked, "Have you ever studied phonetics or grammar?"

"No," said the boatman, "I've no use for those tools."

"Too bad," said the scholar, "You've wasted half your life. It's useful to know the rules."

Later, as the rickety boat crashed into a rock in the middle of the river, the boatman turned to the scholar and said, "Pardon my humble mind that to you must seem dim, but, wise man, tell me, have you ever learned to swim?"

"No," said the scholar, "I've never learned. I've immersed myself in thinking."

"In that case," said the boatman, "you've wasted all your life. Alas, the boat is sinking."

http://www.storyarts.org/library/nutshell/stories/boatman.html

 

[vela]

Interesting article. At first I thought the article was saying that for many parents, helping their kids with their math homework was the hardest because maybe they weren't all that great at math themselves.

But the real reason is that the "Common Core" math methods used in primary school these days involve new tricks ("grouping") to solve the problems. The methods that we are all used to have been replaced, so if you're going to help your kids with their work, you'll need to learn the new paradigm. Sigh.

The choice of grouping as an example is a particularly poor one. If a parent can't understand grouping, I would say it's because they aren't great at math. Plus I remember a form of grouping taught when I was in elementary school, but to be fair, not a lot of time was spent on it, if I recall correctly.

What's sad is the attitude of one mother quoted in the article, who said "I refused to do the math portion of anything," leaving that responsibility to her husband. It tends to foster the idea that it's okay to give up on math because "it's too hard!" and the stereotype that girls aren't good at math.

The sense I got from the article is that these frustrated parents view learning as an unpleasant chore, and that's what they're teaching their kids. I get that it can be annoying to parents that they can't answer questions off the top of their head, but the flip side is that it's the good opportunity for parents to show their children how to deal with subjects they don't already know.

 

[vela]

One interesting thing to note is that US teachers stop at 10x10 multiplication tables whereas in other countries they extend it to 12x12

Is that true? I was taught to memorize up to 12x12 (back in the '70s), and I seem to recall a year or two ago my niece (from Wisconsin) telling me they had to learn up to 12x12.

 

[Mark44]

I was taught to memorize up to 12x12 (back in the '70s),

Same for me, but it was back in the mid-50s.

 

[berkeman]

Quick story about multiplication tables and an "aha" moment that I had as a kid...

I was having trouble memorizing the multiplication table (10x10 for me), I think in 1st or 2nd grade. There were just some combinations of numbers that I got confused on and sometimes I was off by a digit or two. I was watching TV at home one night when the "Beverley Hillbillies" TV show came on, and I started singing the theme song at the start of the show. When I was done, my dad said, "So you can memorize that song, but you can't memorize the multiplication table?"

Whelp, that gave me pause, and I realized that I just wasn't putting enough effort into memorizing the multiplication table. Things went much more smoothly soon after that.

 

[Fervent Freyja]

Is that true? I was taught to memorize up to 12x12 (back in the '70s), and I seem to recall a year or two ago my niece (from Wisconsin) telling me they had to learn up to 12x12.

Yes, they still do this in some places to 12x12. My nine year old little girl demands that I quiz her on the table! She’s obsessed. She even quizzes me in the car while I’m driving. Bath time is her math, physics, and philosophy (she’s curious about death right now) free-for-all! I love it.

 

[hmmm27]

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.

 

[mathwonk]

Some of my best friends were intimately involved with creating the common core math stuff, and they meant well, but I am motivated to recall my comment about the consequences of changing calculus books when we taught math in the department: namely, no matter how much better a given new book might be in theory, whenever we changed books, we had one year of confusion while the teachers learned the new book themselves. Hence since the teacher is more important than the book, it is always better to stay with the book they know. I.e. no matter how bad the current book is, the teachers have learned how to compensate for it, and when you change it, you throw the whole system out of balance for one or more years.

Of course this is not true for the tiny subset of good students who actually read the book and teach themselves from it. These may be the ones the professors who write the core identify with. But I am conscious that my friends involved in the common core have no chance to defend their work here. By the way, one of those friends has a math PhD in arithmetic geometry from Penn and held the prestigious Gibbs instructorship in the math dept. at Yale. My impression is that when she had children and observed what was being taught them in schools, her commitment switched to improving math instruction, and she is author of a nationally recognized book for teaching elementary school math. Although it is true that education majors score notoriously low in math, the math ed department at UGA requires their students to take significant amounts of substantive math courses from the math department, including say foundations of geometry, graduate algebra (Galois theory etc.)

My friend's book is excellent in my opinion, and could help frustrated parents. Almost the only negative amazon reviews are referring to the weight of the book for carrying around or the physical condition of the used copy received. It is quite expensive though.
https://www.amazon.com/dp/0321825721/?tag=pfamazon01-20

@berkeman: you made me laugh and recall my kid's experience in second grade. We splurged for the expensive private school and on the days before class started, I emphasized that he should know his multiplication tables at least up to 12 x 12, so as not to be embarrassed. When he came home I asked him if he had indeed been ready for this high end school, and he burst out laughing. When he calmed down, he explained that at this progressive school, they did the times tables one month at a time, and the first month was the zeroes times table. I had a little conversation with myself over the loans I had taken out to pay the tuition at this "school", but at least in terms of English training it did prove valuable.

 

[Mark44]

When he calmed down, he explained that at this progressive school, they did the times tables one month at a time, and the first month was the zeroes times table.

Then in a 9-month school year, they must have gotten only to the "8 times" row. Unbelievable.

 

[mathwonk]

When I had to teach the grouping stuff, I tried to make up real world examples, like using old fashioned English money: i.e. given 5,000 pence say, how many pounds, shillings, and pence would that make, if you changed it all into the largest denominators possible? Or given 650 empty bottles, how many cardboard cartons holding 24 bottles, and cardboard six packs, and cases holding 4 cartons, do they fill ?

Of course the basic purpose of memorizing multiplication tables is to be able to actually multiply multiple digit numerals using positional notation, and since we use the decimal system, you only need to know them up to 10 x 10. For doing problems in your head of course, and for fun, it would be nice to know them up to 16 x 16, or 20 x 20. In our strange world you mark yourself off as an egg head if you happen to know what 17x17 is. I used to try to impress my class by flaunting such esoteric knowledge, or by using tricks like 19x19 = 20x20 - 40 + 1.

 

[mathwonk]

@Mark44: yes when our son came home he had this guilty look, like he had a license to steal, since there were no challenging expectations on him at that school where he got to essentially play all day long.

 

[gmax137]

Of course the basic purpose of memorizing multiplication tables is to be able to actually multiply multiple digit numerals using positional notation, and since we use the decimal system, you only need to know them up to 10 x 10.

Well, up to 9x9 really. That's what I had in grade school, mid 1960s.

For someone who ended up doing more math than average, I had a difficult time memorizing the "times tables." I still remember in class one day, the teacher calling on us one by one, she asked me "seven times eight?" and all I could say was "uh... uh... uh.." then the kid behind me blurted out "fifty six" and I remembered that one from then on. Weird.

 

[mathwonk]

Technically I agree, but it is hard to imagine someone who needs to use that method to multiply 10 by 10 (i.e. one digit at a time). I.e. I would have said up to 9x9, but I couldn't imagine someone not knowing what it means to move a decimal point.

By the way as I read your post I could not remember what 7x8 is either. That was always a hard one for me. I prefer 8x8 or 9x9, or 7x7, maybe I like the squared ones.

 

[gmax137]

if it is asked as 8 times 7, I see it as "twice 28" which I know is forty plus sixteen.

 

[Greg Bernhardt]

An important thing about elementary grade school teachers is that they very seldom have strong math skills.

Perhaps, but my wife is Montessori primary teacher and she might say, most people don't understand child development. Just because you're an expert in mathematics doesn't mean you know how to teach it to young children. She is more in favor of common core as it's closer to how they teach in Montessori which is curriculum designed with the science of whole child development in mind.

 

[berkeman]

Perhaps, but my wife is Montessori primary teacher and she might say, most people don't understand child development. Just because you're an expert in mathematics doesn't mean you know how to teach it to young children. She is more in favor of common core as it's closer to how they teach in Montessori which is curriculum designed with the science of child development in mind.

That's very interesting, thanks Greg. Do you know of any papers or other references for more reading? I wonder how my brain changed from those early days to now, in terms of how I process math problems.

 

[TeethWhitener]

An important thing about elementary grade school teachers people is that they very seldom have strong math skills.

Fixed that for you.

Common core is the way most everyone I know in science and engineering thinks about math, it's just that they're trying to teach that mode of thinking to kids now (as opposed to the algorithmic arithmetic that we all grew up with). I personally think it's great: kids actually get to engage with math concepts instead of being drilled on how to multiply fractions quickly or whatever.

The practicality of knowing how to do tedious arithmetic by hand is largely in the past--calculators exist for a reason (After all, how many of us actually remember how to compute a square root by hand, if we ever knew in the first place?). Sure, an intuition for when the result of a computation doesn't "smell" right is important, but I'm not convinced that knowing how to do algorithmic arithmetic quickly is the exclusive path to developing that intuition.

I think it's probably more a function of familiarity and comfort with numbers, which is where the parents and teachers come in. Since a large fraction of adults in general (including parents and teachers) are functionally innumerate and even math-phobic--I'm mainly talking about the US here--it's not particularly surprising that they would wrestle with changes to a basic curriculum that they perhaps never truly understood to begin with.

 

[Mark44]

Just because you're an expert in mathematics doesn't mean you know how to teach it to young children.

And conversely, just because someone is an expert in child development, he or she doesn't necessarily know how to teach math skills to young children. This thread is about distance learning and Common Core math teaching, so my point is that elementary and middle-school teachers should also have something beyond one year of high school math. I'm not sure that getting a certificate to teach at the elementary level requires even one college-level math class, or even any science class, for that matter.

 

[Greg Bernhardt]

I'm not sure that getting a certificate to teach at the elementary level requires even one college-level math class, or even any science class, for that matter.

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.

 

[Mark44]

My wife teaches 3,4,5 year olds. She needs to know calculus why?

My remarks were about the primary and middle-school grades. Obviously, someone teaching kindergarten and below doesn't need calculus, but those teaching math at, say, 4th grade and up should have more than one year of high-school math, IMO.

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, 50 years ago puts us at 1970, and which was quite a while after I had graduated from high school. What exactly was so bad about the educational methods back then?

BTW, the educational methods at that time had produced scientists and engineers who were about to put men on the moon in 1969.

 

[mathwonk]

Greg's comment about understanding how a child's mind develops reminded me of reading works by Piaget where he pointed out that his experiments showed that at a certain age children do not realize that the amount of milk does not change when you pour it from one glass to another different shaped glass. I would probably not have realized that when trying to teach volumes. So as always, teaching requires more of us, both knowledge of the subject and knowledge of instruction.

 

[Mark44]

So as always, teaching requires more of us, both knowledge of the subject and knowledge of instruction.

I don't disagree with that at all. It just seems to me that too many teachers at the K-12 level have very lopsided levels of subject knowledge vs. instruction knowledge.

 

[mathwonk]

Indeed what to do? Maybe if we raised the pay...?

 

[Greg Bernhardt]

I don't disagree with that at all. It just seems to me that too many teachers at the K-12 level have very lopsided levels of subject knowledge vs. instruction knowledge.

Might be a matter of how much we pay them. If you want them to be a master at instruction and knowledge you better pay them more than $58k/yr. :)

 

[Greg Bernhardt]

BTW, the educational methods at that time had produced scientists and engineers who were about to put men on the moon in 1969.

And in 550BC Pythagoras was kicking butt, but do we want to go back to 550BC education system? There will always be some that rise in any occasion give circumstances are right. For a national education system we need to keep evolving with the science of child development and pedagogy to rise the educational floor for everyone. I can tell you straight up the traditional education system did not work for me. It was by a grace of some deity that was I privileged to have so many opportunities even after really struggling in school. Dice roll in any other way and I my life is a different story.

Kids naturally need is to move and communicate. So what do, we put them in a chair all day, doing countless worksheets and telling them to raise their hand if they want to speak. Then we they don't we scold and drug them with Ritalin.

 

[berkeman]

My wife teaches 3,4,5 year olds. She needs to know calculus why?

So she can calculate accurately when a child's behavior is going to go asymptotic...

https://mathblog.com/reference/algebra/asymptotes/

https://en.wikipedia.org/wiki/Asymptote

 

[Mark44]

kids actually get to engage with math concepts instead of being drilled on how to multiply fractions quickly or whatever.

It's not an either/or situation. There's been a controversy amongst math teachers for at least the past 25 years, with one side denigrating any rote memorization such as the "times" table or even such basics as adding or multiplying single-digit numbers. This seems foolish to me, to toss out such foundational knowledge. One of my fellow math instructors used the term "vacuous drill" so often that I offered an abbreviation -- VD. My point is that both are important - math concepts and a certain set of basic knowledge. After all, you can't build a house without a solid foundation.

The practicality of knowing how to do tedious arithmetic by hand is largely in the past--calculators exist for a reason (After all, how many of us actually remember how to compute a square root by hand, if we ever knew in the first place?). Sure, an intuition for when the result of a computation doesn't "smell" right is important, but I'm not convinced that knowing how to do algorithmic arithmetic quickly is the exclusive path to developing that intuition.

And even calculators are becoming passé -- everyone has a smart phone with calculator abilities. But what do you do when your phone dies? And if you don't have some vague sense of what the calculated answer should be, and you key in an incorrect number, it's just garbage in-garbage out, and you have no way to get a ballpark estimate to confirm or toss what the calculator gave.
Instead of calculators, I think kids would be better off working with slide rules, and there are a lot of math teachers at the middle-school and high school level who agree with this.
@TeethWhitener, What are you calling "algorithmic arithmetic"? The only things I can think of, aside from the square root calculation, are multiplication and division, and the arithmetic of fractions. Surely these concepts don't need to be the sole focus of math classes - creative teachers could come up with applied problems that use these concepts.
Going back to what I said earlier about the need for teachers to have better math skills, at least from about 4th or 5th grade on up, there are a lot of teachers who wind up being assigned to teach a math class, but who aren't qualified to do so.
As an anecdote, I have a close friend who was a welder for his entire career, and is now retired. He told me about being in 9th grade at a public school in Idaho. The math teacher was a coach pressed into teaching the math class. The teacher spent most of the time talking with the guys who were on whatever team the teacher coached, and virtually no time teaching anyone in the class mathematics. These are the kinds of teachers who rely on "drill and kill" because they don't know anything else to do. As a result of this experience, my friend has an extremely limited knowledge of arithmetic, and nothing beyond that. He wanted to learn trig so that he could qualify for an advanced ham radio license, but gave up because it was too hard for him.
I've met lots of people, who, when they found out that I taught math, had similar experiences along about junior high or high school.
BTW, I learned how to do square roots by hand in, I think, 8th grade. Still know how to do them.

Kids naturally need is to move and communicate. So what do, we put them in a chair all day, doing countless worksheets and telling them to raise their hand if they want to speak. Then we they don't we scold and drug them with Ritalin.

What you said especially applies to boys. Our education system tries to pretend that there is no difference between boys and girls, with the result that boys are dosed with Ritalin at twice the rate that girls are (https://www.statista.com/statistics/814592/adha-among-us-kids-by-gender/). There's a lot of lip service given to being able to teach diverse groups, but educators seem to believe in a one-size-fits-all sort of methodology with regard to sex differences.

nd in 550BC Pythagoras was kicking butt, but do we want to go back to 550BC education system?

Well that's quite a jump, from 50 years back, to 2570 years back.

For a national education system we need to keep evolving with the science of child development and pedagogy to rise the educational floor for everyone.

One important factor that would prevent this from happening is when parents don't see the value of education, don't read to their young children, and don't make sure that the children are keeping up with their schoolwork.

 

[PAllen]

On the issue of “the right method” versus “the right answer”, Feynman had an extreme position. The only thing that matters is the right answer. Any method that works for a student should get full credit if answer is right; and no credit should be given for the wrong answer, no matter how well the “right method” is used.

Of course, this is not surprising since Feynman specialized in using methods no one else used at the time.

 

[TeethWhitener]

My point is that both are important - math concepts and a certain set of basic knowledge.

I agree. Common core was meant to solve the problem that math had basically just become drills for years on end (that’s certainly my recollection of elementary and middle school—I was interested in math in spite of, not because of, my experience in school). It’s fine to teach the standard multiplication algorithm. Common core does this! But they also spend lots of time explaining (in a variety of ways) why the algorithm works the way it does. I personally think that’s a much better use of time than drills. In the past, you would simply have to rely on being lucky enough to have a stellar teacher who would engage with “supplementary” material. Common core at least attempts to make this supplementary material requisite. Is there still going to be a difference between stellar teachers and the rest? Of course. But I think it’s a good thing that we’re trying to move toward a system where substellar teachers are adopting at least some of the techniques the stellar teachers are already using.

But what do you do when your phone dies?

Use one of the other 9 or so digital devices nearby? Plug in your phone? If you can’t figure out how to solve this problem, I doubt you’ll be proficient at multiplying 5-digit numbers by hand.

And if you don't have some vague sense of what the calculated answer should be, and you key in an incorrect number, it's just garbage in-garbage out, and you have no way to get a ballpark estimate to confirm or toss what the calculator gave.

And if you don't have some vague sense of what the calculated answer should be, and you forget to carry a 1, it's just garbage in-garbage out, and you have no way to get a ballpark estimate to confirm or toss what the calculator gave.

My point is that I’m not convinced that the countless drills kids encounter in elementary school are particularly effective at instilling that ballpark estimation process. In fact, given most adults’ relationship with math, I’d venture to say that they aren’t very effective for most of the population. (A quick google search on innumeracy gives a 2003 estimate of 1/4 of Americans being functionally innumerate and another 1/3 of Americans being substandard at basic math. Source: https://www.self.inc/blog/innumeracy-biggest-problem ) The hope with common core is that if kids are given a wide variety of ways to think about math problems, they’ll be more likely to develop the mathematical intuition that only a very few people develop when exposed to interminable drills.

What are you calling "algorithmic arithmetic"? The only things I can think of, aside from the square root calculation, are multiplication and division, and the arithmetic of fractions. Surely these concepts don't need to be the sole focus of math classes - creative teachers could come up with applied problems that use these concepts.

They could. Some of them certainly do. But my experience is that the majority don’t. And the data suggests that whatever methods people were using before common core (2010) just weren’t doing the trick for at least 55% of the population. Will common core work out for the kids who are coming through school now? That’s a great question and the jury is still out.

Going back to what I said earlier about the need for teachers to have better math skills, at least from about 4th or 5th grade on up, there are a lot of teachers who wind up being assigned to teach a math class, but who aren't qualified to do so.

Teacher qualifications have become significantly stricter since NCLB and Every Student Succeeds (edit: my mistake, ESSA eliminates the “highly qualified” clause from NCLB. It pushes standardization of teacher qualifications down to the state level. In VA, for instance, grade 7-12 math teachers have to pass a https://www.ets.org/s/praxis/pdf/5161.pdf that includes calculus and statistics questions, among other topics). Of course, there is still wide variability among jurisdictions.

 

[bhobba]

Then in a 9-month school year, they must have gotten only to the "8 times" row. Unbelievable.

Reminds me of how Feynman got really mad when asked to look at some science and math school textbooks. Math is in fact about concepts yet even these days they teach it like a cookbook you must remember, and not a well thought through one at that. The sooner they get onto algebra, geometry, then calculus the better. Especially geometry because a cookbook approach is not that useful there - you must actually think - and the way they teach elementary math will hinder rather than help. As per the title of this thread out here in Australia we have the University of Open Learning, where you can do university level subjects online. For math, once your kids have done algebra and geometry I would enrol them in - Essential Math 1 (in the US called precalculus):
https://www.open.edu.au/subjects/un...ematics-1-algebra-and-trigonometry-usa-enr101

Then Essential Math 2 - Calculus
https://www.open.edu.au/subjects/un...a-essential-mathematics-2-calculus-usa-enr102

Two university subjects at a credit average are enough here in Australia to get you automatic entrance in many universities, or simply continue with a degree at the University of Open Learning - trouble is, and I have rang them up about it, they do not have many STEM related degrees. They say they are working on it. A math degree would be a good start because of the government changes here to cost - math degrees only cost about $3,000 py - the prices in the links I gave above obviously do not reflect these recent changes. But other institutions here in Aus do have online STEM related degrees. The sooner they are out of 'public education' the better - schools have gone in a 'trendy' direction detrimental to teaching critical thinking IMHO. But even at uni, as the Sokal affair demonstrated, you still need to be carefull.

Thanks
Bill

 

[bhobba]

On the issue of “the right method” versus “the right answer”, Feynman had an extreme position. The only thing that matters is the right answer. Any method that works for a student should get full credit if answer is right; and no credit should be given for the wrong answer, no matter how well the “right method” is used.

Of course, this is not surprising since Feynman specialized in using methods no one else used at the time.

Trouble is that will not work in geometry where you must show your steps in reasoning. What I think concerned Feynman most was it being taught 'mechanically'. He was entirely correct in that. As a preparation for having to do actual proofs you must be taught also how to justify your answer. It would matter naught to Feynman who could always do that when required, but just did not see the need to bother if not required - which of course is most of the time. But having to at least occasionally do it, especially at school, teaches critical thinking skills.

Thanks
Bill

 

[etotheipi]

Reminds me of how Feynman got really mad when asked to look at some science and math school textbooks

Details in the "Judging Books by Their Covers" chapter in the book "Surely you're joking Mr Feynman!". A fun example:

For example, there was a book that started out with four pictures: first there was a wind-up toy; then there was an automobile; then there was a boy riding a bicycle; then there was something else. And underneath each picture it said, "What makes it go?"

I thought, "I know what it is: They're going to talk about mechanics, how the springs work inside the toy; about chemistry, how the engine of the automobile works; and biology, about how the muscles work." It was the kind of thing my father would have talked about: "What makes it go? Everything goes because the sun is shining." And then we would have fun discussing it:

"No, the toy goes because the spring is wound up," I would say.
"How did the spring get wound up?" he would ask.
"I wound it up."
"And how did you get moving?"
"From eating."
"And food grows only because the sun is shining. So it's because the sun is shining that all these things are moving." That would get the concept across that motion is simply the transformation of the sun's power.

I turned the page. The answer was, for the wind-up toy, "Energy makes it go." And for the boy on the bicycle, "Energy makes it go." For everything, "Energy makes it go."

Now that doesn't mean anything. Suppose it's "Wakalixes." That's the general principle: "Wakalixes makes it go." There's no knowledge coming in. The child doesn't learn anything; it's just a word!

What they should have done is to look at the wind-up toy, see that there are springs inside, learn about springs, learn about wheels, and never mind "energy." Later on, when the children know something about how the toy actually works, they can discuss the more general principles of energy. It's also not even true that "energy makes it go," because if it stops, you could say, "energy makes it stop" just as well. What they're talking about is concentrated energy being transformed into more dilute forms, which is a very subtle aspect of energy. Energy is neither increased nor decreased in these examples; it's just changed from one form to another. And when the things stop, the energy is changed into heat, into general chaos.

But that's the way all the books were: They said things that were useless, mixed-up, ambiguous, confusing, and partially incorrect. How anybody can learn science from these books, I don't know, because it's not science

 

[bhobba]

I agree. Common core was meant to solve the problem that math had basically just become drills for years on end (that’s certainly my recollection of elementary and middle school—I was interested in math in spite of, not because of, my experience in school).

I had zero interest in math, and was not even good at it at primary school. During later primary years I was interested in electronics - but could never understand the explanations of what parts like inductors and capacitors did. I pulled my hair out trying - but to no avail. We started school at 5 during those days and when I hit grade 8 at 12 we did algebra and geometry (we combine them here in Aus rather than separate subjects) it all just clicked. I raced through the textbook in about a week. Then I applied it to understanding feedback in electronics which had totally defeated me. I just wrote down the equation of the circuit, solved it, and low and behold you saw how as the gain of the amp increased it became more and more determined by the resistors in the feedback path. I was hooked. A bit later - 13 or 14 - I actually forget - I learned calculus, and low and behold, the voltage across an inductor was proportional to the differential of the current, and for a capacitor it was proportional differential of the voltage the inductance and capacitance was just the constant of proportionality. Then after learning complex numbers, phase shift and all of that stuff was trivial. Of course my school math suffered - I did not learn for example the simple proof of the quadratic equation formula using completing the square - I did that much later by myself. But boy I had a lot of fun just following what took my fancy at the time.

Thanks
Bill

 

[bhobba]

Details in the "Judging Books by Their Covers" chapter in the book "Surely you're joking Mr Feynman!". A fun example:

Yes - Feynman should have written the books.

Thanks
Bill

 

[Mark44]

On the issue of “the right method” versus “the right answer”, Feynman had an extreme position. The only thing that matters is the right answer. Any method that works for a student should get full credit if answer is right; and no credit should be given for the wrong answer, no matter how well the “right method” is used.

My grading style wasn't as extreme as Feynman's, especially for problems with very many steps. If students knew what they were doing, but made a bonehead mistake toward the end of their calculations, I gave partial credit.
In one class I taught, Intermediate Algebra (really just 9th grade algebra sped up to 10 weeks), one of my students complained that I had marked her correct answer to a homework problem wrong, but her friend got the wrong answer, and received half credit. She thought that was very unfair.
I told her that her friend's work was logical and easy to follow, but with a mistake, while hers was a mishmash of random, unconnected gibberish that miraculously ended up with the correct answer, which was given in the answers section of the book.

 

[Averagesupernova]

From what I understand common core is perceived as, and I may be a bit off, they show kids seemingly odd ways of solving a simple math problem. I probably figured out various forms of this on my own many years ago. Example: What is 9 x 15? Heck I have no idea off the top of my head but I know 10 x 15 is 150 and it's easy to subtract 15 off that to come up with the correct answer of 135. Now this certainly isn't worth bragging about but I do find it interesting that according to my dad, my grandpa was taught math like this in a country school more than a century ago. He was born in 1903. I sometimes think my grandpa's generation had stronger math skills per hour of school than anything today. I have seen many posts on Facebook by people with kids in school with examples similar to the example I gave and people completely tear this concept to pieces and give the 'correct' way to do the math problem.

 

[Haelfix]

I had read a lot of concern about common core, but when I had to teach a lightning review of grade 8-12 to a family member to pass an equivalency test, I was pleasantly surprised. In my opinion, it’s a perfectly acceptable system and indeed even has many advantages over the old way.
My only complaint is the really aggravating amount of online multiple choice tests, often with ambiguous syntax. Things like whether 1/2 (x+1) is a linear expression or a fraction.

The amount of absolutely phenomenal videos on YouTube and Khan is also a game changer. I watched a video the other day teaching the central limit theorem, and there is absolutely no way I could ever teach it that succinctly or as well. The visuals alone is well out of reach of a normal teacher with a blackboard.

What I think remains necessary and vastly undervalued by educators, is speed training. Especially for advanced students. Its quite unglamorous to teach (drills and more drills), but learning to be lightning quick with your integrals, and evaluating expressions and arithmetic quickly is a skill that stays with you for the rest of your life, and saves countless hours (and is one of the skills that translates directly into research success).

 

[mathwonk]

I predict this thread will easily surpass 100 posts. At least in the past, these discussions of pedagogy seem to live on forever. I taught for a long time in a college math dept and listened to many of these arguments. I noticed that no one ever agreed about how teaching should be done, when discussing it in the abstract in the coffee room, but usually these same people, when actually observing each other in the classroom, tended to agree that what was being done was good. I don't know why this is, maybe it is impossible to convey in words just what one thinks should be imparted, whereas in practice, an experienced teacher has learned how to make his/her vision work. Of course I used to have a reputation as a thread killer, but this time, I doubt it.

 

[bhobba]

At the risk of adding to the possibility of exceeding 100 posts, I must say I personally like, for primary school math (here in Aus defined as years 1-6) the Montessori method - its very experiential:
https://hollismontessori.org/blog/2018/3/19/montessori-basics-how-math-progresses-through-the-levels

Years 7, 8 and 9 can cover algebra, geometry and trigonometry, then in grade 10 you start calculus. Good students can do calculus in grade 9. I actually do not believe in grade 11 and 12 at all - you go to university, trade school or technical training of some sort after that - here in Aus called TAFE - Technical and Further Education. Entrance to university is via a few university subjects done in grade 10. TAFE generally requires no entrance requirements and you can start in grade 10. You proceed from certificate 3 to certificate 4 to diploma to advanced diploma stopping at the level that suits your goals. Certificate 4 or beyond is enough for university admission. Diploma's are generally considered equivalent to the first year of university, while advanced diplomas are equivalent to second year. Many careers have overlap eg you do a diploma in Nursing and you are considered an enrolled nurse, start working, and with 4 years part time study while working you become a full registered nurse.

Thanks
Bill

 

[mathwonk]

I always wonder what it means to "do calculus", in say the 9th grade. The hard concept is mainly the definition of "completeness" the real numbers, and without such a definition one cannot prove the existence of the things calculus is designed to find, such as maxima and minima, and areas of curved regions.

So does such an early calculus course just assume the real numbers are in one to one correspondence with the points of an intuitive continuous "number line", or perhaps even state that they are represented by infinite decimals, or (unlikely) give the axioms for them as an Archimedean ordered field?

Are limits introduced in an intuitive way, or are they defined carefully? If they are defined, are basic facts about them proved, such as the limit of a product is the product of the limits, or maybe that the limit represented by the infinite decimal .9999... is 1?

I.e. one can teach "calculus" roughly along the lines of Calculus made easy, by Silvanus P. Thompson, where nothing at all is defined precisely and nothing is proved, (and several erroneous statements are made), but numerous examples in using calculus techniques are illustrated;

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;

or one can teach it as presented in Spivak or Courant, where real numbers are described either as infinite decimals with a certain equivalence relation, or axiomatically as an ordered field. In this last case, everything is precisely defined and every theorem is proved.

Now I think most people call the last approach "analysis" rather than calculus, but some of us actually were introduced to calculus by such an approach and hence do not know what others mean when they say "calculus".

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?

Would he/she see a rigorous definition of the sine, cosine and exponential functions using infinite series? I would think not, but that is how they were presented in my first (college honors) calculus course. Would one just assume the properties of trig functions from a (non rigorous) trig course, as in both Thompson and apparently even Courant?

Come to think of it I actually once taught a high school short course (elaborating an exercise from Spivak) on rigorous real numbers as infinite decimals, proving all the axioms for an Archimedean ordered field hold for them; moreover some students were 10th graders, so I know it can be done. (only 57 more posts to go.)

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.

 

[PAllen]

I always wonder what it means to "do calculus", in say the 9th grade. The hard concept is mainly the definition of "completeness" the real numbers, and without such a definition one cannot prove the existence of the things calculus is designed to find, such as maxima and minima, and areas of curved regions.

So does such an early calculus course just assume the real numbers are in one to one correspondence with the points of an intuitive continuous "number line", or perhaps even state that they are represented by infinite decimals, or (unlikely) give the axioms for them as an Archimedean ordered field?

Are limits introduced in an intuitive way, or are they defined carefully? If they are defined, are basic facts about them proved, such as the limit of a product is the product of the limits, or maybe that the limit represented by the infinite decimal .9999... is 1?

I.e. one can teach "calculus" roughly along the lines of Calculus made easy, by Silvanus P. Thompson, where nothing at all is defined precisely and nothing is proved, (and several erroneous statements are made), but numerous examples in using calculus techniques are illustrated;

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;

or one can teach it as presented in Spivak or Courant, where real numbers are described either as infinite decimals with a certain equivalence relation, or axiomatically as an ordered field. In this last case, everything is precisely defined and every theorem is proved.

Now I think most people call the last approach "analysis" rather than calculus, but some of us actually were introduced to calculus by such an approach and hence do not know what others mean when they say "calculus".

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?

Would he/she see a rigorous definition of the sine, cosine and exponential functions using infinite series? I would think not, but that is how they were presented in my first (college honors) calculus course. Would one just assume the properties of trig functions from a (non rigorous) trig course, as in both Thompson and apparently even Courant?

Come to think of it I actually once taught a high school short course (elaborating an exercise from Spivak) on rigorous real numbers as infinite decimals, proving all the axioms for an Archimedean ordered field hold for them; moreover some students were 10th graders, so I know it can be done. (only 57 more posts to go.)

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.

First, I’m not sure there is any likelihood that a first college course in calculus is necessarily better on these points than one in high school. The problem is not the age of the student but attitudes on what a ‘first course ‘ in calculus should be like.

However, as an exercise I pulled out the book I used to self teach calculus in 10th grade (never took a course in it, just self taught, getting 5 on BC exam in 10th grade, which exam was never before offered by my rural school).

1) A rigorous definition of completeness of the reals is given on page 10 of the first chapter. Limits are very carefully defined and used in all proofs.

2) Real numbers are defined via infinite decimals, then the axioms of an archimedean ordered field are developed (without using the term).

3) Your preliminary result on a continuous function of a closed interval having a maximum and minimum is discussed, presented as an important theorem relying on previously introduced notions of completeness of reals and rigorous limit definitions, but the full proof is left for a future course. Many issues and subtleties are presented in the text and exercises, but the full proof is deferred. Given this one cop out, the other related results you mention are rigorously proved (slope being zero at a maximum, Rolle’s theorem, then mean value theorem, extended theorem of the mean).

4) trig functions were defined in terms of a unit circle as functions of reals, with all properties derived from definitions, including properties of their derivatives. Later, infinite series for them were derived. Arguably, the notion of a unit circle is not developed with full rigor.

fyi, the book was “calculus and analytic geometry” by Fisher and Ziebur.

 

[mathwonk]

Thanks, yes that was considered a good book in my day (by me). My guess is your self taught course excels what many high school courses offer. I myself was not offered calculus in high school, (and did not study it myself), indeed I did not even learn trig. Upon entering college I was enrolled in a Spivak type calculus course, but unfortunately that was several years before the book by Spivak was available, so every 9am lecture I missed was lost material. About one year later I was out working in a factory, and wondering what had gone amiss.

By the way, many years later, I observed the following proof (much improved by a brilliant colleague) of the basic theorem omitted from Fisher and Ziebur and most other books. Let f be defined on the closed interval [0,1], and consider its values on each of the subintervals, [0, .1], [.1, .2], ...,[.9, 1], of length 1/10. Since you say your book defined the notion of complete ordered field I assume it defined a "least upper bound", i.e. a number that is an upper bound of a given set and yet no smaller number is. the fundamental axiom of the reals is that every non empty subset of reals has a least upper bound, which is either infinity if the set is unbounded above, or is a finite real number if the set has a finite upper bound.

Subdivide the interval [0,1] into smaller intervals of length 1/10. Then choose such a subinterval, say [.4, .5], such that the least upper bound of the values of f on this subinterval is as large as that on any other such subinterval. Hence if there is a maximum value of f on [0,1], it must occur in this subinterval [.4, .5]. Then assign the first decimal place of our desired maximum to be .4.

Now subdivide the interval [.4, .5] further into subintervals of length 1/100, and choose one, say [.43, .44] where again the least upper bound of f is at least as high as that on any other such subinterval. If there is a maximum of f on [0,1], then again it must occur in this subinterval. Thus the first two decimal places of our desired maximum point are .45.

Continuing, we obtain an infinite decimal of form c = .43xxxx..., hence a real number in [0,1], and it is a straightforward exercise in the definition of continuity to show that, if f is continuous at c, then the finite real number f(c) is a maximum for the values of f on the whole interval [0,1]. In particular f is bounded by a finite real number on [0,1] and does in fact attain a finite maximum at some point of [0,1].

Now why would such a simple proof be omitted from a college calculus course? Notice it only requires the knowledge that an infinite decimal does define a real number, plus the definition of continuity, things which supposedly are included in the course. (One easily extends this proof to the case of f defined on any closed bounded interval [a,b], by sending [0,1] to [a,b], by t --> a + t.(b-a), and composing this map with f.)

 

[PAllen]

thanks, yes that was considered a good book in my day (by me). my guess is your self taught course excels what many high school courses offer. I myself did not have calculus in high school, (and did not study it myself), indeed I did not even learn trig. Upon entering college I was enrolled in a Spivak type calculus course, but unfortunately that was several years before the book by Spivak was available, so every 9am lecture I missed was lost. About one year later I was out working in a factory, and wondering what had gone amiss.

A funny part of my story was how difficult it was to convince my parents to get me a calculus textbook for my birthday toward the end of 9th grade (I had picked it visiting the nearest real library by bus and comparing texts as to apparent completeness of explanation and number of exercises, both with and without solutions given). It took weeks of persuasion that this was what I really wanted, and nothing else would do. They just didn’t accept that a math textbook was an appropriate birthday gift.

 

[mathwonk]

I think you chose very well. That book was not as well known as many others, such as Thomas, but was excellent. As parent stories go, my mom wanted me to practice the piano and be the next Liberace. Instead I was reading Theory of Sets, at the nearby university library, by Erich Kamke. Funny how you never forget those formative sources, even 60+ years later.

 

[PAllen]

Thanks, yes that was considered a good book in my day (by me). My guess is your self taught course excels what many high school courses offer. I myself was not offered calculus in high school, (and did not study it myself), indeed I did not even learn trig. Upon entering college I was enrolled in a Spivak type calculus course, but unfortunately that was several years before the book by Spivak was available, so every 9am lecture I missed was lost material. About one year later I was out working in a factory, and wondering what had gone amiss.

By the way, many years later, I observed the following proof (much improved by a brilliant colleague) of the basic theorem omitted from Fisher and Ziebur and most other books. Let f be defined on the closed interval [0,1], and consider its values on each of the subintervals, [0, .1], [.1, .2], ...,[.9, 1], of length 1/10. Since you say your book defined the notion of complete ordered field I assume it defined a "least upper bound", i.e. a number that is an upper bound of a given set and yet no smaller number is. the fundamental axiom of the reals is that every non empty subset of reals has a least upper bound, which is either infinity if the set is unbounded above, or is a finite real number if the set has a finite upper bound.

Subdivide the interval [0,1] into smaller intervals of length 1/10. Then choose such a subinterval, say [.4, .5], such that the least upper bound of the values of f on this subinterval is as large as that on any other such subinterval. Hence if there is a maximum value of f on [0,1], it must occur in this subinterval [.4, .5]. Then assign the first decimal place of our desired maximum to be .4.

Now subdivide the interval [.4, .5] further into subintervals of length 1/100, and choose one, say [.43, .44] where again the least upper bound of f is at least as high as that on any other such subinterval. If there is a maximum of f on [0,1], then again it must occur in this subinterval. Thus the first two decimal places of our desired maximum point are .45.

Continuing, we obtain an infinite decimal of form c = .43xxxx..., hence a real number in [0,1], and it is a straightforward exercise in the definition of continuity to show that, if f is continuous at c, then the finite real number f(c) is a maximum for the values of f on the whole interval [0,1]. In particular f is bounded by a finite real number on [0,1] and does in fact attain a finite maximum at some point of [0,1].

Now why would such a simple proof be omitted from a college calculus course? Notice it only requires the knowledge that an infinite decimal does define a real number, plus the definition of continuity, things which supposedly are included in the course. (One easily extends this proof to the case of f defined on any closed bounded interval [a,b], by sending [0,1] to [a,b], by t --> a + t.(b-a), and composing this map with f.)

Actually, the form of theorem they omitted proof of was more general. That if f is a continuous function on [a,b], then its range over this closed interval is some closed interval [A,B]. This includes, as immediate corollaries, that it has a maximum and minimum (B and A, respectively, and possibly the same).

 

[mathwonk]

yes, pardon me, the proof i gave yields half of that result almost immediately, with my c equal to your B. Replacing f by -f gives the existence of a minimum, say d, for f. Then the image interval is at least contained in [d,c].

The theorem they stated is actually the composition of two basic results, the other one being the intermediate value theorem, i.e. the continuous image of an interval is an interval, which strengthens this theorem to show that the continuous image of a closed bounded interval is a closed bounded interval.

Their statement is actually less general than this however, since in fact the continuous image of any interval is an interval, not just that of a closed bounded interval. However, it is true that their more restricted theorem can be used to prove the more general theorem about the continuous image of any interval, since in any interval where f changes sign, we can choose two points a,b where it does so and apply the restricted theorem to [a,b].

One can give a similar elementary argument for the IVT. Namely it suffices to show that if f is continuous on [0,1] and is negative at 0 and positive at 1, then at some intermediate point c, we have f(c) = 0. Just subdivide again and choose a subinterval such that f changes signs at the endpoints of the subinterval...continue and get a point c where f changes sign at the endpoints of infinitely many subintervals shrinking to c, hence by continuity at c, f(c) = 0.

My point is that there are two general concepts which are proved differently but which they are conflating in their statement. I.e. the continuous image of every closed and bounded set is closed and bounded, and the continuous image of every interval is an interval, but there are many closed and bounded sets which are not intervals, and many intervals which are neither closed nor bounded. Since they omit the proofs, this is not made clear. (In abstract topology these concepts are called compactness and connectedness. Then the continuous image of a compact set is compact and the continuous image of a connected set is connected.)

I must say however that since they are not giving the proofs, their choice of a combined statement seems an intelligent and useful one.

 

[PAllen]

@mathwonk , I like your proofs because they are constructive in flavor. What I encountered later, were proofs of generalizations of these theorems beyond reals (e.g. arbitrary compact sets, connected sets, etc.). These proofs were all nonconstructive proofs by contradiction.