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

[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.

 

[mathwonk]

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.

 

[mathwonk]

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.

 

[mathwonk]

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.

 

[Averagesupernova]

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 I am 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.

 

[gmax137]

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.

 

[mathwonk]

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 -" let's just get through this, I hate it". And those were future teachers. The challenge is keep the fun in.

 

[gmax137]

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?

 

[bhobba]

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

 

[jedishrfu]

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

 

[PAllen]

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".